Stability Of Weak Solutions Research Articles

Solutions of evolutionary $${\varvec{p(x)}}$$-Laplacian equation based on the weighted variable exponent space

The paper studies the equation $$\begin{aligned} u_t= {\text {div}} (a(x)\left| \nabla u \right| ^{p(x) - 2}\nabla u) +\sum _{i=1}^N\frac{\partial b_i(u)}{\partial x_i},\ \ (x,t) \in \Omega \times (0,T), \end{aligned}$$ with the boundary degeneracy due to $$a(x)\mid _{x\in \partial \Omega }=0$$ , and $$\Omega \subset \mathbb {R}^{N}$$ , where N is a positive integer. By the theory of the weighted variable exponent Sobolev spaces, the well posedness of weak solutions of this equation is discussed. The novelty of our results lies in the fact that under certain conditions, if a(x) satisfies $$\int \nolimits _{\Omega }a^{-\frac{1}{p(x)-1}}\mathrm{d}x<\infty $$ , the global stability of weak solutions can be established without any boundary value condition. While $$\int \nolimits _{\Omega }a^{-\frac{1}{p(x)-1}}\mathrm{d}x=\infty $$ , the local stability of weak solutions can be obtained without any boundary value condition.

The stability of the anisotropic parabolic equation with the variable exponent

Consider the equation \t\t\tut=div(a(x)|∇u|p(x)−2∇u→)=∑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}$${u_{t}}= \\operatorname{div} \\bigl(\\overrightarrow{a(x){ \\vert{ \\nabla u} \\vert^{p(x) - 2}}\\nabla u} \\bigr)=\\sum_{i=1}^{N} \\bigl(a_{i}(x) \\vert u_{x_{i}} \\vert^{p_{i}(x)-2}u_{x_{i}} \\bigr)_{x_{i}}, $$\\end{document} with a_{i}(x), p_{i}(x)in C^{1}(overline{Omega}), p_{i}(x)>1, a_{i}(x)geq 0. Basing on the weighted variable exponent Sobolev space, a new kind of weak solutions of the equation is introduced. Whether the usual Dirichlet homogeneous boundary value condition can be imposed depends on whether a_{i}(x) is degenerate on the boundary or not. If some of { a_{i}(x)} are degenerate on the boundary, a partial boundary value condition is imposed. If every a_{i}(x) is degenerate on the boundary, by the new definition of a weak solution, the stability of weak solutions can be proved without any boundary value condition.

Global Convergence of a Sticky Particle Method for the Modified Camassa--Holm Equation

In this paper, we prove convergence of a sticky particle method for the modified Camassa--Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions $u$ with regularity: $u$ and $u_x$ are space-time BV functions. The total variation of $m(\cdot ,t)=u(\cdot, t)-u_{xx}(\cdot,t)$ is bounded by the total variation of the initial data $m_0$. We also obtain $W^{1,1}(\mathbb{R})$-stability of weak solutions when solutions are in $ L^\infty(0,\infty;W^{2,1}(\mathbb{R}))$. (Notice that peakon weak solutions are not in $W^{2,1}(\mathbb{R})$.) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.

On the compressible Navier-Stokes-Korteweg equations

In this paper, we consider compressible Navier-Stokes-Korteweg (N-S-K) equations with more general pressure laws, that is the pressure $P$ is non-monotone. We prove the stability of weak solutions in the periodic domain $\Omega=\mathbb{T}^{N}$, when $N = 2,3$. Utilizing an interesting Sobolev inequality to tackle the complicated Korteweg term, we obtain the global existence of weak solutions in one dimensional case. Moreover, when the initial data is compactly supported in the whole space $\mathbb{R}$, we prove the compressible N-S-K equations will blow-up in finite time.

The global stabilization of the Camassa-Holm equation with a distributed feedback control

The global stabilization of the Camassa-Holm equation with a distributed feedback control of the form $-(\lambda u-\beta u_{xx}-\lambda[u])$ is investigated. The existence and uniqueness of global strong solutions and global weak solutions to the closed loop control system are obtained. The exponential asymptotical stabilization of weak solutions to the problem is established. Namely, the weak solutions to the problem exponentially uniformly decay to a constant. The main novelty in this paper is that the effects of the coefficients λ and β on the global existence and exponential asymptotical stabilization of solutions are given.

A Semilinear Wave Equation with a Boundary Condition of Many-Point Type: Global Existence and Stability of Weak Solutions

This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.

Flow-plate interactions: Well-posedness and long-time behavior

We consider flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are treated with Neumann type flow conditions, and a novel treatment of the so called Kutta-Joukowsky flow conditions are given in the subsonic case. The goal of the paper is threefold: (i) to provide an accurate review of recent resultson existence, uniqueness, and stability of weak solutions, (ii) to present a construction of finite dimensional, attracting setscorresponding to the structural dynamics and discuss convergence of trajectories, and (iii) to state several open questions associated with the topic. This second task is based on a decoupling technique which reduces the analysis of the full flow-structure system to aPDE system with delay.

Stability of weak solutions for the compressible Navier-Stokes-Poisson equations

In this paper, we consider the isentropic compressible Navier-Stokes-Poisson equations arising from transport of charged particles or motion of gaseous stars in astrophysics. We are interested in the case that the viscosity coefficients depend on the density and shall degenerate in the appearance of (density) vacuum, and show the L1-stability of weak solutions for arbitrarily large data on spatial multi-dimensional bounded or periodic domain or whole space.

A new approach to the stabilization of a fluid–structure interaction model

This article is a continuation of our work on a linear fluid–structure interaction model [Grobbelaar-Van Dalsen, On a fluid–structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech. 10(3) (2008), pp. 388–401; Grobbelaar-Van Dalsen, Strong stability for a fluid––structure model, Math. Methods Appl. Sci., 32(2009) pp. 1452–1466]. The model describes the interaction between a 3-D incompressible fluid and a 2-D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with the inclusion of the shear stress that the fluid exerts on the plate. A dissipative damping mechanism of Kelvin–Voigt type is applied to the interior of the plate. While our earlier work shows that weak solutions in a space of finite energy are strongly asymptotically stable under no-slip transmission conditions at the interface with uniform exponential stability only attainable under an additional domination condition, the present research is directed at achieving uniform exponential stability of weak solutions without imposing the domination condition. Using energy methods we establish uniform exponential decay under a modified transmission condition at the interface. This condition entails that the fluid velocity at the interface is coupled to a linear combination of the plate velocity and displacement.

Existence, uniqueness and stability of weak solutions of parabolic systems with discontinuous nonlinearities

This paper is devoted to the study of the initial value problem for parabolic systems with discontinuous nonlinearities from the viewpoint of the existence, uniqueness and stability of weak solutions. Also the relationship with the solutions of the corresponding differential inclusions is studied.

A new nonlinear functional for general scalar conservation laws

The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427–1442] for the L 1 stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, we introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [J.-L. Hua, Z.-H. Jiang, T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, preprint] for general scalar conservation laws. This functional improves the one given in [H.-X. Liu, T. Yang, A nonlinear functional for general scalar hyperbolic conservation laws, J. Differential Equations 235 (2) (2007) 658–667] and it can be viewed as a better attempt for the generalized entropy functional for general equations.

Asymptotic stability for the Navier–Stokes equations in L n

Considering the Navier–Stokes equations in $$\Omega \subset {\mathbb{R}}^n$$ , we prove the asymptotic stability for weak solutions in the marginal class u ∈ C B (0, ∞; L n ) with arbitrary initial and external perturbations.

Asymptotic Stability for the 3D Navier–Stokes Equations

ABSTRACT In this paper we consider the 3D Navier–Stokes equations in Ω ⊂ ℝ3, not necessarily bounded. We prove the asymptotic stability for weak solutions in the class ∇u ∈ L α(0, ∞;L γ(Ω)) for with arbitrary initial and external perturbations.

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equations $$ - {\text{div}}\;A_{p} {\left( {x,u} \right)} + a_{p} {\left( {x,u} \right)} = f{\left( x \right)} $$ in space W (θ, p)(Ω), which is bigger than W (1, p)(Ω). Next, using revise reverse Holder inequality we prove that if ω c is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈ U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.