Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media
115
0
0.0
(
0
)
Authors
Stephane Mischler
Ask ChatGPT about the research
No Arabic abstract
We consider a space-homogeneous gas of {it inelastic hard spheres}, with a {it diffusive term} representing a random background forcing (in the framework of so-called {em constant normal restitution coefficients} $alpha in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $alpha in [alpha_*,1)$ for some constructive $alpha_* in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $alpha in [alpha_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.
rate research
Read More
The paper is concerned with the steady-state Burgers equation of fractional dissipation on the real line. We first prove the global existence of viscosity weak solutions to the fractal Burgers equation driven by the external force. Then the existence and uniqueness of solution with finite $H^{frac{alpha}{2}}$ energy to the steady-state equation are established by estimating the decay of fractal Burgers solutions. Furthermore, we show that the unique steady-state solution is nonlinearly stable, which means any viscosity weak solution of fractal Burgers equation, starting close to the steady-state solution, will return to the steady state as $trightarrowinfty$.
It is known that in the parameters range $-2 leq gamma <-2s$ spectral gap does not exist for the linearized Boltzmann operator without cutoff but it does for the linearized Landau operator. This paper is devoted to the understanding of the formation of spectral gap in this range through the grazing limit. Precisely, we study the Cauchy problems of these two classical collisional kinetic equations around global Maxwellians in torus and establish the following results that are uniform in the vanishing grazing parameter $epsilon$: (i) spectral gap type estimates for the collision operators; (ii) global existence of small-amplitude solutions for initial data with low regularity; (iii) propagation of regularity in both space and velocity variables as well as velocity moments without smallness; (iv) global-in-time asymptotics of the Boltzmann solution toward the Landau solution at the rate $O(epsilon)$; (v) continuous transition of decay structure of the Boltzmann operator to the Landau operator. In particular, the result in part (v) captures the uniform-in-$epsilon$ transition of intrinsic optimal time decay structures of solutions that reveals how the spectrum of the linearized non-cutoff Boltzmann equation in the mentioned parameter range changes continuously under the grazing limit.
It is expected in physics that the homogeneous quantum Boltzmann equation with Fermi-Dirac or Bose-Einstein statistics and with Maxwell-Boltzmann operator (neglecting effect of the statistics) for the weak coupled gases will converge to the homogeneous Fokker-Planck-Landau equation as the Planck constant $hbar$ tends to zero. In this paper and the upcoming work cite{HLP2}, we will provide a mathematical justification on this semi-classical limit. Key ingredients into the proofs are the new framework to catch the {it weak projection gradient}, which is motivated by Villani cite{V1} to identify the $H$-solution for Fokker-Planck-Landau equation, and the symmetric structure inside the cubic terms of the collision operators.
In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at $y=pm L$ moving relative to each other with opposite velocities $(pm alpha L,0,0)$ along the $x$-direction. Assuming that the stationary state takes the specific form of $F(y,v_x-alpha y,v_y,v_z)$ with the $x$-component of the molecular velocity sheared linearly along the $y$-direction, such steady flow is governed by a boundary value problem on a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous non-moving diffuse reflection boundary condition. In case of the Maxwell molecule collisions, we establish the existence of spatially inhomogeneous non-equilibrium stationary solutions to the steady problem for any small enough shear rate $alpha>0$ via an elaborate perturbation approach using Caflischs decomposition together with Guos $L^inftycap L^2$ theory. The result indicates the polynomial tail at large velocities for the stationary distribution. Moreover, the large time asymptotic stability of the stationary solution with an exponential convergence is also obtained and as a consequence the nonnegativity of the steady profile is justified.
The solutions of the one-dimensional homogeneous nonlinear Boltzmann equation are studied in the QE-limit (Quasi-Elastic; infinitesimal dissipation) by a combination of analytical and numerical techniques. Their behavior at large velocities differs qualitatively from that for higher dimensional systems. In our generic model, a dissipative fluid is maintained in a non-equilibrium steady state by a stochastic or deterministic driving force. The velocity distribution for stochastic driving is regular and for infinitesimal dissipation, has a stretched exponential tail, with an unusual stretching exponent $b_{QE} = 2b$, twice as large as the standard one for the corresponding $d$-dimensional system at finite dissipation. For deterministic driving the behavior is more subtle and displays singularities, such as multi-peaked velocity distribution functions. We classify the corresponding velocity distributions according to the nature and scaling behavior of such singularities.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
