This paper deals with a Cauchy problem of the full compressible Hall-magnetohydrodynamic flows. We establish the existence and uniqueness of global solution, provided that the initial energy is suitably small but the initial temperature allows large oscillations. In addition, the large time behavior of the global solution is obtained.

We study positive solutions to the steady state reaction diffusion equation of the form: − Δ u=λ f(u); Ω ∂ u∂ η +λ u=0; ∂ Ω $$\begin{array}{} \displaystyle \left\lbrace \begin{matrix} -{\it\Delta} u =\lambda f(u);~ {\it\Omega} \\ \frac{\partial u}{\partial \eta}+ \sqrt{\lambda} u=0;~\partial {\it\Omega}\end{matrix} \right. \end{array}$$ where λ > 0 is a positive parameter, Ω is a bounded domain in

We study the wave inequality with a Hardy potential ∂ ttu− Δ u+λ |x|2u≥ |u|pin (0,∞ )× Ω , $$\begin{array}{} \displaystyle \partial_{tt}u-{\it\Delta} u+\frac{\lambda}{|x|^2}u\geq |u|^p\quad \mbox{in } (0,\infty)\times {\it\Omega}, \end{array}$$ where Ω is the exterior of the unit ball in ℝ N , N ≥ 2, p > 1, and λ ≥ − N− 222 $\begin{array}{} \displaystyle \left(\frac{N-2}{2}\right)^2 \end{array}$, under

We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in ℝ N without growth restrictions at infinity. A comparison result in terms of the half-space

In this paper, we consider a class of semilinear elliptic equation with critical exponent and -1 growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: (− Δ )su− γ u|x|2s=K(x)|u|2s∗ (t)− 2u|x|t+f(x)inRN,u∈ H˙ s(RN), $$\begin{array}{} \displaystyle \begin{cases} (-{\it\Delta})^s u -\gamma\dfrac{u}{|x|^{2s}}=K(x)\dfrac{|u|^{2^*_s(t)-2}u}{|x|^t}+f(x) \quad\mbox{in}\quad\mathbb R^N,\\ \qquad\qquad\qquad\quad u\in

In this paper, we prove the existence of multiple solutions for the following sixth-order p ( x )-Kirchhoff-type problem − M∫ Ω 1p(x)|∇ Δ u|p(x)dxΔ p(x)3u=λ f(x)|u|q(x)− 2u+g(x)|u|r(x)− 2u+h(x)inΩ ,u=Δ u=Δ 2u=0,on∂ Ω , $$\begin{array}{} \displaystyle \begin{cases} -M\left( \int\limits_{\it\Omega} \frac{1}{p(x)}|\nabla {\it\Delta} u|^{p(x)}dx\right){\it\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u

We consider a nonlinear Robin problem driven by the ( p , q )-Laplacian and a parametric reaction exhibiting the competition of a concave term and of a resonant perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on ℝ̊ + = (0, +∞). Also, we determine the continuity properties of the solution multifunction.

Various strategies are available to construct iteratively a common fixed point of nonexpansive operators by activating only a block of operators at each iteration. In the more challenging class of composite fixed point problems involving operators that do not share common fixed points, current methods require the activation of all the operators at each iteration, and the question of maintaining convergence

In this article, our aim is to establish a generalized version of Lions-type theorem for the p -Laplacian. As an application of this theorem, we consider the existence of ground state solution for the quasilinear elliptic equation with the critical growth.

The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary. The case of the boundary value with a nonzero flow rate is considered. In this case there is a source/sink in O and the solution necessary has infinite energy integral. In the first part of the paper the formal asymptotic expansion of

The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary. We consider the case where the boundary value has a nonzero flux over the boundary. In this case there is a source/sink in O and the solution necessary has infinite energy integral. In the first part of the paper the formal asymptotic

We consider the following p -harmonic problem Δ (|Δ u|p− 2Δ u)+m|u|p− 2u=f(x,u),x∈ RN,u∈ W2,p(RN), $$\begin{array}{} \displaystyle \left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\ u \in W^{2,p}({\mathbb R}^N), \end{array} \right. \end{array}$$ where m > 0 is a constant, N > 2 p ≥ 4 and limt→ ∞ f(x,t)|t|p−

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space ( X , ∣·∣, ⪯), T : C → C a monotone 1-Lipschitz mapping and x ⪯ T ( x ), then the sequence of averages 1n∑i=0n−1Ti(x) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T . As a consequence

We prove that a strong solution u to the Navier-Stokes equations on (0, T ) can be extended if either u ∈ L θ (0, T ; U˙ ∞ ,1/θ ,∞ − α $\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$) for 2/ θ + α = 1, 0 < α < 1 or u ∈ L 2 (0, T ; V˙ ∞ ,∞ ,20 $\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$), where U˙ p,β ,σ s $\begin{array}{} \displaystyle

Existence of fixed point of a Frobenius-Perron type operator P : L 1 ⟶ L 1 generated by a family { φ y } y ∈ Y of nonsingular Markov maps defined on a σ -finite measure space ( I , Σ , m ) is studied. Two fairly general conditions are established and it is proved that they imply for any g ∈ G = { f ∈ L 1 : f ≥ 0, and ∥ f ∥ = 1}, the convergence (in the norm of L 1 ) of the sequence {Pjg}j=1∞ $\begin{array}{}

This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented

This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen

In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter

In the present paper, we consider the nonlinear periodic systems involving variable exponent driven by p ( t )-Laplacian with a locally Lipschitz nonlinearity. Our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue-Sobolev space. Applying the non-smooth critical point theory, we establish some new existence results.

It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class

In this paper we introduce and study the commutators of the local multilinear fractional maximal operators and a vector-valued function b⃗ = ( b 1 , …, b m ). Under the condition that each b i belongs to the first order Sobolev spaces, the bounds for the above commutators are established on the first order Sobolev spaces.

We obtain a critical imbedding and then, concentration-compactness principles for fractional Sobolev spaces with variable exponents. As an application of these results, we obtain the existence of many solutions for a class of critical nonlocal problems with variable exponents, which is even new for constant exponent case.

The aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.

In this paper, we study the singularly perturbed fractional Choquard equation ε 2s(− Δ )su+V(x)u=ε μ − 3(∫R3|u(y)|2μ ,s∗ +F(u(y))|x− y|μ dy)(|u|2μ ,s∗ − 2u+12μ ,s∗ f(u))inR3, $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3

The Keller-Segel-Stokes system (*) n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n ∇ c ) + ρ n − μ n α , c t + u ⋅ ∇ c = Δ c − c + n , u t = Δ u + ∇ P − n ∇ Λ , ∇ ⋅ u = 0 , $$\begin{eqnarray*} \left\{ \begin{array}{lcll} n_t + u\cdot\nabla n &=& \it\Delta n - \nabla \cdot (n\nabla c) + \rho n - \mu n^\alpha, \\[1mm] c_t + u\cdot\nabla c &=& \it\Delta c-c+n, \\[1mm] u_t &=& \it\Delta u + \nabla P - n\nabla \it\Lambda

In the present paper, we introduce a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term. Denoting by 𝓢 the solution set of the obstacle problem and by 𝓢 n the solution sets of approximating problems, we prove the following convergence relation ∅ ≠ w-lim supn→ ∞ Sn=s-lim supn→ ∞ Sn⊂ S, $$\begin{array}{}

This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation (ϵ i∇ − A(x))2u+V(x)u+ϵ − 2(|x|− 1∗ |u|2)u=f(|u|2)u+|u|4uin R3,u∈ H1(R3,C), $$\begin{array}{} \displaystyle \left\{ \begin{aligned} &\Big(\frac{\epsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u=f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3,\\ &u\in

We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type ( Φ ( k ( t ) x ′ ( t ) ) ) ′ + f ( t , G x ( t ) ) ρ ( t , x ′ ( t ) ) = 0 , $$\big({\it \Phi}(k(t)\,x'(t))\big)' + f(t,{{\mathcal{G}}}_x(t))\,\rho(t, x'(t)) = 0,$$ on a compact interval [ a , b ]. These equations are quite general due to the presence of a strictly increasing

In this article, we consider a class of Kirchhoff equations with critical Hardy-Sobolev exponent and indefinite nonlinearity, which has not been studied in the literature. We prove very nicely that this equation has at least two solutions in ℝ 3 . And some known results in the literature are improved.

In this paper, we study following Kirchhoff type equation: − a+b∫ Ω |∇ u|2dxΔ u=f(u)+h in Ω ,u=0 on ∂ Ω . $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ We consider first the case that Ω ⊂ ℝ 3 is a bounded domain. Existence

The transonic channel flow problem is one of the most important problems in mathematical fluid dynamics. The structure of solutions near the sonic curve is a key part of the whole transonic flow problem. This paper constructs a local classical hyperbolic solution for the 3-D axisymmetric steady compressible full Euler equations with boundary data given on the degenerate hyperbolic curve. By introducing

In this paper, we consider the following nonlinear Schrödinger equation: iut+Δ gu+ia(x)u− |u|p− 1u=0(x,t)∈ M× (0,+∞ ),u(x,0)=u0(x)x∈ M, $$\begin{array}{} \displaystyle \begin{cases}iu_t+{\it\Delta}_g u+ia(x)u-|u|^{p-1}u=0\qquad (x,t)\in \mathcal{M} \times (0,+\infty), \cr u(x,0)=u_0(x)\qquad x\in \mathcal{M},\end{cases} \end{array}$$(0.1) where (𝓜, g ) is a smooth complete compact Riemannian manifold

In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth: − Δ u+λ 1u=μ 1u3+β uv2+2qpyu2qp− 1v2inΩ ,− Δ v+λ 2v=μ 2v3+β u2v+2yu2qpvinΩ ,u,v> -->0inΩ ,u,v=0on∂ Ω , $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v=\mu_2

We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity v , coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor 𝔹. By a proper choice of the constitutive

In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems

In this paper, the equations and systems of Monge-Ampère with parameters are considered. We first show the uniqueness of of nontrivial radial convex solution of Monge-Ampère equations by using sharp estimates. Then we analyze the existence and nonexistence of nontrivial radial convex solutions to Monge-Ampère systems, which includes some new ingredients in the arguments. Furthermore, the asymptotic

When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f -plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained

Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + ∑ i=2m $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim V λ i pairs of nontrivial solutions if a parameter involved in the equation is large enough, where V λ i denotes the eigenspace associated to the i-th eigenvalue λ i of laplacian operator with homogeneous

Our purpose of this paper is to consider Liouville property for the fractional Lane-Emden equation (− Δ )α u=upinΩ ,u=0inRN∖ Ω , $$\begin{array}{} \displaystyle (-{\it\Delta})^\alpha u = u^p\quad {\rm in}\quad {\it\Omega},\qquad u = 0\quad {\rm in}\quad \mathbb{R}^N\setminus {\it\Omega}, \end{array}$$ where α ∈ (0, 1), N ≥ 1, p > 0 and Ω ⊂ ℝ N –1 × [0, +∞) is an unbounded domain satisfying that Ω t

In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, − a+b∫ RN|∇ u|2dxΔ u=α k(x)|u|q− 2u+β ∫ RN|u(y)|2μ ∗ |x− y|μ dy|u|2μ ∗ − 2u,x∈ RN, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\

The paper deals with a posteriori analysis of the spectral element discretization of a non linear heat equation. The discretization is based on Euler’s backward scheme in time and spectral discretization in space. Residual error indicators related to the discretization in time and in space are defined. We prove that those indicators are upper and lower bounded by the error estimation.

We consider nonlinear sub-elliptic systems with VMO-coefficients for the case 1 < p < 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique of 𝓐-harmonic approximation introduced by Duzaar-Grotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg

This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive

In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense.

Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when

This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term − Δ u+b→ (x)⋅ ∇ u+V(x)u=Hv(x,u,v)inRN,− Δ v− b→ (x)⋅ ∇ v+V(x)v=Hu(x,u,v)inRN. $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u

In this paper, we study blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation i∂ tψ − (− Δ )sψ +(Iα ∗ |ψ |p)|ψ |p− 2ψ =0. $$\begin{array}{} \displaystyle i\partial_t\psi- (-{\it\Delta})^s \psi+(I_\alpha \ast |\psi|^{p})|\psi|^{p-2}\psi=0. \end{array}$$ By using localized virial estimates, we firstly establish general blow-up criteria for non-radial

In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial

In this paper, we study the following nonlinear Hamiltonian elliptic system with gradient term − ϵ 2Δ ψ +ϵ b→ ⋅ ∇ ψ +ψ +V(x)φ =f(|η |)φ in RN,− ϵ 2Δ φ − ϵ b→ ⋅ ∇ φ +φ +V(x)ψ =f(|η |)ψ in RN, $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\epsilon^{2}{\it\Delta} \psi +\epsilon \vec{b}\cdot \nabla \psi +\psi+V(x)\varphi=f(|\eta|)\varphi~~\hbox{in}~\mathbb{R}^{N},\\ -\epsilon^{2}{\it\Delta}

We consider a nonlinear parametric Dirichlet problem driven by the (p, q)-Laplacian (double phase problem) with a reaction exhibiting the competing effects of three different terms. A parametric one consisting of the sum of a singular term and of a drift term (convection) and of a nonparametric perturbation which is resonant. Using the frozen variable method and eventually a fixed point argument based

In this paper, we prove a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients to the zero-Dirichlet problem of general nonlinear elliptic equations with the nonlinearities satisfying Orlicz growth. It is mainly assumed that the variable exponents p ( x ) satisfy the log-Hölder continuity, while the nonlinearity and underlying domain ( A , Ω

The aim of the paper is to investigate the solvability of an infinite system of nonlinear integral equations on the real half-axis. The considerations will be located in the space of function sequences which are bounded at every point of the half-axis. The main tool used in the investigations is the technique associated with measures of noncompactness in the space of functions defined, continuous and

We refine a recent result on the structure of measures satisfying a linear partial differential equation 𝓐 μ = σ , μ , σ are Radon measures, considering the measure μ ( x ) = g ( x ) d x + μ us ( x̃ )( μ s ( x̄ ) + d x̄ ) where x = ( x̃ , x̄ ) ∈ ℝ k × ℝ d − k , μ us is a uniformly singular measure in x̃ 0 and the measure μ s is a singular measure. We proved that for μ us -a.e. x̃ 0 the range of the

Article Editorial to Volume 10 of ANA was published on January 1, 2021 in the journal Advances in Nonlinear Analysis (volume 10, issue 1).

In this paper, we study the following nonlinear magnetic Schrödinger-Poisson type equation (ε i∇ − A(x))2u+V(x)u+ϵ − 2(|x|− 1∗ |u|2)u=f(|u|2)uin R3,u∈ H1(R3,C), $$\begin{array}{} \displaystyle \left\{\!\begin{aligned}&\Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u = f(|u|^{2})u\quad\hbox{in }\mathbb{R}^3,\\&u\in H^{1}(\mathbb{R}^{3}, \mathbb{C})

In this paper, we study the existence of peaked traveling wave solution of the generalized μ -Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ -version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

We provide new sufficient conditions for the existence of T -periodic solutions for the ϕ -laplacian pendulum equation ( ϕ ( x ′))′ + k x ′ + a sin x = e ( t ), where e ∈ C͠ T . Our main tool is a continuation theorem due to Capietto, Mawhin and Zanolin and we improve or complement previous results in the literature obtained in the framework of the classical, the relativistic and the curvature pendulum

In this note, we study isolated singular positive solutions of Kirchhoff equation Mθ (u)(− Δ )u=upinΩ ∖ {0},u=0on∂ Ω , $$\begin{array}{} \displaystyle M_\theta(u)(-{\it\Delta}) u =u^p \quad{\rm in}\quad {\it\Omega}\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial {\it\Omega}, \end{array}$$ where p > 1, θ ∈ ℝ, M θ ( u ) = θ + ∫ Ω |∇ u | dx , Ω is a bounded smooth domain containing the origin in

We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions