Global exponential stability of classical solutions to the hydrodynamic model for semiconductors

In this paper, the global well-posedness and stability of classical solutions to the multidimensional hydrodynamic model for semiconductors on the framework of Besov space are considered. We weaken the regularity requirement of the initial data, and improve some known results in Sobolev space. The local existence of classical solutions to the Cauchy problem is obtained by the regularized means and compactness argument. Using the high- and low- frequency decomposition method, we prove the global exponential stability of classical solutions (close to equilibrium). Furthermore, it is also shown that the vorticity decays to zero exponentially in the 2D and 3D space. The main analytic tools are the Littlewood-Paley decomposition and Bony's para-product formula.Comment: 18 page

Blow-up of the hyperbolic Burgers equation

The memory effects on microscopic kinetic systems have been sometimes modelled by means of the introduction of second order time derivatives in the macroscopic hydrodynamic equations. One prototypical example is the hyperbolic modification of the Burgers equation, that has been introduced to clarify the interplay of hyperbolicity and nonlinear hydrodynamic evolution. Previous studies suggested the finite time blow-up of this equation, and here we present a rigorous proof of this fact

Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

We study the Hilbert expansion for small Knudsen number $\varepsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{\rho_{0}(t,x)}{(2\pi \theta_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2\theta_{0}(t,x)},\text{\ }\theta_{0}(t,x)=K\rho_{0}^{2/3}(t,x).$Our main result states that if the Hilbert expansion is valid at$t=0$for well-prepared small initial data with irrotational velocity$u_0$, then it is valid for$0\leq t\leq \varepsilon ^{-{1/2}\frac{2k-3}{2k-2}},$where$\rho_{0}(t,x)$and$ u_{0}(t,x)$satisfy the Euler-Poisson system for monatomic gas$\gamma=5/3$

Development of singularities for the compressible Euler equations with external force in several dimensions

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem by Sideris on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.Comment: 23 page

Dispersion and collapse of wave maps

We study numerically the Cauchy problem for equivariant wave maps from 3+1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures. The first conjecture states that singularities which are produced in the evolution of sufficiently large initial data are approached in a universal manner given by the profile of a stable self-similar solution. The second conjecture states that the codimension-one stable manifold of a self-similar solution with exactly one instability determines the threshold of singularity formation for a large class of initial data. Our results can be considered as a toy-model for some aspects of the critical behavior in formation of black holes.Comment: 14 pages, Latex, 9 eps figures included, typos correcte

The Cosmic No-Hair Theorem and the Nonlinear Stability of Homogeneous Newtonian Cosmological Models

The validity of the cosmic no-hair theorem is investigated in the context of Newtonian cosmology with a perfect fluid matter model and a positive cosmological constant. It is shown that if the initial data for an expanding cosmological model of this type is subjected to a small perturbation then the corresponding solution exists globally in the future and the perturbation decays in a way which can be described precisely. It is emphasized that no linearization of the equations or special symmetry assumptions are needed. The result can also be interpreted as a proof of the nonlinear stability of the homogeneous models. In order to prove the theorem we write the general solution as the sum of a homogeneous background and a perturbation. As a by-product of the analysis it is found that there is an invariant sense in which an inhomogeneous model can be regarded as a perturbation of a unique homogeneous model. A method is given for associating uniquely to each Newtonian cosmological model with compact spatial sections a spatially homogeneous model which incorporates its large-scale dynamics. This procedure appears very natural in the Newton-Cartan theory which we take as the starting point for Newtonian cosmology.Comment: 16 pages, MPA-AR-94-

Black Hole Critical Phenomena Without Black Holes

Studying the threshold of black hole formation via numerical evolution has led to the discovery of fascinating nonlinear phenomena. Power-law mass scaling, aspects of universality, and self-similarity have now been found for a large variety of models. However, questions remain. Here I briefly review critical phenomena, discuss some recent results, and describe a model which demonstrates similar phenomena without gravity.Comment: 13 pages, 6 figures; Submission for the proceedings of ICGC 2000 in the journal Preman

Dynamical elastic bodies in Newtonian gravity

Well-posedness for the initial value problem for a self-gravitating elastic body with free boundary in Newtonian gravity is proved. In the material frame, the Euler-Lagrange equation becomes, assuming suitable constitutive properties for the elastic material, a fully non-linear elliptic-hyperbolic system with boundary conditions of Neumann type. For systems of this type, the initial data must satisfy compatibility conditions in order to achieve regular solutions. Given a relaxed reference configuration and a sufficiently small Newton's constant, a neigborhood of initial data satisfying the compatibility conditions is constructed

Interplay Between Chaotic and Regular Motion in a Time-Dependent Barred Galaxy Model

We study the distinction and quantification of chaotic and regular motion in a time-dependent Hamiltonian barred galaxy model. Recently, a strong correlation was found between the strength of the bar and the presence of chaotic motion in this system, as models with relatively strong bars were shown to exhibit stronger chaotic behavior compared to those having a weaker bar component. Here, we attempt to further explore this connection by studying the interplay between chaotic and regular behavior of star orbits when the parameters of the model evolve in time. This happens for example when one introduces linear time dependence in the mass parameters of the model to mimic, in some general sense, the effect of self-consistent interactions of the actual N-body problem. We thus observe, in this simple time-dependent model also, that the increase of the bar's mass leads to an increase of the system's chaoticity. We propose a new way of using the Generalized Alignment Index (GALI) method as a reliable criterion to estimate the relative fraction of chaotic vs. regular orbits in such time-dependent potentials, which proves to be much more efficient than the computation of Lyapunov exponents. In particular, GALI is able to capture subtle changes in the nature of an orbit (or ensemble of orbits) even for relatively small time intervals, which makes it ideal for detecting dynamical transitions in time-dependent systems.Comment: 21 pages, 9 figures (minor typos fixed) to appear in J. Phys. A: Math. Theo

Concerning the Wave equation on Asymptotically Euclidean Manifolds

We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $x ^{- \rho}$ with $\rho>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $\rho> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $\rho>0$ for $d= 3$.Comment: Final version. To appear in Journal d'Analyse Mathematiqu