Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

The Acoustic Limit for the Boltzmann Equation

The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L ¹) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L ² initial data of the acoustic equations. The associated local conservation laws are recovered in the limit. Accepted: October 22, 1999     

Long-time Solvability of the Navier–Stokes Equations in a Rotating Frame with Spatially Almost Periodic Large Data

We consider existence of solutions, for large times, to the Navier–Stokes equations in a rotating frame with spatially almost periodic large data provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. To show existence of solutions for large times, we use the ℓ ¹-norm of amplitudes. Existence for large times is proven by means of techniques of fast singular oscillating limits and bootstrapping from a global-in-time unique solution to the limit equation.     

A Density Result in Two-Dimensional Linearized Elasticity,and Applications

We show that in a two-dimensional bounded open set whose complement has a finite number of connected components, the vector fields uH ¹ (Ωℝ²) are dense in the space of fields whose symmetrized gradient e(u) is in L ² (Ωℝ⁴). This allows us to show the continuity of some linearized elasticity problems with respect to variations of the set, with applications to shape optimization or the study of crack evolution. (Accepted September 18, 2002) Published online February 4, 2003 Commmunicated by V. Šverák     

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

Levitan Almost Periodic and Almost Automorphic Solutions of <Emphasis Type="Italic">V</Emphasis>-monotone Differential Equations

In the present article we consider a special class of equations

Loeb solutions of the boltzmann equation

Existence problems for the Boltzmann equation constitute a main area of research within the kinetic theory of gases and transport theory. The present paper considers the spatially periodic case with L¹ initial data. The main result is that the Loeb subsolutions obtained in a preceding paper are shown to be true solutions. The proof relies on the observation that monotone entropy and finite energy imply Loeb integrability of non-standard approximate solutions, and uses estimates from the proof of the H-theorem. Two aspects of the continuity of the solutions are also considered.     

Decay and Continuity of the Boltzmann Equation in Bounded Domains

Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the L ^∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a new L ² decay theory and its interplay with delicate L ^∞ decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.     

Justification of the Shallow-Water Limit for a Rigid-Lid Flow with Bottom Topography

The so-called lake equations arise as the shallow-water limit of the rigid-lid equations—three-dimensional Euler equations with a rigid-lid upper boundary condition—in a horizontally periodic basin with bottom topography. We prove an a priori estimate in the Sobolev space H ^m for m≥ 3 which shows that a solution to the rigid-lid equations can be approximated by a solution of the lake equations for an interval of time which can be estimated in terms of the initial deviation from a columnar configuration and the magnitude of the initial data in H ^m , the gradient of the bottom topography in H ^m+1 , and the aspect ratio of the basin. In particular, any solution to the lake equations remains close to some solution of the rigid-lid equations for an interval of time that can be made arbitrarily large by choosing the aspect ratio of the basin small. Received 10 October 1996 and accepted 15 May 1997     

Linear Stability Analysis of Resonant Periodic Motions in the Restricted Three-Body Problem

The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses 1 −μ and μ, 0≤μ≤ 1/2, that circle each other with period equal to 2π. When μ=0, the problem admits orbits for the massless particle that are ellipses of eccentricity e with the primary of mass 1 located at one of the focii. If the period is a rational multiple of 2π, denoted 2π p/q, some of these orbits perturb to periodic motions for μ > 0. For typical values of e and p/q, two resonant periodic motions are obtained for μ > 0. We show that the characteristic multipliers of both these motions are given by expressions of the form in the limit μ→ 0. The coefficient C(e,p,q) is analytic in e at e=0 and C(e,p,q)=O(e^|p-q|). The coefficients in front of e^|p-q|, obtained when C(e,p,q) is expanded in powers of e for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass 1 −μ.     

Evolution of single elliptic vortex rings

The evolution of single elliptic vortex rings for initial aspect ratio (AR)=2,4,6 has been studied. The incompressible Navier-Stokes equations are solved by a dealiased pseudo-spectral method with 64³ grid points in a periodic cube. We find that there are three kinds of vortex motion asAR increases and bifurcation occurs at certainAR. The processes of advection, interaction and decay of vortex ring are discussed. Numerical results coincide with experiments and other authors' numerical simulation. The project is supported by National Natural Science Foundation of China and Doctoral Program of Institution of Higher Education     

Vorticities in a LES Model for 3D Periodic Turbulent Flows

This paper is devoted to the study of a LES model to simulate turbulent 3D periodic flow. We focus our attention on the vorticity equation derived from this LES model for small values of the numerical grid size δ. We obtain entropy inequalities for the sequence of corresponding vorticities and corresponding pressures independent of δ, provided the initial velocity u₀ is in L_x² while the initial vorticity ω₀ = ∇ × u₀ is in L_x¹. When δ tends to zero, we show convergence, in a distributional sense, of the corresponding equations for the vorticities to the classical 3D equation for the vorticity.     

The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer

We show that, in general, the solutions to the initial-boundary value problem for the Navier-Stokes equations under a widely adopted Navier-type slip boundary condition do not converge, as the viscosity goes to zero, to the solution of the Euler equations under the classical zero-flux boundary condition, and same smooth initial data, in any arbitrarily small neighborhood of the initial time. Convergence does not hold with respect to any space-topology which is sufficiently strong as to imply that the solution to the Euler equations inherits the complete slip type boundary condition. In our counter-example Ω is a sphere, and the initial data may be infinitely differentiable. The crucial point here is that the boundary is not flat. In fact (see Beir?o da Veiga et al. in J Math Anal Appl 377:216–227, 2011) if  W = \mathbb R³₊,{\,\Omega = \mathbb R^3_+,} convergence holds in C([0,T]; W^k,p(\mathbb R³₊)){C([0,T]; W^{k,p}(\mathbb R^3_+))}, for arbitrarily large k and p. For this reason, the negative answer given here was not expected.     

Stability and exponential convergence for the Boltzmann equation

We prove existence, uniqueness and stability for solutions of the nonlinear Boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function. The results are given for a range of spaces, including L ¹, and extend previous results in L for the non-homogeneous equation, as well as the more developed L ^p -theory for the spatially homogeneous Boltzmann equation.We also give new L -estimates for the spatially homogeneous equation in the case of Maxwellian interactions.     

Linear Stability of Steady States for Thin Film and Cahn-Hilliard Type Equations

We study the linear stability of smooth steady states of the evolution equation

A note on the delayed heat equation: Instability with respect to initial data

Working in the context of a simple, one-dimensional, initial-boundary value problem involving homogeneous Dirichlet boundary data, we show that the time delayed heat equation can exhibit a type of instability with respect to the initial condition (IC); specifically, we show that a slight (in the L² sense) change in the IC can change a well-posed problem to an ill-posed one. We also establish that a physically realistic solution is possible only if the IC is of a (very) specific form. The main implication of this study is that the single and dual phase lag models, which have been put forward as possible alternatives to Fourier’s law, are not valid constitutive relations for the thermal flux vector.     

Homogenization of Periodic Systems with Large Potentials

We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by the period, the potential is scaled as ^–2. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second-order equation. This result applies to various types of equations such as parabolic, hyperbolic or eigenvalue problems, as well as fourth-order plate equation. We also prove that, for well-prepared initial data concentrating at the bottom of a Bloch band, the resulting homogenized tensor depends on the chosen Bloch band. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.     

Density-Dependent Incompressible Fluids in Bounded Domains

This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of with boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in W^1,q for some q > N, and the initial velocity has fractional derivatives in L^r for some r > N and arbitrarily small. Assuming in addition that the initial density is bounded away from 0, we prove existence and uniqueness on a short time interval. This result is shown to be global in dimension N = 2 regardless of the size of the data, or in dimension N ≥ 3 if the initial velocity is small. Similar qualitative results were obtained earlier in dimension N = 2, 3 by O. Ladyzhenskaya and V. Solonnikov in [18] for initial densities in W^1,∞ and initial velocities in with q > N.     

Parabolic Systems with Nowhere Smooth Solutions

We construct smooth 2×2 parabolic systems with smooth initial data and C^α right-hand side which admit solutions that are nowhere C¹. The elliptic part is in variational form and the corresponding energy ϕ is strongly quasiconvex and in particular satisfies a uniform Legendre-Hadamard (or strong ellipticity) condition.     

Symmetric Trajectories for the 2N-Body Problem with Equal Masses

We consider the problem of 2N bodies of equal masses in for the Newtonian-like weak-force potential r ^−σ, and we prove the existence of a family of collision-free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations. In addition, the rotation number with respect to the vertical axis ranges in a suitable interval. These solutions have the hip-hop symmetry, a generalization of that introduced in [19], for the case of many bodies and taking account of a topological constraint. The argument exploits the variational structure of the problem, and is based on the minimization of Lagrangian action on a given class of paths.