Stability Of Weak Solutions Research Articles

On a degenerate parabolic equation from double phase convection

The initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let a(x) and b(x) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that a(x)+b(x)>0, xin overline{Omega } and the boundary value condition should be imposed. In this paper, the condition a(x)+b(x)>0, xin overline{Omega } is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and u_{t}in L^{2}(Q_{T}) is shown. The stability of weak solutions is studied according to the different integrable conditions of a(x) and b(x). To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by a(x)b(x)|_{xin partial Omega }=0 is found for the first time.

Stability of non-Newtonian fluid and electrorheological fluid mixed-type equation

In this paper, we consider a non-Newtonian fluid and electrorheological fluid mixed-type equation where p>1, and . The existence of weak solution is investigated by means of the parabolically regularized method. Due to , the stability of weak solution is established by choosing a suitable test function and the local stability is also discussed without any boundary value condition.

On a doubly degenerate parabolic equation with a nonlinear damping term

Consider a double degenerate parabolic equation arising from the electrorheological fluids theory and many other diffusion problems. Let v_{varepsilon } be the viscous solution of the equation. By showing that |nabla v_{varepsilon }|in L^{infty }(0,T; L_{mathrm{loc}}^{p(x)}(Omega )) and nabla v_{varepsilon }rightarrow nabla v almost everywhere, the existence of weak solutions is proved by the viscous solution method. By imposing some restriction on the nonlinear damping terms, the stability of weak solutions is established. The innovation lies in that the homogeneous boundary value condition is substituted by the condition a(x)| _{xin partial Omega }=0, where a(x) is the diffusion coefficient. The difficulties come from the nonlinearity of vert {nabla v} vert ^{p(x)-2} as well as the nonlinearity of |v|^{alpha (x)}.

Existence and stability of the doubly nonlinear anisotropic parabolic equation

In this paper, we are concerned with a doubly nonlinear anisotropic parabolic equation, in which the diffusion coefficient and the variable exponent depend on the time variable t. Under certain conditions, the existence of weak solution is proved by applying the parabolically regularized method. Based on a partial boundary value condition, the stability of weak solution is also investigated.

On Solutions of a Parabolic Equation with Nonstandard Growth Condition

A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficient bx,t on the time variable t, and the partial boundary value condition based on a submanifold of ∂Ω×0,T. How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.

The nonnegative weak solution of a degenerate parabolic equation with variable exponent growth order

A degenerate parabolic equation of the form (|v|β−1v)t=div(b(x,t)|∇v|p(x,t)−2∇v)+∇g→⋅∇γ→(v)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\bigl( \\vert v \\vert ^{\\beta-1}v\\bigr)_{t}= \\operatorname{div} \\bigl(b(x,t) \\vert \\nabla v \\vert ^{p(x,t)-2}\\nabla v \\bigr)+\\nabla\\vec{g}\\cdot\\nabla\\vec{\\gamma}(v) $$\\end{document} is considered, where vec{g}={g^{i}(x,t)}, vec{gamma}(v)={ gamma_{i}(v)}. If the diffusion coefficient b(x,t)geq0 is degenerate on the boundary, by adding some restrictions on b(x,t) and g⃗, the existence and uniqueness of weak solutions are proved. Based on the uniqueness, the stability of weak solutions can be proved without any boundary condition.

On a Partial Boundary Value Condition of a Porous Medium Equation with Exponent Variable

The initial-boundary value problem of a porous medium equation with a variable exponent is considered. Both the diffusion coefficient ax,t and the variable exponent px,t depend on the time variable t, and this makes the partial boundary value condition not be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition matching up with the equation is based on a submanifold of ∂Ω×0,T. By this innovation, the stability of weak solutions is proved.

Стабилизация решений уравнения теории пластин с переменным по времени запаздыванием и слабой вязкоупругостью в пространстве Rn

This article considers a dynamical system with delay described by a differential equation with partial derivatives of hyperbolic type and delay with respect to a time variable. We establish in Theorem 3.1 the k(t)-stability of weak solution under suitable initial conditions in $\mathbb{R}^n, n>4$ by introducing appropriate Lyapunov functions.

The weak solutions of a doubly nonlinear parabolic equation related to the p(x)-Laplacian

A nonlinear degenerate parabolic equation related to the p(x)-Laplacianut=div(b(x)|∇a(u)|p(x)−2∇a(u))+∑i=1N∂bi(u)∂xi+c(x,t)−b0a(u)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {u_{t}}= \\operatorname{div} \\bigl({b(x)} { \\bigl\\vert {\\nabla a(u)} \\bigr\\vert ^{p(x) - 2}}\\nabla a(u) \\bigr)+\\sum _{i=1}^{N}\\frac{\\partial b_{i}(u)}{ \\partial x_{i}}+c(x,t) -b_{0}a(u) $$\\end{document} is considered in this paper, where b(x)|_{xin varOmega }>0, b(x)|_{x in partial varOmega }=0, a(s)geq 0 is a strictly increasing function with a(0)=0, c(x,t)geq 0 and b_{0}>0. If int _{varOmega }b(x)^{-frac{1}{p ^{-}-1}},dxleq c and vert sum_{i=1}^{N}b_{i}'(s) vert leq c a'(s), then the solutions of the initial-boundary value problem is well-posedness. When int _{varOmega }b(x)^{-(p(x)-1)},dx<infty , without the boundary value condition, the stability of weak solutions can be proved.

The partial boundary value condition for a polytropic filtration equation with variable exponents

A polytropic filtration with variable exponents is considered, where p>1, , , , . The existence of the weak solutions is proved. By showing that , the stability of weak solutions is proved when a partial boundary value condition is assumed and also under assumptions which do not involve the boundary value condition.

On the stability of a non-Newtonian polytropic filtration equation

The non-Newtonian polytropic filtration equation with a convection term \t\t\tvt=div(a(x)|v|α|∇v|p−2∇v)+∑i=1N∂ai(v,x,t)∂xi\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ v_{t}= \\operatorname{div} \\bigl(a(x) \\vert v \\vert ^{\\alpha }{ \\vert {\\nabla v} \\vert ^{p-2}}\\nabla v \\bigr)+ \\sum_{i=1}^{N}\\frac{\\partial a_{i}(v,x,t)}{\\partial x_{i}} $$\\end{document} is considered, where p>1, alpha >0, a(x)geq 0 with a(x) | _{xin partial varOmega }=0. This kind of equation is degenerate on the boundary, the usual boundary value condition may be overdetermined. Some conditions depending on a(x) and a_{i}(cdot ,x,t), which can take place of the boundary value condition, are found. Moreover, how the nonlinear term |v|^{alpha } affects the stability of weak solutions is revealed.

On the Solutions of a Porous Medium Equation with Exponent Variable

The paper studies the initial-boundary value problem of a porous medium equation with exponent variable. How to deal with nonlinear term with the exponent variable is the main dedication of this paper. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of weak solutions is studied. In some special cases, the stability of weak solutions can be proved without any boundary value condition.

Evolutionary p(x)-Laplacian Equation with a Convection Term

The paper studies an evolutionary p(x)-Laplacian equation with a convection term $$u_{t}=\operatorname{div}\left(\rho^{\alpha}|\nabla u|^{p(x)-2} \nabla u\right)+\sum_{i=1}^{N} \frac{\partial b_{i}(u)}{\partial x_{i}},$$ where ρ(x)= dist(x, ∂Ω), essinf p(x) = p− > 2. To assure the well-posedness of the solutions, the paper shows only a part of the boundary, ∑p ⊂ ∂Ω, on which we can impose the boundary value. ∑p is determined by the convection term, in particular, when \(1<\alpha<\frac{p^{-}-2}{2}\), ∑p = {x ∈ ∂Ω: bi′(0)ni(x) < 0}. So, there is an essential difference between the equation and the usual evolutionary p-Laplacian equation. At last, the existence and the stability of weak solutions are proved under the additional conditions \(\alpha<\frac{p^{-}-2}{2}\) and ∑p = ∂Ω.

Solutions of evolutionary equation based on the anisotropic variable exponent Sobolev space

In this paper, we are concerned with the equation $$\begin{aligned} u_t= \sum _{i=1}^N\frac{\partial }{\partial x_i} \left( a_i(x)\left| u_{x_i} \right| ^{p_i(x) - 2} u_{x_i} \right) +\sum _{i=1}^N\frac{\partial b_i(u)}{\partial x_i},\ \ (x,t) \in \Omega \times (0,T), \end{aligned}$$ where $$a_i(x)|_{x\in \partial \Omega }=0$$ and $$a_i(x)|_{x\in \Omega }>0$$ . By the theory of anisotropic variable exponent Sobolev spaces, we study the well-posedness of weak solutions of this equation. Since $$a_i(x)$$ is degenerate on the boundary, the stability of weak solutions may be established without any boundary value condition. The main feature which distinguishes this paper from other related works lies in the fact that we propose a novel analytical method to deal with the stability of weak solutions.

The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise

In this article, we study the stability of weak solutions to a stochastic version of a coupled Cahn-Hilliard-Navier-Stokes model with multiplicative noise. The model consists of the Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard model for the order (phase) parameter. We prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.

Global Weak Solutions for the Cauchy Problem to One-Dimensional Heat-Conductive MHD Equations of Viscous Non-resistive Gas

Magnetohydrodynamics is concerned with the mutual interactions between moving, electrically conductive fluids and the magnetic field. It is widely applied in astrophysics, thermonuclear reactions and industry. The Cauchy problem to one-dimensional heat-conductive magnetohydrodynamic equations of viscous non-resistive gas is considered under the framework of Lagrangian coordinates. Based on the crucial upper and lower bounds of the density, we first obtain global well-posedness of strong solutions with regular initial data. Then existence of global weak solutions is established via the compactness analysis and the method of weak convergence. Stability of weak solutions is also verified by making full use of the specific mathematical structure of the equations. All results are obtained without any restriction to the size of the initial data.

Global stability for the fractional Navier‐Stokes equations in the Fourier‐Herz space

We consider global stability for the fractional incompressible Navier‐Stokes equations in a 3‐D critical Fourier‐Herz space. By introducing a weighted norm space and using Fourier localization technique, the stability of mild solutions with small initial perturbation is established. With the Friedrichs method, the stability of weak solutions is proved under arbitrary large initial perturbation.

On stability with respect to boundary conditions for anisotropic parabolic equations with variable exponents

The anisotropic parabolic equations with variable exponents are considered. If some of diffusion coefficients {b_{i}(x)} are degenerate on the boundary, the others are always positive, then how to impose a suitable boundary value condition is researched. The existence of weak solutions is proved by the parabolically regularized method. The stability of weak solutions, based on the partial boundary value condition, is established by choosing a suitable test function.

Stability of weak solutions to climate dynamics model with effects of topography and non-constant external force

In this paper, a climate dynamics model with the effects of topography and a non-constant external force, which consists of the Navier-Stokes equations and a temperature equation arising from the evolution process of the atmosphere, was considered. Under certain assumptions imposed on the initial data and by using some delicate estimates and compactness arguments, we proved the L 1-stability of weak solutions to the atmospheric equations.

The well-posedness of an anisotropic parabolic equation based on the partial boundary value condition

Consider the anisotropic parabolic equation with the variable exponent ut=∑i=1N(ai(x)|uxi|pi(x)−2uxi)xi,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {u_{t}}=\\sum_{i=1}^{N} \\bigl(a_{i}(x)|u_{x_{i}}|^{p_{i}(x)-2}u_{x_{i}} \\bigr)_{x _{i}}, $$\\end{document} with a_{i}(x), p_{i}(x)in C^{1}(overline{Omega}), p_{i}(x)>1, a_{i}(x)geq0. If some of {a_{i}(x)} are degenerate on the boundary, a partial boundary value condition is imposed, the stability of weak solutions can be proved based on the partial boundary value condition.