We prove the propagation of regularity, uniformly in time, for the scaled solutions of one-dimensional dissipative Maxwell models. This result together with the weak convergence towards the stationary state proven by Pareschi and Toscani in 2006 implies the strong convergence in Sobolev norms and in the L^1 norm towards it depending on the regularity of the initial data. In the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity. This generalizes a recent result of Carlen, Carrillo and Carvalho (arXiv:0805.1051v1), in which, for weak inelasticity, propagation of regularity for the scaled inelastic Boltzmann equation was found by means of a precise control of the growth of the Fisher information.
Due to the dissipative structure of \textit{regularity-loss}, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions to dissipative systems. The aim of this paper is to seek the lowest regularity index for the optimal decay rate of $L^{1}(\mathbb{R}^n)$-$L^2(\mathbb{R}^n)$. Consequently, a notion of minimal decay regularity for dissipative systems of regularity-loss is firstly proposed. To do this, we develop a new time-decay estimate of $L^p(\mathbb{R}^n)$-$L^{q}(\mathbb{R}^n)$-$L^{r}(\mathbb{R}^n)$ type b…
Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup generated by Maxwell's equations in an exterior domain $Ω\subset {\mathbb R}^3$ with dissipative boundary condition $E_{tan}- γ(x) (ν\wedge B_{tan}) = 0, γ(x) > 0, \forall x \in Γ= \partial Ω.$ We study the case when $Ω= \{x \in {\mathbb R^3}:\: |x| > 1\}$ and $γ\neq 1$ is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of $G_b.$
Let $Ω= \mathbb R^3 \setminus \bar{K}$, where $K$ is an open bounded domain with smooth boundary $Γ$. Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup related to Maxwell's equations in $Ω$ with dissipative boundary condition $ν\wedge (ν\wedge E)+ γ(x) (ν\wedge H) = 0, γ(x) > 0, \forall x \in Γ.$ We study the case when $γ(x) \neq 1, \: \forall x \in Γ,$ and we establish a Weyl formula for the counting function of the eigenvalues of $G_b$ in a polynomial neighbourhood of the negative real axis.
Dissipative hyperbolic systems of \textit{regularity-loss} have been recently received increasing attention. Usually, extra higher regularity is assumed to obtain the optimal decay estimates, in comparison with that for the global-in-time existence of solutions. In this paper, we develop a new frequency-localization time-decay property, which enables us to overcome the technical difficulty and improve the minimal decay-regularity for dissipative systems. As an application, it is shown that the optimal decay rate of $L^1(\mathbb{R}^3)$-$L^2(\mathbb{R}^3)$ is available for Euler-Maxwell equation…
We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.
Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup generated by Maxwell's equations in an exterior domain $Ω\subset {\mathbb R}^3$ with dissipative boundary condition $E_{tan}- γ(x) (ν\wedge B_{tan}) = 0, γ(x) > 0, \forall x \in Γ= \partial Ω.$ We prove that if $γ(x)$ is nowhere equal to 1, then for every $0 < ε\ll 1$ and every $N \in {\mathbb N}$ the eigenvalues of $G_b$ lie in the region $Λ_ε \cup {\mathcal R}_N,$ where $Λ_ε = \{ z \in {\mathbb C}:\: |\Re z | \leq C_ε (|\Im z|^{\frac{1}{2} + ε} + 1), \: \Re z < 0\},$ ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-…
For Maxwell operators $(E,H) \to (i ε^{-1} \nabla\times H, -i μ^{-1} \nabla \times E)$ in Lipschitz domains, we describe all m-dissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted for the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a tran…
In this paper we discuss the dissipative property of near-equilibrium classical solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann System in the whole space $\R^3$ when the positive charged ion flow provides a spatially uniform background. The most key point of studying this coupled degenerately dissipative system here is to establish the dissipation of the electromagnetic field which turns out to be of the regularity-loss type. Precisely, for the linearized non-homogeneous system, some $L^2$ energy functionals and $L^2$ time-frequency functionals which are equivalent with the nat…
We prove that in contrast to the free wave equation in $\R^3$ there are no incoming solutions of Maxwell's equations in the form of spherical or modulated spherical waves. We construct solutions which are corrected by lower order incoming waves. With their aid, we construct dissipative boundary conditions and solutions to Maxwell's equations in the exterior of a sphere which decay exponentially as $t \to +\infty$. They are asymptotically disappearing. Disappearing solutions which are identically zero for $t \geq T > 0$ are constructed which satisfy maximal dissipative boundary conditions which…
Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (I) A frequency dependent Lyapunov function approach
Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel（艾克斯-马赛大学、法国国家科学研究中心、马赛中央理工学院、费斯尼尔研究所）；ENSTA Paris (Ecole Nationale Supérieure de Techniques Avancées)（巴黎国立高等先进技术学院）；CNRS (Centre National de la Recherche Scientifique)（法国国家科学研究中心）；INRIA (Institut National de Recherche en Informatique et en Automatique)（法国国家信息与自动化研究所）
For the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm's law, we prove the uniform estimates with respect to Knudsen number $\eps$ for the fluctuations. As consequences, the existence of the global in time classical solutions of VMB with all $\eps \in (0,1]$ is established. Furthermore, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm's law is rigoro…
We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional for which we prove explicit inequality between the relative entropy and the production of entropy to get …
We show that the Smoluchowski coagulation equation with the solvable kernels $K(x,y)$ equal to $2$, $x+y$ or $xy$ is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self-similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann-type equations, and extend already existing results on exponential convergence to self-similarity for Smoluchowski's coagulation equation.
We consider the spatially homogeneous Boltzmann equation for ballistic annihilation in dimension d 2. Such model describes a system of ballistic hard spheres that, at the moment of interaction, either annihilate with probability $α$ $\in$ (0, 1) or collide elastically with probability 1 -- $α$. Such equation is highly dissipative in the sense that all observables, hence solutions, vanish as time progresses. Following a contribution , by two of the authors, considering well posedness of the steady self-similar profile in the regime of small annihilation rate $α$ $\ll$ 1, we prove here that such…
In this paper, we discuss the large--time behavior of solution of a simple kinetic model of Boltzmann--Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time--monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model…
