Seiji Ukai

Sur la solution à support compact de l’equation d’Euler compressible

The Cauchy problem for the compressible Euler equation is discussed with compactly supported initials. To establish the localexistence of classical solutions by the aid of the theory of quasilinear symmetric hyperbolic systems, a new symmetrization is introduced which works for initials having compact support or vanishing at infinity. It is further shown that as far as the classical solution is concerned, its support does not change, and that the life span is finite for any solution except for the trivial zero solution.

OPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH POTENTIAL FORCES

For the viscous and heat-conductive fluids governed by the compressible Navier–Stokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the Lp - Lq estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L2-norm are obtained when the L1-norm of the perturbation is bounded.

Solutions of the Boltzmann Equation

Abstract This note is devoted to the review of the major results obtained in these ten years on the existence problem for the Boltzmann equation. The topics cover the local and global existence theorems for a variety of the Cauchy and initial boundary value problems, the existence and stability of the stationary solution for an exterior problem and the relation between the Boltzmann equation and fluid dynamics. They will be presented with emphasis on mathematical ideas of proofs. Also, important open problems will be pointed out.

THE CLASSICAL INCOMPRESSIBLE NAVIER-STOKES LIMIT OF THE BOLTZMANN EQUATION

The convergence hypothesis of Bardos, Golse, and Levermore,1 which leads to the incompressible Navier-Stokes equation as the limit of the scaled Boltzmann equation, is substantiated for the Cauchy problem with initial data small but independent of the Knudsen number e. The uniform (in e) existence of global strong solutions and their strong convergence as e→0 are proved. A necessary and sufficient condition for the uniform convergence up to t=0, which implies the absence of the initial layer, is also established. The proof relies on sharp estimates of the linearized operators, which are obtained by the spectral analysis and the stationary phase method.

Local solutions in gevrey classes to the nonlinear Boltzmann equation without cutoff

The nonlinear Bolzmann equation is discussed without cutoff approximations on potentials of infinite range. The Cauchy problem is solved locally in time, for both the spatially homogeneous and inhomogeneous cases. For the former case, this is done in function spaces of Gevrey classes in the velocity variables, and for the latter, in spaces of functions which are analytic in the space variables and of Gevrey classes in the velocity variables. The obtained existence theorem is of Cauchy-Kowalewski type. Also, the convergence of Grad’s angular cutoff approximations is established.

The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.

Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C∞ regularity for any positive time.

GLOBAL SOLUTIONS TO THE BOLTZMANN EQUATION WITH EXTERNAL FORCES

For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space dependent density, zero velocity and constant temperature. In this paper, we will study the nonlinear stability of these stationary solutions by using the energy method. The analysis combines the analytic techniques used for the conservation laws using the fluid-type system derived from the Boltzmann equation (cf. [14]) and the dissipative effects on the fluid and non-fluid components of the Boltzmann equation through the celebrated H-theorem. To our knowledge, this is the first result on the global classical solutions to the Boltzmann equation with external force and non-trivial large time behavior in the whole space.

The Boltzmann Equation Without Angular Cutoff in the Whole Space: Qualitative Properties of Solutions

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions; the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium, to be precise. Together with the results of Parts I and II about the well-posedness of the Cauchy problem around the Maxwellian, we conclude this series with a satisfactory mathematical theory for the Boltzmann equation without angular cutoff.

THE BOLTZMANN EQUATION IN THE SPACE

We present a function space in which the Cauchy problem for the Boltzmann equation is well-posed globally in time near an absolute Maxwellian in a mild sense without any regularity conditions. The asymptotic stability of the absolute Maxwellian is also established in this space and, moreover, it is shown that the higher order spatial derivatives of the solutions vanish in time faster than the lower order derivatives. No smallness assumptions are imposed on the derivatives of the initial data, and the optimal decay rates are derived. Furthermore, the Boltzmann equation with a time-periodic source term is solved in the same space on the unique existence and stability of a time-periodic solution which has the same period as the source term. The proof is based on the spectral analysis of the linearized Boltzmann operator.