L2 Initial Data Research Articles

Large Time Behavior for Convection-Diffusion Equations in [formula omitted]N with Periodic Coefficients

We describe the large time behavior of solutions of the convection-diffusion equationut−div(a(x)∇u)=d·∇(|u|q−1u)in(0, ∞)×RNwhere d∈RN and a=a(x) is a symmetric periodic matrix satisfying suitable ellipticity assumptions. We also assume that a∈W1, ∞(RN). First, we consider the linear problem (d=0) and prove that the large time behavior of solutions is given by the fundamental solution of the diffusion equation with a≡ah where ah is the homogenized matrix. In the nonlinear case, when q=1+1N, we prove that the large time behavior of solutions with initial data in L1(RN) is given by a uniparametric family of self-similar solutions of the convection-diffusion equation with constant homogenized diffusion a≡ah. When q>1+1N, we prove that the large time behavior of solutions is given by the fundamental solution of the linear-diffusion equation with a≡ah.

On the Zero Relaxation Limit for a System Modeling the Motions of a Viscoelastic Solid

We consider a simple model of the motions of a viscoelastic solid. The model consists of a two-by-two system of conservation laws including a strong relaxation term. We establish the existence of a BV-solution of this system for any positive value of the relaxation parameter. We also show that this solution is stable with respect to the perturbations of the initial data in L1 . By deriving the uniform bounds, with respect to the relaxation parameter, on the total variation of the solution, we obtain the convergence of the solutions of the relaxation system towards the solutions of a scalar conservation law as the relaxation parameter $\delta$ goes to zero. Due to the $\lip^+$ bound on the solutions of the relaxation system, an estimate on the rate of convergence towards equilibrium is derived. In particular, an ${\cal O}(\sqrt{\delta})$ bound on the L1 -error is established.

Long-time behaviour for diffusion equations with fast convection

We investigate the long-time behaviour of the solutions to the Cauchy problem $$u_t + (u^q )_x - (u^m )_{xx} = 0$$ with a nonnegative initial data in L1 (ℝ),when q ∈ (0, 1)and m ⩾ 1. We prove that the long-time profile in L1(ℝ)of these solutions is given by the unique nonnegative entropy sourcetype solution to the conservation law ut}+(uq)x=0 with the same mass. Uniqueness of such a solution is previously established. These results extend previous results obtained for the case q>1 and m⩾1.

Lattice Gases, Large Deviations, and the Incompressible Navier-Stokes Equations

We study the incompressible limit for a class of stochastic particle systems on the cubic lattice Zd, d = 3. For initial distributions corresponding to arbitrary macroscopic L2 initial data, the distributions of the evolving empirical momentum densities are shown to have a weak limit supported entirely on global weak solutions of the incompressible Navier-Stokes equations. Furthermore, explicit exponential rates for the convergence (large deviations) are obtained. The probability to violate the divergence-free condition decays at rate at least exp{-E-d+l}, while the probability to violate the momentum conservation equation decays at rate exp{fE-d+2} with an explicit rate function given by an HI1 norm.

On the Initial Boundary Value Problems for the Enskog Equation in Irregular Domains

The paper is concerned with the Enskog equation with a constant high density factor for large initial data in L1(Rn). The initial boundary value problem is investigated for bounded domains with irregular boundaries. The proof of an H-theorem for the case of general domains and boundary conditions is given. The main result guarantees the existence of global solutions of boundary value problems for large initial data with all v-moments initially finite and domains having boundary with finite Hausdorff measure and satisfying a cone condition. Existence and uniqueness are first proved for the case of bounded velocities. The solution has finite norm \(\int {(\sup _{0 \leqslant t \leqslant T} f(t_0 + t,x + vt))\sqrt {1 + v^2 } dq{\text{ }}dv,} \) where q = (t0, x) is taken on all possible n-dimensional planes Q(v) in Rn+l intersecting a fixed point and orthogonal to vectors (1, v), v ∈ Rn.

On the Global Dynamics of a Controlled Viscous Burgers' Equation

In this paper we consider a boundary control problem for a forced Burgers' equation in a Hilbert state space consisting of square integrable functions on a finite interval. Our first main result consists in proving the global in time existence of solutions of the closed loop boundary control system for arbitrary L2 initial data and quite general forcing terms (disturbances). For the unforced problem, our main interest is in stability of equilibria, while for the forced problem we are interested in steady state behavior of solutions. We note that the uncontrolled (open loop), unforced problem is not asymptotically stable on L2. However, for positive gains we show that the unforced closed loop system is globally Lyapunov stable and locally exponentially stable. In addition, for nonzero stationary forcing, we show that there is a local absorbing ball and that the corresponding nonlinear semigroup is compact for all t > 0. Using this fact, we are able to deduce several consequences regarding the existence and properties of local attractors. Our method of proof is based on a systematic investigation of the smoothing properties of the controlled dynamics. In particular we establish various regularity results for the dynamics of the closed loop system.

Self-similar solutions for navier-stokes equations in

We construct self-similar solutions for three-dimensional incompressible Navier-Stokes equations, providing some examples of functional spaces where this can be done. We apply our results to a Particular case of L2 initial data.

Time dependent Ginzburg-Landau equations of superconductivity

We study in this article the existence, uniqueness and long time behavior of the solutions of a nonstationary Ginzburg-Landau superconductivity model. We first prove the existence and uniqueness of solutions with H1 initial data, which are crucial for the study of the global attractor. We also obtain, for the first time, the existence of global weak solutions of the model with L2 initial data. It is then proved that the Ginzburg-Landau system admits a global attractor, which represents exactly all the long time dynamics of the system. The global attractor obtained consists of exactly the set of steady state solutions and its unstable manifold. Its Hausdorff and fractal dimensions are estimated in terms of the physically relevant Ginzburg-Landau parameter, diffusion parameter and applied magnetic field. We construct explicitly absorbing sets for some abstract semigroups having a Lyapunov functional and consequently prove the existence of global attractors. This abstract result is applied to the Ginzburg-Landau system, for which the existence of global attractor does not seem to be the direct consequence of some a priori estimates of solutions.

Solutions with Singular Initial Data for a Model of Electrophoretic Separation

AbstractUnique global strong solutions of a Cauchy problem arising in electrophoretic separation are constructed with arbitrary initial data in L1, thus generalizing an earlier global existence result. For small diffusion coefficients, the solutions can be viewed as approximate solutions for the corresponding zero-diffusion Riemann problem.

Global existence inL 1 for the modified nonlinear Enskog equation in ?3

A global existence theorem with large initial data inL1 is given for the modified Enskog equation in ℝ3. The method, which is based on the existence of a Liapunov functional (analog of theH-Boltzmann theorem), utilizes a weak compactness argument inL1 in a similar way to the DiPerna-Lions proof for the Boltzmann equation. The existence theorem is obtained under certain condition on the behavior of the geometric factorY. The condition onY amounts to the fact that theL1 norm of the collision term grows linearly when the local density tends to infinity.

Finite-dimensional attractor for the laser equations

The laser equations of Risken and Nummedal (1968) govern the dynamics of a ring laser cavity. They form a system of hyperbolic, semilinear, damped and driven partial differential equations with periodic boundary conditions. The Lorenz system of ordinary differential equations is an invariant subsystem of the laser equations corresponding to solutions without spatial dependence. The authors prove that the laser system admits global weak solutions for arbitrary L2 initial data. Despite the absence of parabolic diffusion they prove that the laser system enjoys a remarkable property of hyperbolic smoothing for t to infinity : the universal attractor for the L2 evolution consists of Cinfinity functions. They show, moreover, that the universal attractor is finite dimensional and estimate its dimension.