On the asymptotic stability of steady solutions of the Navier–Stokes equations in unbounded domains

We consider the problem of the asymptotic behaviour in the L²‐norm of solutions of the Navier–Stokes equations. We consider perturbations to the rest state and to stationary motions. In both cases we study the initial‐boundary value problem in unbounded domains with non‐compact boundary. In particular, we deal with domains with varying and possibly divergent exits to infinity and aperture domains. Copyright © 2007 John Wiley & Sons, Ltd.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : I. Asymptotic Symmetry for the Cauchy Problem

We consider quasilinear parabolic equations on ? ^N satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry.     

Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles

Abstract

We study the long-time behavior of a linear inhomogeneous Boltzmann equation. The collision operator is modeled by a simple relaxation towards the Maxwellian distribution with zero mean and fixed lattice temperature. Particles are moving under the action of an external potential that confines particles, i.e., there exists a unique stationary probability density. Convergence rate towards global equilibrium is explicitly measured based on the entropy dissipation method and apriori time independent estimates on the solutions. We are able to prove that this convergence is faster than any algebraic time function, but we cannot achieve exponential convergence.     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

A note on conjugate gradient convergence – Part III

Summary. In this paper we again consider the rate of convergence of the conjugate gradient method. We start with a general analysis of the conjugate gradient method for uniformly bounded solutions vectors and matrices whose eigenvalues are uniformly bounded and positive. We show that in such cases a fixed finite number of iterations of the method gives some fixed amount of improvement as the the size of the matrix tends to infinity. Then we specialize to the finite element (or finite difference) scheme for the problem . We show that for some classes of function we see this same effect. For other functions we show that the gain made by performing a fixed number of iterations of the method tends to zero as the size of the matrix tends to infinity. Received July 9, 1998 / Published online March 16, 2000     

Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data

This paper concerns the large time behavior of strong and classical solutions to the two-dimensional Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the two-dimensional Stokes approximation equations for the compressible flows with large external potential force, together with a Navier-slip boundary condition, for arbitrarily large initial data. Under the conditions that the corresponding steady state exists uniquely with the steady state density away from vacuum, we prove that the density is bounded from above independently of time, consequently, it converges to the steady state density in L^p and the velocity u converges to the steady state velocity in W^1,p for any 1p<∞ as time goes to infinity; furthermore, we show that if the initial density contains vacuum at least at one point, then the derivatives of the density must blow up as time goes to infinity.     

Renormalization group and asymptotics of solutions of nonlinear parabolic equations

We present a general method for studying long-time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to higher (possibly fractional) differential linear terms. We present in detail the analysis for nonlinear diffusion-type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity. In an accompanying paper, [5], the method is applied to systems of equations where some variables are “slaved,” such as the complex Ginzburg-Landau equation. © 1994 John Wiley & Sons, Inc.     

A concentration-compactness principle at infinity and positive solutions of some quasilinear elliptic equations in unbounded domains

In this paper, we formulate a concentration-compactness principle at infinity which extends a result introduced by J. Chabrowski [Calc. Var. Partial Differential Equations 3 (1995) 493-512]. Then we consider some quasilinear elliptic equations in some classes of unbounded domains by solving their corresponding constrained minimization problems under certain conditions. We show the existence of positive solutions of those equations via the concentration-compactness principle at infinity, which extends some results in [Differential Integral Equations 6 (1993) 1281-1298].     

Cross-Diffusion Limit for a Reaction-Diffusion System with Fast Reversible Reaction

We consider a reaction-diffusion system which models a fast reversible reaction of type C ₁ + C ₂?C ₃ between mobile reactants inside an isolated vessel. Assuming mass action kinetics, we study the limit when the reaction speed tends to infinity in case of unequal diffusion coefficients and prove convergence of a subsequence of solutions to a weak solution of an appropriate limiting pde-system, where the limiting problem turns out to be of cross-diffusion type. The proof combines the L ²-approach to reaction-diffusion systems having at most quadratic reaction terms with a thorough exploitation of the entropy functional for mass action systems. The limiting cross-diffusion system has unique local strong solutions for sufficiently regular initial data, while uniqueness of weak solutions is in general open but is shown to be valid under restrictions on the diffusivities.     

On a nonlinear elliptic-parabolic integro-differential equation with L-data

We generalize the concept of entropy solutions for parabolic equations with L¹-data and consider a class of nonlinear history-dependent degenerated elliptic-parabolic equations including problems with a fractional time derivative such as with Dirichlet boundary conditions and initial condition, where 0<γ?1. We show uniqueness of entropy solutions for general L¹-data by Kruzhkov's method of doubling variables. Moreover, existence in the nondegenerated case, i.e. b≡id, is shown by using the concept of generalized solutions of a corresponding abstract Volterra equation.     

A solution for the local plane strain problems using elliptic integrals of Cauchy type

We consider a parallel fiber‐reinforced periodic elastic composite that present an imperfect contact of spring type between the fiber and the matrix. We use the elliptic integral of Cauchy type for solving the plane strain local problems that arise from the asymptotic homogenization method. Several general conditions are assumed, which include that the fibers are disposed of arbitrary manner in the local cell, that all fibers present contact perfect with different constants of imperfection, and that their cross section is a smooth closed arbitrary curve. We find that there are infinity solutions for these problems, and we find relations between these solutions and effective coefficients of the composite. Finally, we obtain analytic formulae for the circular fiber case and show some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear artificial boundary conditions with pointwise error estimates for the exterior three dimensional Navier–Stokes problem

On a three–dimensional exterior domain Ω we consider the Dirichlet problem for the stationary Navier–Stokes system. We construct an approximation problem on the domain Ω_R, which is the intersection of Ω with a sufficiently large ball, while we create nonlinear, but local artificial boundary conditions on the truncation boundary. We prove existence and uniqueness of the solutions to the approximating problem together with asymptotically precise pointwise error estimates as R tends to infinity.     

A PDE Approach to Large-Time Asymptotics for Boundary-Value Problems for Nonconvex Hamilton–Jacobi Equations

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton–Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case.     

A general method for proving sharp energy decay rates for memory-dissipative evolution equations

This Note is concerned with stabilization of hyperbolic systems by a distributed memory feedback. We present here a general method which gives energy decay rates in terms of the asymptotic behavior of the kernel at infinity. This method, which allows us to recover in a natural way the known cases (exponential, polynomial, …), applies to a large quasi-optimal class of kernels. It also provides sharp energy decay rates compared to the ones that are available in the literature. We give a general condition under which the energy of solutions is shown to decay at least as fast as the kernel at infinity. To cite this article: F. Alabau-Boussouira, P. Cannarsa, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Fundamental sequences of functions in the approximation of solutions to exterior problems in the bending of thin micropolar plates

In this paper we show how to construct fundamental sequences for approximate solutions to exterior Dirichlet and Neumann-type problems in the bending of micropolar plates. The sequences are based on singular solutions of the governing equilibrium equations. These singular solutions are, however, unbounded at infinity. This leads to difficulties when applying the usual methods for proving linear independence and completeness. By decomposing the sequences into divergent and convergent parts we show that they can be accommodated in a more general framework developed in previous work. This allows us to overcome the difficulties mentioned above. Kupradze's method of generalised Fourier series is then modified and used to construct approximations which converge uniformly to the corresponding exact solutions     

On regularity of a weak solution to the Navier–Stokes equation with generalized impermeability boundary conditions

We consider the 3D Navier–Stokes equation with generalized impermeability boundary conditions. As auxiliary results, we prove the local in time existence of a strong solution (‘strong’ in a limited sense) and a theorem on structure. Then, taking advantage of the boundary conditions, we formulate sufficient conditions for regularity up to the boundary of a weak solution by means of requirements on one of the eigenvalues of the rate of deformation tensor. Finally, we apply these general results to the case of an axially symmetric flow with zero angular velocity.     

Traveling wave solutions in a delayed lattice competition-cooperation system

In this paper, we first reduce the existence of traveling wave solutions in a delayed lattice competition-cooperation system to the existence of a pair of upper and lower solutions by means of Schauder’s fixed point theorem and the cross iteration scheme, and then we construct a pair of upper and lower solutions to obtain the existence and nonexistence of traveling wave solutions. We also consider the asymptotic behaviour of any nonnegative traveling wave solutions at negative infinity.     

Fractional Retarded Differential Equations Involving Mixed Nonlocal Plus Local Initial Conditions

Abstract

The aim of this paper is to discuss the existence of mild solutions and positive mild solutions for a general class of semilinear fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions on infinite dimensional Banach spaces. Under the situation that the nonlinear term and nonlocal function satisfy some appropriate growth conditions and a noncompactness measure condition, we obtained the existence of mild solutions and positive mild solutions by utilizing a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness. In addition, the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example about the retarded parabolic partial differential equation involving a general mixed nonlocal plus local initial conditions is also given to illustrate the feasibility of our abstract results.     

Large Mass Global Solutions for a Class of L 1-Critical Nonlocal Aggregation Equations and Parabolic-Elliptic Patlak-Keller-Segel Models

We consider a class of L ¹ critical nonlocal aggregation equations with linear or nonlinear porous media-type diffusion which are characterized by a long-range interaction potential that decays faster than the Newtonian potential at infinity. The fast decay breaks the L ¹ scaling symmetry and we prove that all ‘sufficiently spread out’ initial data, even with supercritical mass, results in global, decaying solutions. In particular, we produce decaying solutions with arbitrary mass in cases for which finite time blow up solutions or non-decaying solutions are also known to exist for sufficiently large mass. This is in contrast to the classical parabolic-elliptic PKS for which essentially all solutions with supercritical mass blow up in finite time. The results with linear diffusion are proved using properties of the Fokker-Planck semi-group whereas the results with nonlinear diffusion are proved using a more interesting bootstrap argument coupling the entropy-entropy dissipation methods of the porous media equation together with higher L ^p estimates similar to those used in small-data and local theory for PKS-type equations.     

Fluid dynamic limits of kinetic equations II convergence proofs for the boltzmann equation

Using relative entropy estimates about an absolute Maxwellian, it is shown that any properly scaled sequence of DiPerna-Lions renormalized solutions of some classical Boltzmann equations has fluctuations that converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussinesq relations. Moreover, if the initial fluctuations entropically converge to an infinitesimal Maxwellian then the limiting fluid variables satisfy a version of the Leray energy inequality. If the sequence satisfies a local momentum conservation assumption, the momentum densities globaly converge to a solution of the Stokes equation. A similar discrete time version of this result holds for the Navier-Stokes limit with an additional mild weak compactness assumption. The continuous time Navier-Stokes limit is also discussed. © 1993 John Wiley & Sons, Inc.