Abstract
This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in Dahlke and Schneider (Anal Appl 17(2):235–291, 2019, Thms. 4.5, 4.9, 4.12, 4.14) to domains of polyhedral type. In particular, we study the smoothness in the specific scale \(\ B^r_{\tau ,\tau }, \ \frac{1}{\tau }=\frac{r}{d}+\frac{1}{p}\ \) of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.
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1 Introduction
This paper is concerned with regularity estimates of the solutions to evolution equations in nonsmooth domains of polyhedral type \(D \subset \mathbb {R}^3\), cf. Definition 1. In particular, we study linear (\(\varepsilon =0\)) and nonlinear (\(\varepsilon >0\)) equations of the form
with zero initial and Dirichlet boundary conditions, where \(m,M\in \mathbb {N}\), and L denotes a uniformly elliptic operator of order 2m with sufficiently smooth coefficients. Special attention is paid to the spatial regularity of the solutions to (1.1) in specific nonstandard smoothness spaces, i.e., in the so-called adaptivity scale of Besov spaces
Our investigations are motivated by fundamental questions arising in the context of the numerical treatment of Eq. (1.1). In particular, we aim at justifying the use of adaptive numerical methods for parabolic PDEs. Let us explain these relationships in more detail: In an adaptive strategy, the choice of the underlying degrees of freedom is not a priori fixed but depends on the shape of the unknown solution. In particular, additional degrees of freedom are only spent in regions where the numerical approximation is still ‘far away’ from the exact solution. Although the basic idea is convincing, adaptive algorithms are hard to implement, so that beforehand a rigorous mathematical analysis to justify their use is highly desirable.
Given an adaptive algorithm based on a dictionary for the solution spaces of the PDE, the best one can expect is an optimal performance in the sense that it realizes the convergence rate of best N-term approximation schemes, which serves as a benchmark in this context. Given a dictionary \(\Psi =\{\psi _{\lambda }\}_{\lambda \in \Lambda }\) of functions in a Banach space X, the error of best N-term approximation is defined as
i.e., as the name suggests we consider the best approximation by linear combinations of the basic functions consisting of at most N terms. In particular, [23, Thm. 11, p. 586] implies for \(\tau <p\),
Quite recently, it has turned out that the same interrelations also hold for the very important and widespread adaptive finite element schemes. In particular, [27, Thm. 2.2] gives direct estimates,
where \(\sigma _N^{FE}\) denotes the counterpart to the quantity \(\sigma _N(u;X)\), which corresponds to wavelet approximations. It can be seen that the achievable order of adaptive algorithms depends on the regularity of the target function in the specific scale of Besov spaces (1.2). On the other hand it is the regularity of the solution in the scale of Sobolev spaces, which encodes information on the convergence order for nonadaptive (uniform) methods. From this we can draw the following conclusion: adaptivity is justified, if the Besov regularity of the solution in the Besov scale (1.2) is higher than its Sobolev smoothness!
For the case of elliptic partial differential equations, a lot of positive results in this direction are already established [13,14,15,16,17,18,19, 30, 31]. It is well known that if the domain under consideration, the right-hand side and the coefficients are sufficiently smooth, then the problem is completely regular [1], and there is no reason why the Besov smoothness should be higher than the Sobolev regularity. However, on general Lipschitz domains and in particular in polyhedral domains, the situation changes dramatically. On these domains, singularities at the boundary may occur that diminish the Sobolev regularity of the solution significantly [10, 12, 28, 29, 32]. However, the analysis in the above mentioned papers shows that these boundary singularities do not influence the Besov regularity too much, so that the use of adaptive algorithms for elliptic PDEs is completely justified!
In this paper, we study similar questions for evolution equations of the form (1.1) and of associated semilinear versions. To the best of our knowledge, not so many results in this direction are available so far. For parabolic equations, first results for the special case of the heat equation have been reported in [2,3,4], but for a slightly different scale of Besov spaces.
Our results show in the linear case \(\varepsilon =0\) that if the right-hand side as well as its time derivatives are contained in specific Kondratiev spaces, then, for every \(t \in [0,T]\) the spatial Besov smoothness of the solution to (1.1) is always larger than 2m, provided that some technical conditions on the operator pencils are satisfied, see Theorems 8 and 9. The reader should observe that the results are independent of the shape of the polyhedral domain, and that the classical Sobolev smoothness is usually limited by m, see [35]. Therefore, for every t, the spatial Besov regularity is more than twice as high as the Sobolev smoothness, which of course justifies the use of (spatial) adaptive algorithms. Moreover, for smooth domains and right-hand sides in \(L_2,\) the best one could expect would be smoothness order 2m in the classical Sobolev scale. So, the Besov smoothness on polyhedral type domains is at least as high as the Sobolev smoothness on smooth domains.
Afterwards, we generalize this result to nonlinear parabolic equations of the form (1.1). We show that in a sufficiently small ball containing the solution of the corresponding linear equation, there exists a unique solution to (1.1) possessing the same Besov smoothness in the scale (1.2). The proof is performed by a technically quite involved application of the Banach fixed point theorem. The final result is stated in Theorem 10.
The next natural step is to also study the regularity in time direction. For the linear parabolic problem (1.1) with \(\varepsilon =0\) we show that the mapping \(t\mapsto u(t, \cdot )\) is in fact a \(C^l\)-map into the adaptivity scale of Besov spaces, precisely,
see Theorem 11.
In conclusion, the results presented in this paper imply that for each \(t \in (0,T)\) the spatial Besov regularity of the unknown solutions of the problems studied here is much higher than the Sobolev regularity, which justifies the use of spatial adaptive algorithms. This corresponds to the classical time-marching schemes such as the Rothe method. We refer, e.g., to the monographs [34, 42] for a detailed discussion. Of course, it would be tempting to employ adaptive strategies in the whole space-time cylinder. First results in this direction have been reported in [41]. To justify also these schemes, Besov regularity in the whole space-time cylinder has to be established. This case will be studied in a forthcoming paper.
Throughout the paper we use the same notation as in [22], which for the convenience of the reader is recalled in Appendix 1.
2 Sobolev and Kondratiev Spaces
In this section, we briefly collect the basics concerning weighted and unweighted Sobolev spaces needed later on. In particular, we put \(H^m=W^m_2\) and denote by \(\mathring{H}^m\) the closure of test functions in \(H^m\) and its dual space by \(H^{-m}\). Moreover, \(\mathcal {C}^{k,\alpha }\), \(k\in \mathbb {N}_0\), stands for the usual Hölder spaces with exponent \(\alpha \in (0,1]\). The following generalized version of Sobolev’s embedding theorem for Banach-space valued functions will be useful, cf. [40, Thm. 1.2.5].
Theorem 1
(Generalized Sobolev’s embedding theorem) Let \(1<p<\infty \), \(m\in \mathbb {N}\), \(I\subset \mathbb {R}\) be some bounded interval, and X a Banach space. Then
Here the Banach-valued Sobolev spaces are endowed with the norm
whereas for the Hölder spaces we use
where \(\Vert u|C^k(I,X)\Vert =\sum _{j=0}^k\max _{t\in I}\Vert u^{(j)}(t)|X\Vert \) and \(|u^{(k)}|_{C^{\alpha }(I,X)}= {\mathop {\mathop {\sup }\limits _{s,t\in I,}}\limits _{s\ne t}}\frac{\Vert u^{(k)}(t)-u^{(k)}(s)|X\Vert }{|t-s|^{\alpha }}\).
We collect some notation for specific Banach-space valued Lebesgue and Sobolev spaces, which will be used when studying the regularity of solutions of parabolic PDEs.
Let \(\Omega _T:=[0,T]\times \Omega \). Then we abbreviate
Moreover, we put
normed by
2.1 Kondratiev Spaces
In the sequel we work to a great extent with weighted Sobolev spaces, the so-called Kondratiev spaces \(\mathcal {K}^m_{p,a}(\mathcal {O})\), defined as the collection of all \(u\in \mathcal {D}'(\mathcal {O})\), which have m generalized derivatives satisfying
where \(a\in \mathbb {R}\), \(1<p<\infty \), \(m\in \mathbb {N}_0\), \(\alpha \in \mathbb {N}^n_0\), and the weight function \(\varrho : D\rightarrow [0,1]\) is the smooth distance to the singular set of \(\mathcal {O}\), i.e., \(\varrho \) is a smooth function and in the vicinity of the singular set S it is equivalent to the distance to that set. Clearly, if \(\mathcal {O}\) is a polygon in \(\mathbb {R}^2\) or a polyhedral domain in \(\mathbb {R}^3\), then the singular set S consists of the vertices of the polygon or the vertices and edges of the polyhedra, respectively.
It follows directly from (2.2) that the scale of Kondratiev spaces is monotone in m and a, i.e.,
if \(m'<m\) and \(a'<a\).
Moreover, generalizing the above concept to functions depending on the time \(t\in [0,T]\), we define Kondratiev type spaces, denoted by \(L_q((0,T),\mathcal {K}^m_{p,a}(\mathcal {O}))\), which contain all functions u(x, t) such that
with \(0<q\le \infty \) and parameters a, p, m as above.
Kondratiev spaces on domains of polyhedral type
For our analysis we make use of several properties of Kondratiev spaces that have been proved in [20]. Therefore, in our later considerations, we will mainly be interested in the case that \(\mathcal {O}\) is a bounded domain of polyhedral type.
The precise definition below is taken from Maz’ya and Rossmann [36, Def. 4.1.1] (Fig. 1).
Definition 1
A bounded domain \(D\subset \mathbb {R}^3\) is defined to be of polyhedral type if the following holds:
-
(a)
The boundary \(\partial D\) consists of smooth (of class \(C^{\infty }\)) open two-dimensional manifolds \(\Gamma _j\) (the faces of D), \(j=1,\ldots , n\), smooth curves \(M_k\) (the edges), \(k=1,\ldots , l\), and vertices \(x^{(1)}, \ldots , x^{(l')}\).
-
(b)
For every \(\xi \in M_k\) there exists a neighborhood \(U_{\xi }\) and a \(C^{\infty }\)-diffeomorphism \(\kappa _{\xi }\) which maps \(D\cap U_{\xi }\) onto \(\mathcal {D}_{\xi }\cap B_1(0)\), where \(\mathcal {D}_{\xi }\subset \mathbb {R}^3\) is a dihedron, which in polar coordinates can be described as
$$\begin{aligned} \mathcal {D}_{\xi }=K\times \mathbb {R}, \qquad K=\{(x_1,x_2): \ 0<r<\infty , \ -\theta /2<\varphi <\theta /2\}, \end{aligned}$$where the opening angle \(\theta \) of the 2-dimensional wedge K satisfies \(0<\theta \le 2\pi \).
-
(c)
For every vertex \(x^{(i)}\) there exists a neighborhood \(U_i\) and a diffeomorphism \(\kappa _i\) mapping \(D\cap U_i\) onto \(K_i\cap B_1(0)\), where \(K_i\) is a polyhedral cone with edges and vertex at the origin.
Polyhedron
Remark 2
-
(i)
In the literature many different types of polyhedral domains are considered. A more general version which coincides with the above definition when \(d=3\) is discussed in [20]. Further variants of polyhedral domains can be found in Babuška and Guo [7], Bacuta et al. [8], and Mazzucato and Nistor [37].
-
(ii)
Let us point out that ‘smooth’ domains without edges and/or vertices are admissible in Definition 1. We discuss this further in Sect. 3.2.
Some properties of Kondratiev spaces
Concerning pointwise multiplication the following results are proven in [20].
Corollary 1
-
(i)
Let \(m\in \mathbb {N}\), \(a\ge \frac{3}{p}\), and either \(1<p<\infty \) and \(m>\frac{3}{p}\) or \(p=1\) and \(m\ge 3\). Then the Kondratiev space \(\mathcal {K}^m_{a,p}(D)\) is an algebra with respect to pointwise multiplication, i.e., there exists a constant c such that
$$\begin{aligned} \Vert uv| \mathcal {K}^{m}_{a,p}(D)\Vert \le c\Vert u|\mathcal {K}^{m}_{a,p}(D)\Vert \cdot \Vert v|\mathcal {K}^{m}_{a,p}(D)\Vert \end{aligned}$$holds for all \(u,v\in \mathcal {K}^{m}_{a,p}(D)\).
-
(ii)
Let \(\frac{3}{2}<p<\infty \), \(m\in \mathbb {N}\), and \(a\ge \frac{3}{p}-1\). Then there exists a constant c such that
$$\begin{aligned} \Vert uv| \mathcal {K}^{m-1}_{a-1,p}(D)\Vert \le c\Vert u|\mathcal {K}^{m+1}_{a+1,p}(D)\Vert \cdot \Vert v|\mathcal {K}^{m-1}_{a-1,p}(D)\Vert \end{aligned}$$holds for all \(u\in \mathcal {K}^{m+1}_{a+1,p}(D)\) and \(v\in \mathcal {K}^{m-1}_{a-1,p}(D)\).
Our main tool when investigating the Besov regularity of solutions to the PDEs will be the following embedding result between Kondratiev and Besov spaces, which is an extension of [30, Thm. 1]. A proof may be found in [40, Thm. 1.4.12].
Theorem 2
(Embeddings between Kondratiev and Besov spaces) Let \(D\subset \mathbb {R}^3\) be some polyhedral type domain and assume \(k\in \mathbb {N}_0\), \(0<q\le \infty \). Furthermore, let \(s, a\in \mathbb {R}\), \(\gamma \in \mathbb {N}_0\), and suppose \(\min (s,a)>\frac{\delta }{3}\gamma \), where \(\delta \) denotes the dimension of the singular set (i.e., \(\delta =0\) if there are only vertex singularities and \(\delta =1\) if there are edge and vertex singularities). Then there exists some \(0<\tau _0\le p\) such that
for all \(\tau _{*}<\tau <\tau _0\), where \(\frac{1}{\tau _{*}}=\frac{\gamma }{3}+\frac{1}{p}\).
3 Parabolic PDEs and Operator Pencils
In the sequel we deal with two different parabolic settings, Problems 1 and 2, which are of general order and defined on domains of polyhedral type according to Definition 1. In particular, Problem 2 is the nonlinear version of Problem 1 and we investigate the spatial Besov regularity of the solutions of these two problems and to some extent also the Hölder regularity with respect to the time variable of Problem 1.
3.1 The Fundamental Parabolic Problems
Let D denote some domain of polyhedral type in \(\mathbb {R}^d\) according to Definition 1 with faces \(\Gamma _j\), \(j=1,\ldots , n\). For \(0<T<\infty \) put \(D_T=(0,T]\times D\) and \( \Gamma _{j,T}=[0,T]\times \Gamma _j\).
We will investigate the Besov regularity of the following linear parabolic problem.
Problems 1
(Linear parabolic problem in divergence form) Let \(m\in \mathbb {N}\). We consider the following first initial-boundary value problem
Here f is a function given on \(D_T\), \(\nu \) denotes the exterior normal to \(\Gamma _{j,T}\), and the partial differential operator L is given by
where \(a_{\alpha \beta }\) are bounded real-valued functions from \(C^{\infty }(D_T)\) with \(a_{\alpha \beta }=(-1)^{|\alpha |+|\beta |}{a}_{\beta \alpha }\). Furthermore, the operator L is assumed to be uniformly elliptic with respect to \(t\in [0,T]\), i.e.,
Let us denote by
the time-dependent bilinear form.
Moreover, for simplicity we set
Remark 3
(Assumptions on the time-dependent bilinear form) When dealing with parabolic problems it will be reasonable to suppose that \(B(t,\cdot , \cdot )\) satisfies
for all \(u\in \mathring{H}^m(D)\) and a.e. \(t\in [0,T]\). We refer to [40, Rem. 2.3.5] for a detailed discussion.
It is our intention to also study nonlinear versions of Problem 1. Therefore, we modify (3.1) as follows.
Problems 2
(Nonlinear parabolic problem in divergence form) Let \(m,M\in \mathbb {N}\) and \(\varepsilon >0\). We consider the following nonlinear parabolic problem
The assumptions on f and the operator L are as in Problem 1. When we establish Besov regularity results for Problem 2 we interpret (3.6) as a fixed point problem and show that the regularity estimates for Problem 1 carry over to Problem 2, provided that \(\varepsilon \) is sufficiently small.
3.2 Operator Pencils
In order to correctly state the global regularity results in Kondratiev spaces for Problems 1 and 2, we need to work with operator pencils generated by the corresponding elliptic problems in the polyhedral type domain \({D}\subset \mathbb {R}^3\).
We briefly recall the basic facts needed in the sequel. For further information on this subject we refer to [33] and [36, Sects. 2.3, 3.2, 4.1]. On a domain \(D\subset \mathbb {R}^3\) of polyhedral type according to Definition 1 we consider the problem
The singular set S of D then is given by the boundary points \(M_1\cup \cdots \cup M_l\cup \{x^{(1)}, \ldots , x^{(l')}\}\). We do not exclude the cases \(l=0\) (corner domain) and \(l'=0\) (edge domain). In the last case, the set S consists only of smooth nonintersecting edges. Figure 2 gives examples of polyhedral domains without edges or corners, respectively.
Corner domain \(D_c\) (\(l=0\)) and edge domain \(D_e\) (\(l'=0\))
The elliptic boundary value problem (3.7) on D generates two types of operator pencils for the edges \(M_k\) and for the vertices \(x^{(i)}\) of the domain, respectively.
(1) Operator pencil \(A_{\xi }(\lambda )\) for edge points
The pencils \(A_{\xi }(\lambda )\) for edge points \(\xi \in M_k\) are defined as follows: According to Definition 1 there exists a neighborhood \(U_{\xi }\) of \(\xi \) and a diffeomorphism \(\kappa _{\xi }\) mapping \(D\cap U_{\xi }\) onto \(\mathcal {D}_{\xi } \cap B_1(0)\), where \(\mathcal {D}_{\xi }\) is a dihedron (Fig. 3).
Let \(\Gamma _{k_{\pm }}\) be the faces adjacent to \(M_k\). Then by \(\mathcal {D}_{\xi }\) we denote the dihedron which is bounded by the half-planes \(\mathring{\Gamma }_{k_{\pm }}\) tangent to \(\Gamma _{k_{\pm }}\) at \(\xi \) and the edge \(M_{\xi }=\mathring{\Gamma }_{k_{+}}\cap \mathring{\Gamma }_{k_{-}}\). Furthermore, let \(r,\varphi \) be polar coordinates in the plane perpendicular to \(M_{\xi }\) such that
Dihedron \(\mathcal {D}_{\xi }\)
We define the operator pencil \(A_{\xi }(\lambda )\) as follows:
where \(u(x)=r^{\lambda }U(\varphi )\), \(\lambda \in \mathbb {C}\), U is a function on \(I_{\xi }:=\left( \frac{-\theta _{\xi }}{2}, \frac{\theta _{\xi }}{2}\right) \), and
denotes the main part of the differential operator \(L(x,D_x)\) with coefficients frozen at \(\xi \). This way we obtain in (3.8) a boundary value problem for the function U on the 1-dimensional subdomain \(I_{\xi }\) with the complex parameter \(\lambda \). Obviously, \(A_{\xi }(\lambda )\) is a polynomial of degree 2m in \(\lambda \).
The operator \(A_{\xi }(\lambda )\) realizes a continuous mapping
for every \(\lambda \in \mathbb {C}\). Furthermore, \(A_{\xi }(\lambda )\) is an isomorphism for all \(\lambda \in \mathbb {C}\) with the possible exception of a denumerable set of isolated points, the spectrum of \(A_{\xi }(\lambda )\), which consists of its eigenvalues with finite algebraic multiplicities: Here a complex number \(\lambda _0\) is called an eigenvalue of the pencil \(A_{\xi }(\lambda )\) if there exists a nonzero function \(U\in H^{2m}(I_{\xi })\) such that \(A_{\xi }(\lambda _0)U=0\). It is known that the ’energy line’ \(\mathrm {Re}\lambda =m-1\) does not contain eigenvalues of the pencil \(A_{\xi }(\lambda )\). We denote by \(\delta _{\pm }^{(\xi )}\) the largest positive real numbers such that the strip
is free of eigenvalues of the pencil \(A_{\xi }(\lambda )\). Furthermore, we put
For example, concerning the Dirichlet problem for the Poisson equation on a domain \(D\subset \mathbb {R}^3\) of polyhedral type, the eigenvalues of the pencil \(A_{\xi }(\lambda )\) are given by
where \(\theta _{\xi }\) is the inner angle at the edge point \(\xi \), cf. [40, Ex. 2.5.2]. Therefore, the first positive eigenvalue is \(\lambda _1=\frac{\pi }{\theta _{\xi }}\) and we obtain \(\delta _{\pm }=\frac{\pi }{\theta _{\xi }}\), cf. [40, Ex. 2.5.1].
(2) Operator pencil \(\mathfrak {A}_i(\lambda )\) for corner points
Let \(x^{(i)}\) be a vertex of D. According to Definition 1 there exists a neighborhood \(U_i\) of \(x^{(i)}\) and a diffeomorphism \(\kappa _i\) mapping \(D\cap U_i\) onto \(K_i\cap B_1(0)\), where
is a polyhedral cone with edges and vertex at the origin. W.l.o.g. we may assume that the Jacobian matrix \(\kappa _i'(x)\) is equal to the identity matrix at the point \(x^{(i)}\). We introduce spherical coordinates \(\rho =|x|\), \(\omega =\frac{x}{|x|}\) in \(K_i\) and define the operator pencil
where \(u(x)=\rho ^{\lambda }U(\omega )\) and \(U\in \mathring{H}^{m}(\Omega _i)\) is a function on \(\Omega _i\). An eigenvalue of \(\mathfrak {A}_i(\lambda )\) is a complex number \(\lambda _0\) such that \(\mathfrak {A}_i(\lambda _0)U=0\) for some nonzero function \(U\in \mathring{H}^{m}(\Omega _i)\). The operator \(\mathfrak {A}_i(\lambda )\) realizes a continuous mapping
Furthermore, it is known that \(\mathfrak {A}_i(\lambda )\) is an isomorphism for all \(\lambda \in \mathbb {C}\) with the possible exception of a denumerable set of isolated points. The mentioned enumerable set consists of eigenvalues with finite algebraic multiplicities.
Moreover, the eigenvalues of \(\mathfrak {A}_i(\lambda )\) are situated, except for finitely many, outside a double sector \(|\mathrm {Re}\lambda |<\varepsilon |\mathrm {Im}\lambda |\) containing the imaginary axis, cf. [33, Thm. 10.1.1]. In Fig. 4 the situation is illustrated: Outside the yellow area there are only finitely many eigenvalues of the operator pencil \(\mathfrak {A}_i(\lambda )\).
Dealing with regularity properties of solutions, we look for the widest strip in the \(\lambda \)-plane, free of eigenvalues and containing the ‘energy line’ \( \mathrm {Re}\lambda =m-3/2,\) cf. Assumption 1. From what was outlined above, information on the width of this strip is obtained from lower estimates for real parts of the eigenvalues situated over the energy line.
Eigenvalues of operator pencil \(\mathfrak {A}_i(\lambda )\)
Remark 4
(Operator pencils for parabolic problems) Since we study parabolic PDEs, where the differential operator \(L(t,x,D_x)\) additionally depends on the time t, we have to work with operator pencils \(A_{\xi }(\lambda ,t)\) and \(\mathfrak {A}_{i}(\lambda ,t)\) in this context. The philosophy is to fix \(t\in [0,T]\) and define the pencils as above: We replace (3.8) by
and work with \(\delta ^{(\xi )}_{\pm }(t)\) and \(\delta _{\pm }^{(k)}(t)=\inf _{\xi \in M_k}{\delta _{\pm }^{(\xi )}}(t)\) in (3.9) and (3.10), respectively. Moreover, we put
Similar for \(\mathfrak {A}_{i}(\lambda ,t)\), where now (3.11) is replaced by
4 Regularity Results in Sobolev and Kondratiev Spaces
This section presents regularity results for Problems 1 and 2 in Sobolev and Kondratiev spaces. They will form the basis for obtaining regularity results in Besov spaces later on via suitable embeddings. The results in Sobolev and Kondratiev spaces for Problems 1 and 2 on domains of polyhedral type \(D\subset \mathbb {R}^d\) are essentially new and not published elsewhere so far: In [22] we restricted our investigations to polyhedral cones \(K\subset \mathbb {R}^3\) relying on the results from [35].
However, the extension of the regularity results for Problem 1 to polyhedral type domains follows from very similar arguments as in [22], which is why we merely state the results in Sects. 4.1 and 4.2 and give references for the proofs wherever necessary. In contrast to this the regularity results for the nonlinear Problem 2 require some careful adaptations and are carried out in detail in Sect. 4.3.
4.1 Regularity Results in Sobolev Spaces for Problem I
In this subsection, we are concerned with the Sobolev regularity of the weak solution of Problem 1. We start with the following lemma, whose proof is similar to [5, Lem. 4.1].
Lemma 1
(Continuity of bilinear form) Assume that for each \(t\in [0,T]\), \(F(t,\cdot ,\cdot ): \mathring{H}^m(D)\times \mathring{H}^m(D)\rightarrow \mathbb {R}\) is a bilinear map satisfying
for all \(t\in [0,T]\) and all \(u,v\in \mathring{H}^m(D)\), where C is a constant independent of u, v, and t. Assume further that \(F(\cdot , u,v)\) is measurable on [0, T] for each pair \(u,v\in \mathring{H}^m(D)\). Assume that \(u\in {H}^{m,1*}(D_T)\) satisfies \(u(0)\equiv 0\) and
for a.e. \(t\in [0,T]\) and all \(v\in \mathring{H}^m(D)\). Then \(u\equiv 0\) on \([0,T]\times D\).
Using the spectral method the following regularity result now follows.
Theorem 3
(Sobolev regularity without time derivatives) Let \(f\in L_2([0,T],H^{-m}(D))\). Then Problem 1 has a unique weak solution u in the space \({H}^{m,1*}(D_T)\) and the following estimate holds
where C is a constant independent of f and u.
This proof follows [5, Lem. 4.2], which in turn is based on [26, Sect. 7.1.2].
By an application of Theorem 3 and induction we obtain the following regularity result. The proof is similar to [6, Thm. 2].
Theorem 4
(Sobolev regularity with time derivatives) Let \(l\in \mathbb {N}_0\) and assume that the right-hand side f of Problem 1 satisfies
Then the weak solution u in the space \({H}^{m,1*}(D_T)\) of Problem 1 in fact belongs to \({H}^{m,l+1*}(D_T)\), i.e., has derivatives with respect to t up to order l satisfying
and
where C is a constant independent of u and f.
Remark 5
Note that the regularity results for the solution u in [35, Thm. 2.1., Lem. 3.1] are slightly stronger than the ones obtained in Theorem 4 above (with the cost of also assuming more regularity on the right-hand side f). By using similar arguments as in [5, Lem. 4.3] we are probably able to also show in our context that Theorem 3 can be strengthened in the sense that if \(f\in L_2([0,T], L_2(D))\) then the weak solution u of Problem 1 belongs in fact to \(L_2([0,T], \mathring{H}^m)\cap H^1([0,T], L_2(D))\). A corresponding generalization of Theorem 4 should also be possible in the spirit of [5, Thm. 3.1]. However, for our purposes the above results on the Sobolev regularity are sufficient, so these investigations are postponed for the time being.
4.2 Regularity Results in Kondratiev Spaces for Problem I
Concerning weighted Sobolev regularity of Problem 1 first fundamental results on polyhedral cones \(K\subset \mathbb {R}^3\) can be found in [35, Thms. 3.3, 3.4]. In [22] we extended and generalized these results, which we now wish to transfer to our setting of polyhedral type domains \(D\subset \mathbb {R}^3\).
For our regularity assertions we rely on known results for elliptic equations. Therefore, we consider first the following Dirichlet problem for elliptic equations
where \(D\subset \mathbb {R}^3\) is a domain of polyhedral type according to Definition 1 with faces \(\Gamma _j\). Moreover, we assume that
is a uniformly elliptic differential operator of order 2m with smooth coefficients \(A_{\alpha }\). We need the following technical assumptions in order to state the Kondratiev regularity of (4.4).
Assumption 1
(Assumptions on operator pencils) Consider the operator pencils \(\mathfrak {A}_i(\lambda ,t)\), \(i=1,\ldots , l'\) for the vertices and \(A_{\xi }(\lambda ,t)\) with \(\xi \in M_k\), \(k=1,\ldots , l\) for the edges of the polyhedral type domain \(D\subset \mathbb {R}^3\) introduced in Sect. 3.2. For the elliptic problem (4.4) we may drop t from the notation of the pencils, otherwise (for our parabolic problems) we assume \(t\in [0,T]\) is fixed.
Let \(\mathcal {K}^{\gamma }_{p,b}(D)\) and \(\mathcal {K}^{\gamma '}_{p,b'}(D)\) be two Kondratiev spaces, where the singularity set S of D is given by \(S=M_1\cup \cdots \cup M_l\cup \{x^{(1)},\ldots , x^{(l')}\}\) and weight parameters \(b,b'\in \mathbb {R}\). Then we assume that the closed strip between the lines
does not contain eigenvalues of \(\mathfrak {A}_i(\lambda ,t)\). Moreover, b and \(b'\) satisfy
where \(\delta _{\pm }^{(k)}\) are defined in (3.10) (replaced by (3.12) for parabolic problems).
Remark 6
If \(l'=0\) we have an edge domain without vertices, cf. Fig. 2. In this case condition (4.5) is empty. Moreover, if \(l=0\), we have a corner domain without edges, in which case condition (4.6) is empty. For further remarks and explanations concerning Assumption 1 we refer to [22, Rem. 3.3].
The following lemma on the regularity of solutions to elliptic boundary value problems in domains of polyhedral type is taken from [36, Cor. 4.1.10, Thm. 4.1.11]. We rewrite it for our scale of Kondratiev spaces.
Lemma 2
(Kondratiev regularity for elliptic PDEs) Let \(D\subset \mathbb {R}^3\) be a domain of polyhedral type. Moreover, let \(u\in \mathcal {K}^{\gamma }_{2,a+2m}(D)\) be a solution of (4.4), where
Suppose that \(\mathcal {K}^{\gamma }_{2,a}(D)\) and \(\mathcal {K}^{\gamma '}_{2,a'}(D)\) satisfy Assumption 1. Then \(u\in \mathcal {K}^{\gamma '}_{2,a'+2m}(D)\) and
where C is a constant independent of u and F.
Remark 7
In particular, if in Theorem 4 we use the stronger assumption \(\partial _{t^k}f(t) \in L_2(D)\) instead of \(\partial _{t^k}f(t)\in H^{-m}(D)\) for \(k=0,\ldots , l\), then it follows that
where the latter embedding follows from the corresponding duality assertion, i.e., we have \( \mathcal {K}^m_{2,m}(D) \hookrightarrow \mathcal {K}^{0}_{2,0}(D) \) since \(m\ge 0\). In this case, the solution u of Problem 1 satisfies
where the first embedding is taken from [36, Lem. 3.1.6] and the second embedding for Kondratiev spaces holds whenever \(m\ge a\). We additionally require in our considerations that \( \partial _{t^k}u(t)\in \mathcal {K}^0_{2,a}(D) \hookrightarrow \mathcal {K}^{-m}_{2,-m}(D)\) which holds for \(a\ge -m\). From (4.7) and (4.8) we see that it is possible to take \(\gamma =m\) and \(a=-m\) in Lemma 2, i.e., if \(f(t)\in \mathcal {K}^{-m}_{2,-m}(D)\) then \(u(t)\in \mathcal {K}^m_{2,m}(D)\). Note that all our arguments with u(t) and f(t), respectively, hold for a.e. \(t\in [0,T]\). However, since lower order time derivatives are continuous w.r.t. suitable spaces (but not necessarily the highest one, cf. the proof of Thm. 11), we will suppress this distinction in the sequel.
Using similar arguments as in [35, Thm. 3.3] we are now able to show the following regularity result in Kondratiev spaces. The proof follows along the same lines as [22, Thm. 3.6].
Theorem 5
(Kondratiev regularity A) Let \(D\subset \mathbb {R}^3\) be a domain of polyhedral type. Let \(\gamma \in \mathbb {N}\) with \(\gamma \ge 2m\) and put \(\gamma _m:=\left[ \frac{\gamma -1}{2m}\right] \). Furthermore, let \(a\in \mathbb {R}\) with \({a\in [-m,m]}\). Assume that the right-hand side f of Problem 1 satisfies
-
(i)
\(\partial _{t^k} f\in L_2(D_T)\cap L_2([0,T],\mathcal {K}^{2m(\gamma _m-k)}_{2,a+2m(\gamma _m-k)}(D))\), \(k=0,\ldots , \gamma _m\); \(\partial _{t^{\gamma _m+1}} f\in L_2(D_T)\).
-
(ii)
\(\partial _{t^k} f(x,0)=0\), \(k=0,1,\ldots , {\gamma _m}.\)
Furthermore, let Assumption 1 hold for weight parameters \(b=a+2m(\gamma _m-i)\), where \(i=0,\ldots , \gamma _m\), and \(b'=-m\). Then for the weak solution \(u\in {{H}}^{m,\gamma _m+2*}(D_T)\) of Problem 1 we have
for \(l=-1,0,\ldots , \gamma _m\). In particular, for the derivatives \(\partial _{t^{l+1}} u\) up to order \(\gamma _m+1\) we have the a priori estimate
where the constant is independent of u and f.
Remark 8
The existence of the solution \(u\in {H}^{m,{\gamma _m+2}*}(D_T)\) follows from Theorem 4 using \(l=\gamma _m+1\).
The regularity results obtained in Theorem 5 only hold under certain restrictions on the parameter a we are able to choose. In particular, we cannot choose \(\gamma _m>0\) if we have a nonconvex polyhedral type domains D, since there is no suitable a satisfying all of our requirements in this case. In order to treat nonconvex domains as well, we impose stronger assumptions on the right-hand side f, requiring that it is arbitrarily smooth w.r.t. the time. This additional assumption allows for a larger range of a. However, as a drawback, these results are hard to apply to nonlinear equations since the right-hand sides are not taken from a Banach or quasi-Banach space. The proof of the following theorem is similar to [22, Thm. 3.9] adapted to our setting.
Theorem 6
(Kondratiev regularity B) Let \(D\subset \mathbb {R}^3\) be a domain of polyhedral type and \(\eta \in \mathbb {N}\) with \(\eta \ge 2m\). Moreover, let \(l\in \mathbb {N}_0\) and \(a\in \mathbb {R}\) with \(a\in [-m,m]\). Assume that the right-hand side f of Problem 1 satisfies
-
(i)
\( f\in \bigcap _{l=0}^{\infty }H^l([0,T], L_2(D)\cap \mathcal {K}^{\eta -2m}_{2,a}(D))\).
-
(ii)
\(\partial _{t^l} f(x,0)=0\), \(l\in \mathbb {N}_0.\)
Furthermore, let Assumption 1 hold for weight parameters \(b=a\) and \(b'=-m\). Then for the weak solution \(u\in \bigcap _{l=0}^{\infty }{{H}}^{m,l+1*}(D_T)\) of Problem 1 we have
In particular, for the derivative \(\partial _{t^l} u\) we have the a priori estimate
where the constant is independent of u and f.
Remark 9
In Theorem 6 compared to Theorem 5 we only require the parameter a to satisfy \(a\in [-m,m]\) and \(-\delta _{-}^{(k)}<a+m<\delta _{+}^{(k)}\) independent of the regularity parameter \(\eta \) which can be arbitrarily high. In particular, for the heat equation on a domain of polyhedral type D (which for simplicity we assume to be a polyhedron with straight edges and faces where \(\theta _k\) denotes the angle at the edge \(M_k\)), we have \(\delta _{\pm }^{(k)}=\frac{\pi }{\theta _k}\), which leads to the restriction \( -1\le a<\min \left( 1, \frac{\pi }{\theta _k}-1\right) . \) Therefore, even in the extremal case when \(\theta _k=2\pi \) we can still take \(-1\le a<-\frac{1}{2}\) (resulting in \(u\in L_2([0,T], \mathcal {K}^{\eta }_{a+2}(D))\) being locally integrable since \(a+2>0\)). Then choosing \(\eta \) arbitrary high, we also cover nonconvex polyhedral type domains with our results from Theorem 6.
4.3 Regularity Results in Sobolev and Kondratiev Spaces for Problem II
In this subsection, we show that the regularity estimates in Kondratiev and Sobolev spaces as stated in Theorems 5 and 4, respectively, carry over to Problem 2, provided that \(\varepsilon \) is sufficiently small. In order to do this we interpret Problem 2 as a fixed point problem in the following way.
Let \(\widetilde{\mathcal {D}}\) and S be Banach spaces (\(\widetilde{\mathcal {D}}\) and S will be specified in the theorem below) and let \(\tilde{L}^{-1}:\widetilde{\mathcal {D}}\rightarrow S\) be the linear operator defined via
Problem 2 is equivalent to
where \(N:S\rightarrow \widetilde{\mathcal {D}}\) is a nonlinear operator. If we can show that N maps S into \(\widetilde{\mathcal {D}}\), then a solution of Problem 2 is a fixed point of the problem
Our aim is to apply Banach’s fixed point theorem, which will also guarantee uniqueness of the solution, if we can show that \(T:=(\tilde{L}^{-1}\circ N): S_0\rightarrow S_0\) is a contraction mapping, i.e., there exists some \(q\in [0,1)\) such that
where the corresponding subset \(S_0\subset S\) is a small closed ball with center \(\tilde{L}^{-1}f\) (the solution of the corresponding linear problem) and suitably chosen radius \(R>0\).
Our main result is stated in the theorem below.
Theorem 7
(Nonlinear Sobolev and Kondratiev regularity) Let \(\tilde{L}\) and N be as described above. Assume the assumptions of Theorem 5 are satisfied and, additionally, we have \(\gamma _m\ge 1\), \(m\ge 2\), and \(a\ge -\frac{1}{2}\). Let
and consider the data space
Moreover, let
and consider the solution space \(S:=S_1\cap S_2\). Suppose that \(f\in \widetilde{\mathcal {D}}\) and put \(\eta :=\Vert f|\widetilde{\mathcal {D}}\Vert \) and \(r_0>1\). Moreover, we choose \(\varepsilon >0\) so small that
and
where \(c>0\) denotes the constant in (4.28) resulting from our estimates below. Then there exists a unique solution \(u\in S_0\subset S\) of Problem 2, where \(S_0\) denotes a small ball around \(\tilde{L}^{-1}f\) (the solution of the corresponding linear problem) with radius \(R=(r_0-1)\eta \Vert \tilde{L}^{-1}\Vert \).
Proof
Let u be the solution of the linear problem \(\tilde{L}u=f\). From Theorems 5 and 4 we know that \(\tilde{L}^{-1}: \widetilde{\mathcal {D}}\rightarrow S \) is a bounded operator. If \(u^M\in \widetilde{\mathcal {D}}\) (this will immediately follow from our calculations in Step 1 as explained in Step 2 below), the nonlinear part N satisfies the desired mapping properties, i.e., \(Nu=f-\varepsilon u^M\in \widetilde{\mathcal {D}}\) and we can apply Theorem 5 now with right-hand side Nu.
Step 1: Since
one sees that \(\tilde{L}^{-1}\circ N\) is a contraction if, and only, if
where \(u,v\in S_0\) (meaning \(u,v\in B_R(\tilde{L}^{-1}f)\) in S). We analyze the resulting condition with the help of the formula \( u^M-v^M=(u-v)\sum _{j=0}^{M-1} u^jv^{M-1-j}\). This together with Theorem 5 gives
Concerning the derivatives, we use Leibniz’s formula twice and we see that
In order to estimate the terms \(\partial _{t^r}u^j\) and \(\partial _{t^{k-l-r}}v^{M-1-j}\) we apply Faà di Bruno’s formula
where the sum runs over all r-tuples of nonnegative integers \((k_1,\ldots , k_r)\) satisfying
In particular, from (4.15) we see that \(k_{r}\le 1\), where \(r=1,\ldots , k\). Therefore, the highest derivative \(\partial _{t^r}u\) appears at most once. We apply the formula to \(g=u\) and \(f(x)=x^j\) and make use of the embeddings (2.3) and the pointwise multiplier results from Theorem 1 (i) for \(k\le \gamma _m-1\). (Note that the restriction ‘\(a>\frac{d}{p}\)’ for \(d=3\) in Theorem 1 (i) is satisfied since in our situation we have \(a+2m\ge m>\frac{d}{2}\) from the assumptions \(a\in [-m,m]\) and \(m\ge 2\).) This yields
For \(k=\gamma _m\) we use Theorem 1(ii). (Note that in Theorem 1(ii) we require that ’\(a-1\ge \frac{d}{p}-2\)’ with \(d=3\) for the parameter. In our situation below \(a-1\) has to be replaced by a, which leads to our restriction \(a\ge \frac{d}{2}-2=-\frac{1}{2}\).) Similar as above we obtain
Note that we require \(\gamma _m\ge 1\) in the last step. We proceed similarly for \(\partial _{t^{k-l-r}}v^{M-1-j}\). Now (4.13) together with (4.16) and (4.17) inserted in (4.12) together with Theorem 1 give
We give some explanations concerning the estimate above. In (4.18) the term with \(k=\gamma _m\) requires some special care since we have to apply Theorem 1 (ii). In this case, we calculate
The lower order derivatives in the last line cause no problems since we can (again) apply Theorem 1(i) as before. The term \(\left\| u^jv^{M-1-j}|\mathcal {K}^{2}_{2,a+2}(D)\right\| \) can now be further estimated with the help of Theorem 1(i). For the term \(\sum _{r=0}^{\gamma _m}\left\| (\partial _{t^r}u^j) (\partial _{t^{\gamma _m-r}}v^{M-1-j})|\mathcal {K}^{0}_{2,a}(D)\right\| \) we again use Theorem 1(ii), then proceed as in (4.17) and see that the resulting estimate yields (4.18).
Moreover, in (4.19) we use the fact that in the step before we have two sums with \(\kappa _1+\cdots +\kappa _r\le j\) and \(\kappa _1+\cdots +\kappa _{k-l-r}\le M-1-j\), i.e., we have \(k-l\) different \(\kappa _i\)’s which leads to at most k different \(\kappa _i\)’s if \(l=0\). We allow all combinations of \(\kappa _i\)’s and redefine the \(\kappa _i\)’s in the second sum leading to \(\kappa _1', \ldots , \kappa _k'\) with \(\kappa _1'+\cdots +\kappa _k'\le M-1\) and replace the old conditions \(\kappa _1+2\kappa _2+r\kappa _r\le r\) and \(\kappa _1+2\kappa _2+(k-l-r)\kappa _{k-l-r}\le k-l-r\) by the weaker ones \(\kappa _1'+\cdots +\kappa _k'\le k\) and \(\kappa _k'\le 1\). This causes no problems since the other terms appearing in this step do not depend on \(\kappa _i\) apart from the product term. There, the fact that some of the old \(\kappa _i\)’s from both sums might coincide is reflected in the new exponent \(4\kappa _i'\). From Theorem 1 we conclude that
hence, the term involving the maxima, \(\max _{w\in \{u,v\}}\max _{l=0,\ldots , \gamma _m}\max (\ldots )^{M-1}\) in (4.20) is bounded by \(\max (R+\Vert \tilde{L}^{-1}f|S\Vert ,1)^{M-1}\). Moreover, since u and v are taken from \(B_R(\tilde{L}^{-1}f)\) in \(S=S_1\cap S_2\), we obtain from (4.20),
where we put \(\eta :=\Vert f|\mathcal {D}\Vert \) in the last line, \(c_0\) denotes the constant resulting from (4.16) and (4.20) and \(c_2=c_0c_1\) with \(c_1\) being the constant from the estimates in Theorem 5.
We now turn our attention towards the second term \(\Vert \tilde{L}^{-1}\Vert \Vert u^M-v^M|\mathcal {D}_2\Vert \) in (4.12) and calculate
where we used Leibniz’s formula twice as in (4.13) in the second but last line. Again Faà di Bruno’s formula, cf. (4.14), is applied in order to estimate the derivatives in (4.22). We use a special case of the multiplier result from [39, Sect. 4.6.1, Thm. 1(i)], which tells us that for \(m>\frac{3}{2}\) we have
(we remark that this is exactly the point where our assumption \(m\ge 2\) comes into play). With this we obtain
Similar for \(\partial _{t^{k-l-r}}v^{M-1-j}\). As before, from (4.15) we observe \(k_{r}\le 1\), therefore the highest derivative \(u^{(r)}\) appears at most once. Note that since \(H^m(D)\) is a multiplication algebra for \(m> \frac{d}{2}\), we get (4.24) with \(L_2(D)\) replaced by \(H^m(D)\) as well. Now (4.23) and (4.24) inserted in (4.22) give
Similar to (4.20) in the calculations above the term \(k=\gamma _m+1\) required some special care. For the redefinition of the \(\kappa _i\)’s in the second but last line in (4.25) we refer to the explanations given after (4.20). From Theorem 1 we see that
hence the term \({\max _{w\in \{u,v\}}\max _{m=0,\ldots , l}\max (\ldots )^{M-1}}\) in (4.25) is bounded. Moreover, since u and v are taken from \(B_R(\tilde{L}^{-1}f)\) in \(S_2={H}^{m,\gamma _m+2*}(D_T)=H^{\gamma _m+1}([0,T],{\mathring{H}^m}(D))\cap H^{\gamma _m+2}([0,T],H^{-m}(D)) \), as in (4.21) we obtain from (4.25) and (4.26),
where we put \(\eta :=\Vert f|\mathcal {D}\Vert \) and \(c_3\) denotes the constant arising from our estimates (4.25) and (4.26) above. Now (4.12) together with (4.21) and (4.27) yields
where \(c=c_2+c_3\). For \(\tilde{L}^{-1}\circ N\) to be a contraction, we therefore require
cf. (4.11). In case of \(\ \max (R+\Vert \tilde{L}^{-1}\Vert \eta ,1)=1\) this leads to
On the other hand, if \(\max (R+\Vert \tilde{L}^{-1}\Vert \eta ,1)=R+\Vert \tilde{L}^{-1}\Vert \eta \), we choose \(R=(r_0-1)\eta \Vert \tilde{L}^{-1}\Vert \), which gives rise to the condition
Step 2: The calculations in Step 1 show that \(u^M\in \widetilde{\mathcal {D}}\): The fact that \(u^M\in \mathcal {D}_1\cap \mathcal {D}_2\) follows from the estimate (4.28). In particular, taking \(v=0\) in (4.28) we get an estimate from above for \(\Vert u^M|\widetilde{\mathcal {D}}\Vert \). The upper bound depends on \(\Vert u|S\Vert \) and several constants which depend on u but are finite whenever we have \(u\in S\), see also (4.20) and (4.25). The dependence on R in (4.28) comes from the fact that we choose \(u\in B_R(\tilde{L}^{-1}f)\) in S there. However, the same argument can also be applied to an arbitrary \(u\in S\); this would result in a different constant \(\tilde{c}\). In order to have \(u^M\in \widetilde{\mathcal {D}}\), we still need to show that \(\mathrm {Tr}\left( \partial _{t^k}u^M\right) =0\), \(k=0,\ldots , \gamma _m\). This follows from the same arguments as in [22, Thm. 4.10]: Since \(u\in S\hookrightarrow H^{\gamma _m+2}([0,T], H^{-m}(D))\hookrightarrow C^{\gamma _m+1}([0,T], H^{-m}(D))\) we see that the trace operator \(\mathrm {Tr}\left( \partial _{t^k}u\right) :=\left( \partial _{t^k}u\right) (0,\cdot )\) is well defined for \(k=0,\ldots , \gamma _m+1\). Using the initial assumption \(u(0,\cdot )=0\) in Problem 2, by density arguments (\(C^{\infty }(D_T)\) is dense in S) and induction we deduce that \((\partial _{t^k}u)(0,\cdot )=0\) for all \(k=0,\ldots , \gamma _m+1\). Moreover, since by Theorem 1
we see that the trace operator \(\mathrm {Tr}\left( \partial _{t^k}u^M\right) :=\left( \partial _{t^k}u^M\right) (0,\cdot )\) is well defined for \(k=0,\ldots , \gamma _m\). By (4.24) below the term \(\Vert \left( \partial _{t^k}u^M\right) (0,\cdot )|L_2(D)\Vert \) is estimated from above by powers of \(\Vert \left( \partial _{t^l}u\right) (0,\cdot )|H^m(D)\Vert \), \(l=0,\ldots , k\). Since all these terms are equal to zero, this shows that \(u^M\in \widetilde{\mathcal {D}}\).
Step 3: The next step is to show that \((\tilde{L}^{-1}\circ N)(B_R(\tilde{L}^{-1}f))\subset B_R(\tilde{L}^{-1}f)\) in S. Since \((\tilde{L}^{-1}\circ N)(0)=\tilde{L}^{-1}(f-\varepsilon 0^M)=\tilde{L}^{-1}f\), we only need to apply the above estimate (4.28) with \(v=0\). This gives
which, in case that \(\max (R+\Vert \tilde{L}^{-1}\Vert \eta ,1)=1\), leads to
whereas for \(\max (R+\Vert \tilde{L}^{-1}\Vert \eta ,1)=R+\Vert \tilde{L}^{-1}\Vert \eta \) we get
We see that condition (4.31) implies (4.29). Furthermore, since
also condition (4.32) implies (4.30). Thus, by applying Banach’s fixed point theorem in a sufficiently small ball around the solution of the corresponding linear problem, we obtain a unique solution of Problem 2. \(\square \)
Remark 10
The restriction \(m\ge 2\) in Theorem 7 comes from the fact that we require \(s_2=m>\frac{d}{2}=\frac{3}{2}\) in (4.23). This assumption can probably be weakened, since we expect the solution to satisfy \(u\in L_2([0,T], H^{s}(D))\) for all \(s<\frac{3}{2}\), see also Remark 11 and the explanations given there.
Moreover, the restriction \(a\ge -\frac{1}{2}\) in Theorem 7 comes from Theorem 1(ii) that we applied. Together with the restriction \(a\in [-m,m]\) we are looking for \(a\in [-\frac{1}{2},m]\) if the domain is a corner domain, e.g., a smooth cone \(K\subset \mathbb {R}^3\) (subject to some truncation). For polyhedral cones with edges \(M_k\), \(k=1,\ldots , l\), we furthermore require \(-\delta _-^{(k)}<a+2m(\gamma _m-i)+m<\delta ^{(k)}_+\) for \(i=0,\ldots , \gamma _m\) from Theorem 5.
5 Regularity Results in Besov Spaces
With all preliminary work, in this section we finally come to the presentation of the regularity results in Besov spaces for Problems 1 and 2. For this purpose, we rely on the results from Sect. 4 on regularity in Sobolev and Kondratiev spaces for the respective problems and combine these with the embeddings of Kondratiev spaces into Besov spaces. It turns out that in all cases studied the Besov regularity is higher than the Sobolev regularity. This indicates that adaptivity pays off when solving these problems numerically.
The Sobolev regularity we are working with (e.g., see Theorem 3 for Problem 1) canonically comes out from the variational formulation of the problem, i.e., we have spatial Sobolev regularity m if the corresponding differential operator is of order 2m. We give an outlook on how our results could be improved by using regularity results in fractional Sobolev spaces instead. It is planned to do further investigations in this direction in the future.
Moreover, we discuss the role of the weight parameter a appearing in our Kondratiev spaces to some extent.
5.1 Besov Regularity of Problem I
A combination of Theorem 5 (Kondratiev regularity A) and the embedding in Theorem 2 yields the following Besov regularity of Problem 1.
Theorem 8
(Parabolic Besov regularity A) Let D be a bounded polyhedral domain in \(\mathbb {R}^3\). Let \(\gamma \in \mathbb {N}\) with \({\gamma \ge 2m}\) and put \(\gamma _m:=\left[ \frac{\gamma -1}{2m}\right] \). Furthermore, let \(a\in \mathbb {R}\) with \(a\in [-m,m]\). Assume that the right-hand side f of Problem 1 satisfies
-
(i)
\(\partial _{t^k} f\in L_2(D_T)\cap L_2([0,T],\mathcal {K}^{2m(\gamma _m-k)}_{2,a+2m(\gamma _m-k)}(D))\), \(k=0,\ldots , \gamma _m\); \(\partial _{t^{\gamma _m+1}} f\in L_2(D_T)\).
-
(ii)
\(\partial _{t^k} f(x,0)=0\), \(k=0,1,\ldots , {\gamma _m}.\)
Furthermore, let Assumption 1 hold for weight parameters \(b=a+2m(\gamma _m-i)\), where \(i=0,\ldots , \gamma _m\), and \(b'=-m\). Then for the weak solution \(u\in {{H}}^{m,{\gamma _m+2}*}(D_T)\) of Problem 1, we have
where \(\frac{1}{2}<\frac{1}{\tau }<\frac{\alpha }{d}+\frac{1}{2}\) and \(\delta \) denotes the dimension of the singular set of D. In particular, for any \(\alpha \) satisfying (5.1) and \(\tau \) as above, we have the a priori estimate
Proof
According to Theorem 5 by our assumptions we know \( u\in L_2([0,T], {\mathcal {K}^{2m(\gamma _m+1)}_{2,a+2m(\gamma _m+1)}(D)})\). Together with Theorem 2 (choosing \(k=0\)) we obtain
where in the third step we use the fact that \( 2m(\gamma _m+1)\ge 2m\left( \frac{\gamma }{2m}-1+1\right) =\gamma \) and choose \(\alpha \le \gamma \). Moreover, the condition on a from Theorem 2 yields \( {m=}{\min (m,a+2m(\gamma _m+1))}>\frac{\delta }{3}\alpha . \) Therefore, the upper bound for \(\alpha \) is \( \alpha <{\min \left( \gamma ,\frac{3}{\delta }m\right) }. \) Concerning the restriction on \(\tau \), Theorem 2 with \(\tau _0=2\) gives \( \frac{1}{2}<\frac{1}{\tau }<\frac{1}{\tau ^{*}}=\frac{\alpha }{3}+\frac{1}{2}. \) This completes the proof. \(\square \)
Remark 11
(The parameter a) We discuss the role of the weight parameter in our Kondratiev spaces: Note that on the one hand we require \(a+2m(\gamma _m+1)>0\) in order to apply the embedding from Theorem 2. Since we assume \(a\in [-m,m]\) this is always true. On the other hand it should be expected that the derivatives of the solution u have singularities near the boundary of the polyhedral domain. Thus, looking at the highest derivative of \(u(t)\in \mathcal {K}^{2m(\gamma _m+1)}_{2,a+2m(\gamma _m+1)}(D)\) we see that we require
hence, if \(a<0\) the derivatives of the solution u might be unbounded near the boundary of D. From this it follows that the range \(\ {-m}<a<0 \ \) is the most interesting for our considerations.
Remark 12
The above theorem relies on the fact that Problem 1 has a weak solution \(u\in H^{m,{\gamma _m+2}*}(D_T)={H^{\gamma _m+1}}([0,T], \mathring{H}^m(D))\cap {H^{\gamma _m+2}}([0,T],H^{-m}(D))\hookrightarrow L_2([0,T], H^m(D))\), cf. Theorem 4. We strongly believe that (in good agreement with the elliptic case) this result can be improved by studying the regularity of Problem 1 in fractional Sobolev spaces \(H^s(D)\). In this case (assuming that the weak solution of Problem 1 satisfies \(u\in L_2([0,T],H^s(D))\) for some \(s>0\)) under the assumptions of Theorem 8, using Theorem 5 and Theorem 2 (with \(k=0\)), we would obtain
where \(a'=a+ 2m(\gamma _m+1)\ge a+2m\) and again \(\frac{1}{2}<\frac{1}{\tau }<\frac{\alpha }{3}+\frac{1}{2}\) but the restriction on \(\alpha \) now reads as
For general Lipschitz domains \(D\subset \mathbb {R}^3\) we expect that the solution of Problem 1 (for \(m=1\)) is contained in \(H^s(D)\) for all \(s<\frac{3}{2}\) (as in the elliptic case, cf. [32]). This would lead to \(\alpha <\frac{9}{2}\) when \(\delta =1\). For convex domains it probably even holds that \(s=2\) (for the heat equation this was already proven in [45]). First results in this direction can be found in [21].
Alternatively, we combine Theorem 6 (Kondratiev regularity B) and Theorem 2. This leads to the following regularity result in Besov spaces.
Theorem 9
(Parabolic Besov regularity B) Let D be a bounded polyhedral domain in \(\mathbb {R}^3\). Let \(\gamma \in \mathbb {N}\) with \({\gamma \ge 2m}\). Moreover, let \(a\in \mathbb {R}\) with \({a\in [-m,m]}\). Assume that the right-hand side f of Problem 1 satisfies
-
(i)
\(f\in \bigcap _{l=0}^{\infty }H^l([0,T],L_2(D)\cap \mathcal {K}^{\gamma -2m}_{2,a}(D))\).
-
(ii)
\(\partial _{t^l} f(x,0)=0\), \(l\in \mathbb {N}_0\).
Furthermore, let Assumption 1 hold for weight parameters \(b=a\) and \(b'=-m\). Then for the weak solution \(\bigcap _{l=0}^{\infty }u\in {{H}}^{m,{l+1}*}(D_T)\) of Problem 1, we have
where \(\frac{1}{2}<\frac{1}{\tau }<\frac{\alpha }{3}+\frac{1}{2}\) and \(\delta \) denotes the dimension of the singular set of D. In particular, for any \({\alpha }\) satisfying (5.4) and \(\tau \) as above, we have the a priori estimate
Proof
According to Theorem 6 by our assumptions we know \(u\in L_2([0,T], {\mathcal {K}^{\gamma }_{2,a+2m}(D)})\). Together with Theorem 2 (choosing \(k=0\)) we obtain
where \(\alpha \le \gamma \) in the second to last line. Moreover, the condition on the parameter ’a’ from Theorem 2 yields \(m=\min (m,a+2m)>\frac{\delta }{3}\alpha . \) Therefore, the upper bound for \({\alpha }\) is \({\alpha <\min \left( \gamma ,\frac{3}{\delta }m\right) }. \) Concerning the restriction on \(\tau \), Theorem 2 with \(\tau _0=2\) gives \( \frac{1}{2}<\frac{1}{\tau }<\frac{1}{\tau ^{*}}=\frac{{\alpha }}{3}+\frac{1}{2}. \) This finishes the proof. \(\square \)
Remark 13
It might not be obvious at first glance that Assumption 1 is satisfied with the parameter restrictions in Theorems 8 and 9. For a discussion on this subject we refer to [22, Rem. 3.8, Ex 4.8], where this matter was discussed in detail and exemplary illustrated for the heat equation. We do not want to repeat the arguments here.
5.2 Besov Regularity of Problem II
Concerning the Besov regularity of Problem 2, we proceed in the same way as before for Problem 1: Combining Theorem 7 (Nonlinear Sobolev and Kondratiev regularity) with the embeddings from Theorem 2 we derive the following result.
Theorem 10
(Nonlinear Besov regularity) Let the assumptions of Theorems 7 and 5 be satisfied. In particular, as in Theorem 7 for \(\eta :=\Vert f|\widetilde{\mathcal {D}}\Vert \) and \(r_0>1\), we choose \(\varepsilon >0\) so small that
and
Then there exists a solution u of Problem 2, which satisfies \( u\in B_0\subset B\),
for all \(0<\alpha <\min \left( \frac{3}{\delta }m,\gamma \right) \), where \(\delta \) denotes the dimension of the singular set of D, \(\frac{1}{2}<\frac{1}{\tau }<\frac{\alpha }{3}+\frac{1}{2}\), and \(B_0\) is a small ball around \(\tilde{L}^{-1}f\) (the solution of the corresponding linear problem) with radius \(R={C\tilde{C}}(r_0-1)\eta \Vert \tilde{L}^{-1}\Vert \) (Fig. 5).
Nonlinear solution in \(B_0\)
Proof
This is a consequence of the regularity results in Kondratiev and Sobolev spaces from Theorem 7. To be more precise, Theorem 7 establishes the existence of a fixed point u in
This together with the embedding results for Besov spaces from Theorem 2 (choosing \(k=0\)) completes the proof, in particular, we calculate for the solution (cf. the proof of Theorem 8)
Furthermore, it can be seen from (5.7) that new constants C and \(\tilde{C}\) appear when considering the radius R around the linear solution where the problem can be solved compared to Theorem 7. \(\square \)
Remark 14
A few words concerning the parameters appearing in Theorem 10 (and also Theorem 7) seem to be in order. Usually, the operator norm \(\Vert \tilde{L}^{-1}\Vert \) as well as \(\varepsilon \) are fixed; but we can change \(\eta \) and \(r_0\) according to our needs. From this we deduce that by choosing \(\eta \) small enough the condition (5.6) can always be satisfied. Moreover, it is easy to see that the smaller the nonlinear perturbation \(\varepsilon >0\) is, the larger we can choose the radius R of the ball \(B_0\) where the solution of Problem 2 is unique.
5.3 Hölder–Besov Regularity of Problem I
So far we have not exploited the fact that Theorem 5 (Kondratiev regularity A) not only provides regularity properties of the solution u of Problem 1 but also of its partial derivatives \(\partial _{t^k} u\). We use this fact in combination with Theorem 1 in order to obtain some mixed Hölder–Besov regularity results on the whole space-time cylinder \(D_T\).
For parabolic SPDEs, results in this direction have been obtained in [9]. However, for SPDEs, the time regularity is limited in nature. This is caused by the nonsmooth character of the driving processes. Typically, Hölder regularity \(\mathcal {C}^{0,\beta }\) can be obtained, but not more. In contrast to this, it is well known that deterministic parabolic PDEs are smoothing in time. Therefore, in the deterministic case considered here, higher regularity results in time can be obtained compared to the probabilistic setting.
Theorem 11
(Hölder–Besov regularity) Let D be a bounded polyhedral domain in \(\mathbb {R}^3\). Moreover, let \(\gamma \in \mathbb {N}\) with \(\gamma \ge 4m+1\) and put \(\gamma _m:=\left[ \frac{\gamma -1}{2m}\right] \). Furthermore, let \(a\in \mathbb {R}\) with \({a\in [-m,m]}\). Assume that the right-hand side f of Problem 1 satisfies
-
(i)
\(\partial _{t^k} f\in L_2(D_T)\cap L_2([0,T],\mathcal {K}^{2m(\gamma _m-k)}_{2,a+2m(\gamma _m-k)}(D))\), \(k=0,\ldots , \gamma _m\), \(\partial _{t^{\gamma _m+1}} f\in L_2(D_T)\).
-
(ii)
\(\partial _{t^k} f(x,0)=0\), \(k=0,1,\ldots , {\gamma _m}.\)
Let Assumption 1 hold for weight parameters \(b=a+2m(\gamma _m-i)\), where \(i=0,\ldots , \gamma _m\) and \(b'=-m\). Then for the solution \(u\in {H}_2^{m,\gamma _m+2*}(D_T)\) of Problem 1, we have
where \(\frac{1}{2}<\frac{1}{\tau }<\frac{\eta }{3}+\frac{1}{2}\) and \(\delta \) denotes the dimension of the singular set of D. In particular, we have the a priori estimate
where the constant is independent of u and f.
Proof
Theorems 5 and 4 show together with Theorems 2 and 1, that under the given assumptions on the initial data f, we have for \(k\le \gamma _m-2\),
where in the third step we require \(\eta \le 2m(\gamma _m-k)\) and by Theorem 2 we get the additional restriction \( m=\min (m,a+2m(\gamma _m-k))\ge \frac{\delta }{3}\eta , \ \text {i.e.}, \ \eta <\frac{3}{\delta }m. \) Therefore, the upper bound on \(\eta \) reads as \(\eta <\min \left( \frac{3}{\delta }m, 2m(\gamma _m-k)\right) \) since \(k\le \gamma _m-2\), which for \(k=\gamma _m-2\) yields \(\eta <\min \left( \frac{3}{\delta }, 4\right) m\). \(\square \)
Remark 14
-
(i)
For \(\gamma \ge 2m+1\) and \(k=\gamma _m-1\) we have \(\eta \le \min \left( \frac{3}{\delta },2\right) m\) in the theorem above. For \(\gamma \ge 2m\) and \(k=\gamma _m\) we get \(\eta =0\).
-
(ii)
From the proof of Theorem 11 above it can be seen that the solution satisfies \(u\in {\mathcal {C}^{{k},\frac{1}{2}}([0,T], {\mathcal {K}^{2m(\gamma _m-k)}_{2,a+2m(\gamma _m-k)}(D)}), }\) implying that for high regularity in time, which is displayed by the parameter k, we have less spatial regularity in terms of \(2m(\gamma _m-k)\).
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A Appendix
A Appendix
1.1 A.1 Preliminaries
We collect some notation used throughout the paper. As usual, we denote by \(\mathbb {N}\) the set of all natural numbers, \(\mathbb {N}_0=\mathbb N\cup \{0\}\), and \(\mathbb {R}^d\), \(d\in \mathbb {N}\), the d-dimensional real Euclidean space with |x|, for \(x\in \mathbb {R}^d\), denoting the Euclidean norm of x. By \(\mathbb {Z}^d\) we denote the lattice of all points in \(\mathbb {R}^d\) with integer components. For \(a\in \mathbb {R}\), let [a] denote its integer part.
Moreover, c stands for a generic positive constant which is independent of the main parameters, but its value may change from line to line. The expression \(A\lesssim B\) means that \( A \le c\,B\). If \(A \lesssim B\) and \(B\lesssim A\), then we write \(A \sim B\).
Given two quasi-Banach spaces X and Y, we write \(X\hookrightarrow Y\) if \(X\subset Y\) and the natural embedding is bounded. By \(\mathop {\mathrm {supp}\,}\nolimits f\) we denote the support of the function f. For a domain \(\Omega \subset \mathbb {R}^d\) and \(r\in \mathbb {N}\cup \{\infty \}\) we write \(C^r(\Omega )\) for the space of all real-valued r-times continuously differentiable functions, whereas \(C(\Omega )\) is the space of bounded uniformly continuous functions, and \(\mathcal {D}(\Omega )\) for the set of test functions, i.e., the collection of all infinitely differentiable functions with compact support contained in \(\Omega \). Moreover, \(L^1_{\text {loc}}(\Omega )\) denotes the space of locally integrable functions on \(\Omega \).
For a multi-index \(\alpha = (\alpha _1, \ldots ,\alpha _d)\in \mathbb {N}_0^d\) with \(|\alpha | := \alpha _1+\cdots + \alpha _d=r\), \(r\in \mathbb {N}_0\), and an r-times differentiable function \(u:\Omega \rightarrow \mathbb {R}\), we write
for the corresponding classical partial derivative as well as \(u^{(k)}:=D^{(k)}u\) in the one-dimensional case. Hence, the space \(C^r(\Omega )\) is normed by
Moreover, \(\mathcal {S}(\mathbb {R}^d)\) denotes the Schwartz space of rapidly decreasing functions. The set of distributions on \(\Omega \) will be denoted by \(\mathcal {D}'(\Omega )\), whereas \(\mathcal {S}'(\mathbb {R}^d)\) denotes the set of tempered distributions on \(\mathbb {R}^d\). The terms distribution and generalized function will be used synonymously. For the application of a distribution \(u\in \mathcal {D}'(\Omega )\) to a test function \(\varphi \in \mathcal {D}(\Omega )\) we write \((u,\varphi )\). The same notation will be used if \(u\in \mathcal {S}'(\mathbb {R}^d)\) and \(\varphi \in \mathcal {S}(\mathbb {R}^d)\) (and also for the inner product in \(L_2(\Omega )\)). For \(u\in \mathcal {D}'(\Omega )\) and a multi-index \(\alpha = (\alpha _1, \ldots ,\alpha _d)\in \mathbb {N}_0^d\), we write \(D^{\alpha }u\) for the \(\alpha \)-th generalized or distributional derivative of u with respect to \(x=(x_1,\ldots , x_d)\in \Omega \), i.e., \(D^{\alpha }u\) is a distribution on \(\Omega \), uniquely determined by the formula
In particular, if \(u\in L^1_{\text {loc}}(\Omega )\) and there exists a function \(v\in L^1_{\text {loc}}(\Omega )\) such that
we say that v is the \(\alpha \)-th weak derivative of u and write \(D^{\alpha }u=v\). We also use the notation \( \frac{\partial ^k}{\partial x_j^k}u:=D^{\beta }u \) as well as \(\partial _{x_j^k}:=D^{\beta }u\), for some multi-index \(\beta =(0,\ldots , k, \ldots ,0)\) with \(\beta _j=k\), \(k\in \mathbb {N}\). Furthermore, for \(m\in \mathbb {N}_0\), we write \(D^mu\) for any (generalized) m-th order derivative of u, where \(D^0u:=u\) and \(Du:=D^1u\). Sometimes we shall use subscripts such as \(D^m_x\) or \(D^m_t \) to emphasize that we only take derivatives with respect to \(x=(x_1, \ldots , x_d)\in \Omega \) or \(t\in \mathbb {R}\).
1.2 Besov Spaces
Due to the different contexts Besov spaces arose from they can be defined/characterized in several ways, e.g., via higher order differences, the Fourier-analytic approach or decompositions with suitable building blocks, cf. [43, 44] and the references therein. Under certain restrictions on the parameters these different approaches might even coincide. Throughout this paper, we rely on the characterization of Besov spaces via wavelet decompositions and refer in this context to [11, 38]. Let us briefly recall the concept: Wavelets are specific orthonormal bases for \(L_2(\mathbb {R})\) that are obtained by dilating, translating, and scaling one fixed function, the so-called mother wavelet \(\psi \). The mother wavelet is usually constructed by means of a so-called multiresolution analysis, that is, a sequence \(\{V_j\}_{j \in \mathbb {Z}}\) of shift-invariant, closed subspaces of \(L_2(\mathbb {R})\) whose union is dense in \(L_2\) while their intersection is trivial. Moreover, all the spaces are related via dyadic dilation, and the space \(V_0\) is spanned by the translates of one fixed function \(\phi \), called the generator. In her fundamental work [24, 25] I. Daubechies has shown that there exist families of compactly supported wavelets. By taking tensor products, a compactly supported orthonormal basis for \(L_2(\mathbb {R}^d)\) can be constructed.
Let \({\phi }\) be a father wavelet of tensor product type on \(\mathbb {R}^d\) and let \(\Psi '=\{\psi _i: \ i=1,\ldots , 2^d-1\}\) be the set containing the corresponding multivariate mother wavelets such that, for a given \(r\in \mathbb {N}\) and some \(N>0\) the following localization, smoothness, and vanishing moment conditions hold. For all \(\psi \in \Psi '\),
We refer again to [24, 25] for a detailed discussion. The set of all dyadic cubes in \(\mathbb {R}^d\) with measure at most 1 is denoted by
and we set \(\mathcal {D}_j:=\{I\in \mathcal {D}^+: \ |I|=2^{-jd}\}.\) For the dyadic shifts and dilations of the father wavelet and the corresponding wavelets we use the abbreviations
It follows that
is an orthonormal basis in \(L_2(\mathbb {R}^d)\). Denote by Q(I) some dyadic cube (of minimal size) such that \(\mathop {\mathrm {supp}\,}\nolimits \psi _I \subset Q(I)\) for every \(\psi \in \Psi '\). Then, we clearly have \(Q(I)=2^{-j}k+2^{-j}Q\) for some dyadic cube Q. Put \(\Lambda '=\mathcal {D}^{+}\times \Psi '\). Then, every function \(f\in L_2(\mathbb {R}^d)\) can be written as
It will be convenient to include \({\phi }\) into the set \(\Psi '\). We use the notation \({\phi }_I:=0\) for \(|I|<1\), \({\phi }_I={\phi }(\cdot -k)\) for \(I=k+[0,1]^d\), and can simply write
We describe Besov spaces on \(\mathbb {R}^d\) by decay properties of the wavelet coefficients, if the parameters fulfill certain conditions.
Theorem A.1
(Wavelet decomposition of Besov spaces) Let \(0<p,q<\infty \), and \(s>\max \left\{ 0,d(1/p-1)\right\} \). Choose \(r\in \mathbb {N}\) such that \(r>s\) and construct a wavelet Riesz basis as described above. Then a function \(f\in L_p(\mathbb {R}^d)\) belongs to the Besov space \(B^s_{p,q}(\mathbb {R}^d)\) if, and only if,
(convergence in \(\mathcal {S}'(\mathbb {R}^d)\)) with
Remark A.1
In particular, for our adaptivity scale (1.2), i.e., \(B^s_{\tau ,\tau }(\mathbb {R}^d)\) with \(s=d\left( \frac{1}{\tau }-\frac{1}{p}\right) \), we see that the quasi-norm (A.6) becomes
Corresponding function spaces on domains \(\mathcal {O}\subset \mathbb {R}^d\) can be introduced via restriction, i.e.,
Alternative (different or equivalent) versions of this definition can be found, depending on possible additional properties of the distributions g (most often their support). We refer to [44] for details and references.
Remark A.2
We remark that the Besov (and Kondratiev) spaces we are working with are defined in the setting of distributions, i.e., as subsets of \(\mathcal {D}'(\mathcal {O})\), and therefore may contain ‘functions’ which take complex values. However, when considering the fundamental parabolic problems, we restrict ourselves to the real-valued setting: We assume the coefficients of the differential operator L to be real-valued as well as the right-hand side f, therefore, the solutions are real-valued as well.
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Dahlke, S., Schneider, C. Regularity in Sobolev and Besov Spaces for Parabolic Problems on Domains of Polyhedral Type. J Geom Anal 31, 11741–11779 (2021). https://doi.org/10.1007/s12220-021-00700-6
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DOI: https://doi.org/10.1007/s12220-021-00700-6