Asymptotic Behavior Of Weak Solutions Research Articles

Weak solutions to stochastic 3D Navier–Stokes- α model of turbulence: α-Asymptotic behavior

We study the asymptotic behavior of weak solutions to the stochastic 3D Navier–Stokes- α model as α approaches zero. The main result provides a new construction of the weak solutions of stochastic 3D Navier–Stokes equations as approximations by sequences of solutions of the stochastic 3D Navier–Stokes- α model.

Weak solutions of a parabolic system with a discontinuous nonlinearity

This paper is devoted to the study of the initial value problem for a parabolic system with a discontinuous nonlinearity from the viewpoint of the existence and asymptotic behavior of weak solutions.

Global solutions of singular parabolic equations arising from electrostatic MEMS

We study dynamic solutions of the singular parabolic problem ( P) { u t − Δ u = λ ∗ | x | α ( 1 − u ) 2 , ( x , t ) ∈ B × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) ⩾ 0 , x ∈ B , u ( x , t ) = 0 , x ∈ ∂ B , where α ⩾ 0 and λ ∗ > 0 are two parameters, and B is the unit ball { x ∈ R N : | x | ⩽ 1 } with N ⩾ 2 . Our interest is focussed on ( P) with λ ∗ : = ( 2 + α ) ( 3 N + α − 4 ) 9 , for which ( P) admits a singular stationary solution in the form S ( x ) = 1 − | x | 2 + α 3 . This equation models dynamic deflection of a simple electrostatic Micro-Electro-Mechanical-System (MEMS) device. Under the assumption u 0 ≨︀ S ( x ) , we address the existence, uniqueness, regularity, stability or instability, and asymptotic behavior of weak solutions for ( P) . Given α ∗ ∗ : = 4 − 6 N + 3 6 ( N − 2 ) 4 , in particular we show that if either N ⩾ 8 and α > α ∗ ∗ or 2 ⩽ N ⩽ 7 , then the minimal compact stationary solution u λ ∗ of ( P) is stable and while S ( x ) is unstable. However, for N ⩾ 8 and 0 ⩽ α ⩽ α ∗ ∗ , ( P) has no compact minimal solution, and S ( x ) is an attractor from below not from above. Furthermore, the refined asymptotic behavior of global solutions for ( P) is also discussed for the case where N ⩾ 8 and 0 ⩽ α ⩽ α ∗ ∗ , which is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity.

The asymptotic behaviour of weak solutions to the forward problem of electrical impedance tomography on unbounded three‐dimensional domains

AbstractThe forward problem of electrical impedance tomography on unbounded domains can be studied by introducing appropriate function spaces for this setting. In this paper we derive the point‐wise asymptotic behaviour of weak solutions to this problem in the three‐dimensional case. Copyright © 2008 John Wiley & Sons, Ltd.

Global existence and asymptotic behavior of weak solutions to the 1D compressible Navier–Stokes equations with degenerate viscosity coefficient

This paper is concerned with the existence of global weak solutions to the 1D compressible Navier-Stokes equations with density-dependent viscosity and initial density that is connected to vacuum with discontinuities. When the viscosity co- efficient is proportional to ρ θ with 0 <θ< max{3 − γ ,3 /2} where ρ is the density, we prove a global existence theorem, improving thus the result in Meth. Appl. Anal. 12 (2005), 239-252, where 0 <θ< 1 is required. Moreover, we show that the domain occupied by the fluid expands into vacuum at an algebraic rate as time grows up due to the dispersion effect of the total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ = 1, γ = 2).

On a parabolic strongly nonlinear problem on manifolds

In this work we will prove the existence uniqueness and asymptotic behavior of weak solutions for the system (*) involving the pseudo Laplacian operator and the condition ∂u ∂t + n ∑ i=1 ∣∣ ∂u ∂xi ∣∣p−2 ∂u ∂xi νi + |u|u = f on Σ1, where Σ1 is part of the lateral boundary of the cylinder Q = Ω × (0, T ) and f is a given function defined on Σ1.

Non-isothermal lubrication problem with Tresca fluid–solid interface law. Part II Asymptotic behavior of weak solutions

We study in this paper the asymptotic analysis of an incompressible Newtonian and non-isothermal problem, when one dimension of the fluid domain tends to zero. We prove the strong convergence of the unknowns which are the temperature, the velocity and the pressure of the fluid, we obtain the limit problem with the specific Reynolds equation, and we also prove the uniqueness of the limit temperature velocity and pressure distributions.

A weak attractor and properties of solutions for the three-dimensional Bénard problem

In this paper we study the asymptotic behaviour of weak solutions for the three-dimensional Boussinesq equations (also known as the Bénard problem). First, we prove some regularity properties of the weak solutions of the system. Then we construct a one parameter familiy of multivalued semiflows and for them obtain the existence of a global attractor with respect to the weak topology of the phase space. Finally, we obtain a conditional result (valid only under an unproved hypothesis) stating the existence of a global attractor with respect to the strong topology.

Incompressible non‐Newtonian fluids: time asymptotic behaviour of weak solutions

AbstractWe study the asymptotic behaviour in time of incompressible non‐Newtonian fluids in the whole space assuming that initial data also belong to L1. Firstly, we consider the weak solution to the power‐law model with non‐zero external forces and we find the asymptotic behaviour in time of this solution in the same class of existence and uniqueness with p⩾. Secondly, we are interested in the asymptotic behaviour of weak solutions to the second grade model, and finally, we deal with the asymptotic behaviour in time of weak solutions to a simplified model of viscoelastic fluids of the Oldroyd type. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Asymptotic behavior of weak solutions for the relativistic Vlasov–Maxwell equations with large light speed

We study here the behavior of time periodic weak solutions for the relativistic Vlasov–Maxwell boundary value problem in a three-dimensional bounded domain with strictly star-shaped boundary when the light speed becomes infinite. We prove the convergence toward a time periodic weak solution for the classical Vlasov–Poisson equations.

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when p ⩾ max { m , α } , where m + 1 , α + 1 and p + 1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t → ∞ , as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N = 1 , the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.

Global attractors for von Karman equations with nonlinear interior dissipation

In this paper we study the asymptotic behavior of weak solutions for von Karman equations with nonlinear interior dissipation. We prove the existence of a global attractor in the space W˚22(Ω)×L2(Ω).

Some results on a class of degenerate parabolic equations not in divergence form

In this paper, we first prove the existence of weak solutions of the initial and boundary value problem for u t = u div ( | ∇ u | p - 2 ∇ u ) + γ | ∇ u | p with p ⩾ 2 , γ ∈ ( 0 , 1 ) , and then discuss the asymptotic behavior of weak solutions as γ → 0 + .

Global structure and asymptotic behavior of weak solutions to flood wave equations

The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0.

The Existence and Large Time Behavior of Solutions to a System Related to a Phase Transition Problem

We discuss the existence and the asymptotic behavior of weak solutions to a hyperbolic-elliptic mixed type system related to a phase transition problem. As in the hyperbolic case, the weak solutions are not unique. To select the admissible solution, we need to impose admissibility criteria. One of the criteria we use is the entropy rate admissibility criterion. The question is how it can be applied. We examine two alternatives, and the result is used to study the existence and large time behavior of solutions to the perturbed Riemann problem.

SOME UNILATERAL BOUNDARY-VALUE PROBLEMS FOR ELASTIC BODIES WITH RUGGED BOUNDARIES

The system of linear elasticity is considered in a domain whose boundary depends on a small parameter e > 0 and has a part with a rugged structure. The rugged part of the boundary may bend sharply and embrace cavities or channels, and as e → 0, it approaches a limit surface on the boundary of the limit domain. On the rugged part of the boundary, conditions of two types are considered: (I) contact with rigid obstacles (conditions of Signorini type); (II) reaction forces involving the parameter e and nonlinearly depending on displacements. We investigate the asymptotic behavior of weak solutions to such boundary-value problems as e → 0 and construct the limit problem, according to the geometric structure of the rugged part of the boundary and the external surface forces and their dependence on the parameter e. In general, the limit problem has the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone characterizes the boundary conditions of the limit problem and is described in terms of the functions involved in the nonlinear boundary conditions on the rugged boundary. As shown by examples, in the limit, the type of boundary condition may change. To justify these asymptotic results, we give a detailed exposition of some facts about extensions, Korn's inequalities, traces, and nonlinear boundary conditions in partially perforated domains with Lipschitz continuous boundaries. Bibliography: 16 titles.

Asymptotic convergence of the Stefan problem to Hele-Shaw

We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as t → ∞ t\to \infty , and give the precise asymptotic growth rate for the radius.

Global existence and asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions

This paper is concerned with global existence, uniqueness, and asymptotic behavior, as time tends to infinity, of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions. The constitutive assumptions for the Helmholtz free energy include the model for the study of phase transitions in shape memory alloys. To describe phase transitions between different configurations of crystal lattices, we work in a framework in which the strain u u belongs to L ∞ {L^\infty } . It is shown that for any initial data of (strain, velocity, absolute temperature) ( u 0 , v 0 , θ 0 ) ∈ L ∞ × W 0 1 , ∞ × H 1 \left ( {u_0}, {v_0}, {\theta _0} \right ) \in \\ {L^\infty } \times W_0^{1, \infty } \times {H^1} , there is a unique global solution ( u , v , θ ) ∈ C ( [ 0 , + ∞ ] ; L ∞ ) × C ( 0 , + ∞ ) ; W 0 1 , ∞ ) ∩ L ∞ ( [ 0 , + ∞ ) ; W 1 , ∞ ) × C ( [ 0 , + ∞ ) ; H 1 ) \left ( u, v, \theta \right ) \in C\left ( \left [ 0, + \infty \right ]; {L^\infty } \right ) \times C\left ( 0, + \infty \right ); \\ \left . W_0^{1, \infty } \right ) \cap {L^\infty }\left ( \left [ 0, + \infty ); {W^{1, \infty }} \right ) \times C\left ( \left [ 0, + \infty \right ); {H^1} \right ) \right . . Results concerning the asymptotic behavior as time goes to infinity are obtained.

On the asymptotic behavior of semilinear wave equations with degenerate dissipation and source terms

We investigate the asymptotic behavior of weak solutions to the semilinear non autonomous wave equation¶¶ $ u_{tt} - \Delta u + u_t |u_t|^{p-1} = V(t)u |u |^{p-1} + f(.,t), $ ¶¶ where V (t) is a positive time dependent potential satisfying¶¶ $ V(t) = O((1 + t)^{- \lambda})\,\, {\rm as} \, t \rightarrow + \infty $ ¶¶ and f t decays to 0 as $ t \rightarrow + \infty $ . We show that for $ 0 \le \lambda \le p $ there are initial values such that the energy norm of the corresponding solutions grows at least polynomially as $ t \rightarrow + \infty $ , while if $ \lambda \ge p $ the energy norm remains uniformly bounded for any choice of initial values; moreover, in certain cases there is an absorbing ball for the orbits.

Existence and asymptotic behavior of weak solutions to strongly damped semilinear hyperbolic systems

Existence and asymptotic behavior of weak solutions to strongly damped semilinear hyperbolic systems