Dissipative measure-valued solutions for general conservation laws

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Abstract

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued – strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. This property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system et al. and there are also some results concerning general hyperbolic systems. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems, and most importantly, we give examples of equations, for which the aspect of measure-valued – strong uniqueness has not been considered before, like incompressible magnetohydrodynamics and shallow water magnetohydrodynamics.

Introduction

The recent work of Brenier, De Lellis and Székelyhidi [4] significantly ennobled measure-valued solutions of systems of fluid dynamics, as well as hyperbolic systems in general. They postulated a new principle surprisingly stating that measure-valued solutions, which were expected to be non-unique to a large extent, become unique once we know that a strong solution emanating from the same initial data exists. In this case both solutions coincide on the time interval of existence of the strong solution. What they called weak-strong uniqueness for measure-valued solutions is now usually called measure-valued-strong uniqueness, or mv-strong uniqueness for short. We favor the latter term, as it seems more adequate. The analysis in the case of incompressible Euler system is complete, as DiPerna and Majda had shown in [15] existence of measure-valued solutions to the incompressible Euler system exactly in the class which, per the result of Brenier et al., possesses the property of mv-strong uniqueness.

Careful analysis of the incompressible Euler system allowed the authors of [4] to conjecture that an analogous property of mv-strong uniqueness could hold in a more general setting. They had in fact initiated the studies on mv-strong uniqueness for general hyperbolic systems. Following this path, we also direct our interest to a hyperbolic system of the formtA(u)+αFα(u)=0 with an initial condition u(0)=u0. Here u:[0,T]×TdX, where XRn is an open convex set and by X we mean the closure of X. Moreover, A,Fα:XRn, α=1,...,d, we use the Einstein summation convention and we denote Q=[0,T]×Td, where Td is a d−dimensional torus.

In [4] the authors studied system (1.1) with A(u)u, however their result holds in a class where no existence result is available (and seems impossible to be proven). This limitation is not particular only for such general systems, but persists even in special cases, including e.g. compressible Euler system, polyconvex elastodynamics or hyperbolic magnetohydrodynamics. The solution is in the form of a classical Young measure only (even satisfying a technical assumption that the first moment of this measure is in L(Q)), not a triple consisting of a classical Young measure and concentration and concentration angle measures.

In parallel Demoulini et al. [13] proved a corresponding result on mv-strong uniqueness for the system of polyconvex elastodynamics. And again the authors attempted to formulate a more general result for hyperbolic systems. Here the possibility of a concentration measure is allowed in the entropy inequality, not in the weak formulation of the system itself. This approach covers, among others, the case initially considered by the authors, i.e. the system of polyconvex elastodynamics. For this system the mv-strong uniqueness result is in the class coinciding with the class in which one shows existence of solutions. However, this level of generality is still not sufficient to cover the case of abstract hyperbolic system, as well as e.g. Euler equations, where concentration measure appears also in the weak formulation.

Therefore there is still a need to dispose of assumptions that solutions satisfy any a priori bounds, and in particular, that a solution consists only of a classical Young measure. We find it of great importance to include possibilities of concentration measures appearing in all terms A(u), the flux Fα(u) and an entropy function. A result on mv-strong uniqueness shall be deemed complete whenever the class of measure-valued solutions agrees with the class of an existence result.

Finally, we give a couple of examples of systems, for which the general result statement gives an original result of mv-strong uniqueness property, namely a system of shallow water magnetohydrodynamics described in Section 2 and incompressible magnetohydrodynamics described in Section 5. Surely the list of new applications is not complete.

Throughout the paper we will assume the following conditions hold.

  • (H1)

    There exists an open convex set XRn such that the mapping A:XRn is a C2 map on X, continuous and injective from X to Rn and satisfiesA(u) is nonsingular uX.

  • (H2)

    The system (1.1) is endowed with a companion lawtη(u)+αqα(u)=0 with an entropy η:XR+ which is a C2 map on X, continuous on X and satisfies η(u)0 andlim|u|η(u)=. This yields the existence of a smooth function G:XRn such thatη=GAqα=GFα,α=1,...,d. The conditions (1.5)-(1.6) are equivalent toGTA=ATGGTFα=FαTG,α=1,...,d.

  • (H3)

    The symmetric matrix2η(u)G(u)2A(u) is positive definite for all uX.

  • (H4)

    The vector A(u) and the fluxes Fα(u) are bounded by the entropy, i.e.|A(u)|Cη(u)|Fα(u)|Cη(u),α=1,...,d.

  • (H5)

    Defining for a strong solution U taking values in a compact subset of X the relative entropyη(u|U):=η(u)η(U)η(U)A(U)1(A(u)A(U))=η(u)η(U)G(U)(A(u)A(U)) and defining the relative flux asFα(u|U):=Fα(u)Fα(U)Fα(U)A(U)1(A(u)A(U)) for α=1,...,d we assume it holds|Fα(u|U)|Cη(u|U).

Remark 1.1

In comparison with [8] the hypotheses (H1)-(H5) introduce one main novelty, namely the presence of the convex set XRn such that the unknown u takes values in X, contrary to [8], where u takes values in Rn and A:RnRn. This is motivated by applications with an archetype example of compressible Euler system, where we want the density to be nonnegative, see section 2.1. Moreover, we emphasize here that the entropy is not bounded as |u|.

Remark 1.2

Observe that in the above definitions the relative flux Fα(|U) and relative entropy η(|U) are continuous functions in X. This follows directly from the continuity of Fα() and η(). Note that there is an asymmetry, the relative functions are well defined for uX, but for UX.

Remark 1.3

It is not difficult to observe that (H3) implies that the relative entropy η(u|U) is a nonnegative quantity which is equal to zero if and only if u=U, thus it can be used as a sort of a distance functions between u and U. Indeed, note that even though η is not a convex entropy, we can make a change of variables and introduce H(v)=ηA1(v) and thus η(u)=H(A(u)), where H is already a convex entropy as it follows from (H3) that for ξRn and ξ0ξ2H(v)ξ>0. Thus the relative quantity defined in the Appendix (A.2) will vanish if and only if u=U, and since the change of variables is a bijection, then this property is also inherited by η(u|U).

Remark 1.4

Note that if instead of (H4) we assume thatlim|u||A(u)|η(u)=0,|Fα(u)|C(1+η(u)),α=1,...,d, then (H5) follows directly from (1.15), see Lemma A.1 in the appendix.

An analogous lemma under more restrictive assumptionslim|u||A(u)|η(u)=lim|u||Fα(u)|η(u)=0, was proved in [8, Lemma A.1]. Note however that (1.16) is not satisfied e.g. by compressible Euler equations. Any concentration in terms A and Fα are not present due to assumption (1.16), which is a stronger requirement than (H4) assumed in the present paper. This however allowed the authors to omit the general representation of concentrations introduced in [15] and [1], because the concentration effect is considered just for the entropy, which is a non-negative scalar function. Thus one can provide a simple derivation of weak limit as a Young measure and a concentration measure. Under slightly different assumptions on the entropy and in the same formulation as currently considered, i.e., A(u) is not necessarily an identity, as in the aforementioned results, the issue of measure-valued-strong uniqueness was considered in [8]. In the same paper the authors also prove the measure-valued-strong uniqueness for a system of adiabatic thermoelasticity, where the measure valued solutions are defined with a concentration measure in the energy equation and not in the entropy inequality, see [8, Section 5.4]. We note that physical systems with an equation for energy conservation usually violate the hypothesis (H4) and therefore do not fall into the class of systems for which the theory of our paper can be directly applied.

In the spirit of these results, the issue of mv-strong uniqueness was considered for various systems, including compressible Euler system and Savage-Hutter system describing granular media in [23], compressible Navier-Stokes in [18] and complete compressible Euler system in [5]. An overview of these results is provided in [11], [34]. At this moment it is worth mentioning that the result of Březina and Feireisl [5] does not fit in any of the presented frameworks for general hyperbolic systems, including also the framework presented in the current paper. Contrary to the other cases, they consider the full thermo-mechanical system. Thus a new element here is an appearance of the physical entropy. The system consisting of conservation of mass and conservation of momentum is not a closed system, as the pressure depends on the energy. To complete the system additional equation for the energy is considered. Then the role of an entropy η should overtake a physical entropy, not as it was in the case of isentropic compressible Euler (as the system for the variables ϱ,v), when η was the energy (kinetic and potential). In the setting of Březina and Feireisl the entropy inequality does not carry information that would allow to bound the flux Fα(u). We claim that appearance of thermal energy in the system results that the system does not fit into the approach initiated by Brenier et al.

The relative entropy method, which is fundamental for mv-strong uniqueness results, appears to be useful for other areas such as stability studies, asymptotic limits and dimension reduction problems (e.g. [8], [21], [19], [3], [6]). Not only the systems describing phenomena of mathematical physics fall into these applications. Also results on problems arising from biology, cf. [28], [27], [29], [24], can serve as examples. The framework is known in this context as General Relative Entropy (GRE) and applies for showing asymptotic convergence of solutions to steady-state solutions. Finally we would like to underline how these results on measure-valued solutions in fluid mechanics affected certain numerical experiments, cf. [20].

Our interest is directed to the measure-valued-strong uniqueness principle for dissipative measure-valued solutions. We start with the motivation for our definition of measure valued solutions.

Assume we have at hand a sequence of solutions un solving some approximating problemtA(un)+αFα(un)=Pn together with appropriate approximating entropy equationtη(un)+αqα(un)=Qn with Pn,Qn0 in appropriate topologies. Natural a priori bound for such problem is derived through the entropy equation (1.18) and yieldsη(un)L(0,T,L1(Td))C. Due to our assumption (H4), see (1.10), we have the same L(0,T,L1(Td)) bound for quantities A(un) and Fα(un). Therefore due to Lemma A.2 and Remark A.3 we are able to disintegrate concentration measures related to each of these quantities as followsmf(dxdt)=mft(dx)dt.

Before defining solutions let us shortly describe the notation. By P(X) we mean the set of probability measures on X, Lweak((0,T)×Td;P(X)) stands for the space of weakly-star essentially bounded measurable maps with values in P(X). We mean by M([0,T]×Td) the space of measures on [0,T]×Td and M+([0,T]×Td) refers to positive measures.

Definition 1.5

We say that (ν,mA,mFα,mη), α=1,...,d, is a dissipative measure-valued solution of system (1.1) with initial data (ν0,,mA0,mη0) if {νt,x}(t,x)(0,T)×Td, νLweak((0,T)×Td;P(X)) is a parameterized measure and together with concentration measures mA(M([0,T]×Td))n, mFα(M([0,T]×Td))n×n satisfyQνt,x,A(λ)tφdxdt+QtφmA(dxdt)+Qνt,x,Fα(λ)αφdxdt+QαφmFα(dxdt)+Tdν0,x,A(λ)φ(0)dx+Tdφ(0)mA0(dx)=0 for all φCc(Q)n. Moreover, the total entropy balance holds for all ζCc([0,T)), ζ0Qνt,x,η(λ)ζ(t)dxdt+Qζ(t)mη(dxdt)+Tdν0,x,η(λ)ζ(0)dx+Tdζ(0)mη0(dx)0 with a dissipation measure mηM+([0,T]×Td).

Throughout our paper we always assume that there exists a generating sequence of approximate solutions to the system (1.1). Therefore we introduce the following definition.

Definition 1.6

We say that the dissipative measure-valued solution (ν,mA,mFα,mη), α=1,...,d, of system (1.1) is generated by a sequence of approximate solutions if there exist sequences un, Pn and Qn such that (1.17)-(1.18) hold in the sense of distributions, Pn and Qn converge to zero in distributions andf(un(t,x))dxdtνt,x,f(λ)dxdt+mf hold for f=A,Fα and η.

Our main theorem reads as follows.

Theorem 1.7

Assume that hypotheses (H1)-(H5) hold. Let (ν,mA,mFα,mη), α=1,...,d, be a dissipative measure-valued solution to (1.1) generated by a sequence of approximate solutions. Let UW1,(Q) be a strong solution to (1.1) with the same initial data u0L1(Rd), thus ν0,x=δu0(x), mA0=mη0=0. Then νt,x=δU(t,x) a.e. in Q and mA=mFα=mη=0.

One of the key ingredients in the proof of Theorem 1.7 is the following proposition stating relations between different concentration measures.

Proposition 1.8

Assume that the hypotheses (H1)-(H5) hold. Let (ν,mA,mFα,mη), α=1,...,d, be a dissipative measure-valued solution to (1.1) generated by a sequence of approximate solutions. Let UW1,(Q) be a strong solution to (1.1). Then the dissipative measure valued solution (ν,mA,mFα,mη) has the following properties:

  • (i)

    The concentration measure of the relative entropy η(u|U) is equal tomηmAG(U) andmηmAG(U)0.

  • (ii)

    The concentration measure of the relative flux Fα(u|U) is equal tomFαFα(U)A(U)1mA and it is bounded by the concentration measure of the relative entropy, i.e.|mFαFα(U)A(U)1mA|C(mηmAG(U)).

Measure-valued solutions, despite being a relatively weak notion of solutions, play an important role in modern analysis of nonlinear systems of partial differential equations. The basic concept behind this approach is to embed the problem into a wider space. Instead of considering sequences solving approximate problems, which are some measurable functions, one passes to the level of parametrized measures. The benefit of this idea is passing from a nonlinear problem to a linear one. The essence of the proof of existence of such solutions becomes a matter of appropriate estimates rather than subtle weak sequential stability arguments. There is of course a cost to be paid – the result of a limit is only a weak object represented by a Young measure, namely by a parametrized family of measures.

This framework begun with a celebrated paper of Young [35], see also [2] for a summary of the concept of Young measures. Later, Tartar [33] and DiPerna [14] applied this approach to define measure-valued solutions to scalar conservation laws and, as a bystep in the proof of existence of entropy weak solutions, showed uniqueness of entropy measure-valued solutions (we mean by that solutions satisfying in addition a variant of entropy inequality for measures).

The next breakthrough is due to DiPerna and Majda who directed their attention to the incompressible Euler system. Here, sequences of approximate solutions may not only oscillate, but also concentrate. Thus the original Young measure, capable of handling oscillations only, was insufficient to fully characterize weak limits of such sequences. An extension to generalized Young measures (or DiPerna-Majda measures) was later proposed, see [15] and also [1] for some refinements. A measure-valued solution was then defined not only as a Young measure, but a triple describing oscillations, concentrations and concentration angle. Since this framework transfers to other systems and to general case as well we provide the full details in Section 3.2.

We direct our interest to measure-valued solutions to hyperbolic conservation laws. Unlike in the scalar case, for systems of conservation laws we cannot show uniqueness of entropy measure-valued solutions. The main obstacle to formulate analogous result is that, in most cases, we are equipped with only one entropy-entropy flux pair, contrary to a rich family of entropies available in the scalar case. Even more, the corresponding relative entropy inequality lacks appropriate symmetry.

For most systems of mathematical physics it is well known that even weak solutions may fail to be unique. Only some conditional uniqueness can be claimed. This conditional uniqueness property had been studied for many systems of fluid mechanics. First, in their classical papers, Prodi [30] and Serrin [32] had shown that a weak solution to the incompressible Navier-Stokes equations is unique and coincides with the strong solution, provided such a strong solution is known to exist. For conservation laws a conditional uniqueness of weak solutions was established firstly by Dafermos in [9]. This is somehow an extension of the result on uniqueness of strong solutions (cf. [26]), asserting that they are unique not only in the class of strong solutions, but also in the wider class of entropy weak solutions. This property became known as weak-strong uniqueness.

It was discovered, rather surprising, that the class of entropy weak solutions in the above can be widened to the class of measure-valued solutions which satisfy some kind of entropy inequality. One can ask - Is it to the benefit? After all, measure-valued solutions seem a very weak notion and, admittedly, carry hardly any information about the physical problem. Nevertheless, measure-valued solutions, intimately related to Young measures, prove to be a powerful tool in the analysis of nonlinear PDEs.

Numerous results on mv-strong uniqueness for various systems have already been described at the beginning of the introduction, as well as some of the results which concern a general hyperbolic case.

Section snippets

Applications

In this section we provide a short list of applications of the general theory presented above. The first impression is that the general framework cannot cover e.g. incompressible Euler system. In Section 5 we show that a slight refinement allows to include not only incompressible Euler system, but also incompressible magnetohydrodynamics.

Relations between concentration measures

Our aim in this section is to prove Proposition 1.8. We provide two proofs, the first one works with the Radon-Nikodym derivatives of measures, whereas the second one relates our concept of dissipative measure valued solutions to the framework of generalized Young measures and is in its core based on the slicing lemma for products of measures. In particular, in the second proof we have to assume that the modified recession functions (for definition see below) exist for nonlinear functions

Derivation of the relative entropy inequality

We derive the relative entropy inequality. We choose in (1.21) a test function φ=ζ(t)G(U(t,x)) with ζCc([0,T)). As U is a strong solution, thus (1.1) is satisfied by U, we multiply it with the same test function and integrate, finally to subtract it from (1.21) to getQζ(t)G(U)(νt,x,A(λ)A(U))dxdt+Qζ(t)G(U)mA(dxdt)+Qζ(t)αG(U)(νt,x,Fα(λ)Fα(U))dxdt+Qζ(t)αG(U)mFα(dxdt)+Qζ(t)tG(U)(νt,x,A(λ)A(U))dxdt+Qζ(t)tG(U)mA(dxdt)+Tdζ(0)G(U(0))(ν0,x,A(λ)A(U(0)))dx+Tdζ(0)G(U(0))m

Extension

As one may easily observe unfortunately this general framework will not cover systems of conservation laws, which may fail to be hyperbolic, typically incompressible inviscid systems. In the current approach we present a simple extension of the presented framework to cover the case of incompressible fluids, in case of which the assumption that ∇A is a nonsingular matrix is not satisfied. For this reason we distinguish from the flux the part L (Lagrange multiplier) which is perpendicular to the

Declaration of Competing Interest

There is no competing interest.

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    O.K. acknowledges the support of the Neuron Impuls Junior project “Mathematical analysis of hyperbolic conservation laws” in the general framework of RVO: 67985840. This work was partially supported by the Simons Foundation grant 346300 and the Polish Government MNiSW 2015-2019 matching fund. A.Ś.-G. acknowledges the support of the National Science Centre (Poland), DEC-2012/05/E/ST1/02218. The research was partially supported by the Warsaw Center of Mathematics and Computer Science. P.G. received support from the National Science Centre (Poland), 2015/18/M/ST1/00075.

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