BV-Regularity of the Boltzmann Equation in Non-Convex Domains

We consider the Boltzmann equation in a general non-convex domain with the diffuse boundary condition. We establish optimal BV estimates for such solutions. Our method consists of a new W^1,1-trace estimate for the diffuse boundary condition and a delicate construction of an \({\varepsilon}\)-tubular neighborhood of the singular set.     

Fourier Transform in Bounded Domains

The functional analysis, the concept of distributionsu in the sense of Schwartz [7] andtheir extension given by Gelfand and Shilov [5]to ultradistributions u ,enables us to find by the means of the Fourier transform a secondlanguage to characterize physical behaviour. Almost any expressionwith physical meaning can be transformed, even if it isformulated in domains with complicated boundaries and evenif it is not integrable.Numerical procedures in the transformed space can bedeveloped in analogy to those well known in engineeringmechanics like the methods of Finite or BoundaryElements (FEM or BEM). Basis of the approaches presentedhere is the analytical representation of characteristicdistribution of a domain and the theorem of Parseval whichstates the invariance of energy in respect to thetransformation. In addition, the concept of thecharacteristic distribution leads to a very simplederivation of the Green-Gauss formulas fundamental for theBoundary or Finite Elements (e.g. [6]).     

Density-Dependent Incompressible Fluids in Bounded Domains

This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of with boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in W^1,q for some q > N, and the initial velocity has fractional derivatives in L^r for some r > N and arbitrarily small. Assuming in addition that the initial density is bounded away from 0, we prove existence and uniqueness on a short time interval. This result is shown to be global in dimension N = 2 regardless of the size of the data, or in dimension N ≥ 3 if the initial velocity is small. Similar qualitative results were obtained earlier in dimension N = 2, 3 by O. Ladyzhenskaya and V. Solonnikov in [18] for initial densities in W^1,∞ and initial velocities in with q > N.     

Rotating Fluids with Self-Gravitation in Bounded Domains

In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state P=e^S. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant . In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is physically striking and in sharp contrast to the case of the non-rotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.Part of this work was completed when Tao Luo was an assistant professor at the University of Michigan. Joel Smoller was supported in part by the NSF, contract number DMS-010-3998. We are grateful to the referee for his very interesting remarks and comments, which enabled a new section, Section 6, to be added in the final version of the paper.     

Topography Influence on the Lake Equations in Bounded Domains

We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and L ^p perturbations of the depth. As a byproduct, we obtain the existence of a weak solution to the lake equations in the case of singular domains and rough bottoms. Our result thus extends earlier works by Bresch and Métivier treating the lake equations with a fixed topography and by Gérard-Varet and Lacave treating the Euler equations in singular domains.     

The Acoustic Limit for the Boltzmann Equation

The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L ¹) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L ² initial data of the acoustic equations. The associated local conservation laws are recovered in the limit. Accepted: October 22, 1999     

Convergence of Discrete-Velocity Schemes for the Boltzmann Equation

In this paper we prove the convergence of two discrete-velocity deterministic schemes for the Boltzmann equation, namely, Buet's scheme and a new finite-volume scheme that we introduce here. We write the discretized equation in the form of a Boltzmann continuous equation in order to be in the framework of the DiPerna-Lions theory of renormalized solutions. In order to prove convergence we have to overcome two difficulties: the convergence of the discretized collision kernel is very weak and the lemma on the compactness of velocity averages can be recovered only asymptotically when the parameter of discretization tends to zero. (Accepted February 6, 1996)     

Existence of Time-Periodic Solutions for a Magnetoelastic System in Bounded Domains

We investigate the existence of periodic solutions for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domains with boundaries of class C ². The mathematical model includes a nonlinear mechanical dissipation like ρ(u′)=|u′| ^p u′ and a periodic forcing function of period T. We prove the existence of T-periodic weak solutions when p∈[3,4] (p=0 being a simpler case). In the corresponding two-dimensional case, the existence result holds under the assumption that p≥2.     

Global Dynamics of a Coupled Chemotaxis-Fluid Model on Bounded Domains

This paper is concerned with the long-time behavior of large amplitude classical solutions to an initial-boundary value problem of a coupled chemotaxis-fluid model which describes the so-called “chemotactic Boycott effect” arising from the interplay of chemotaxis and diffusion of nutrients or signaling chemicals in bacterial suspensions. The result is proved via energy method.     

Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation

It is well known that the Boltzmann equation is related to the Euler and Navier-Stokes equations in the field of gas dynamics. The relation is either for small Knudsen number, or, for dissipative waves in the time-asymptotic sense. In this paper, we show that rarefaction waves for the Boltzmann equation are time-asymptotic stable and tend to the rarefaction waves for the Euler and Navier-Stokes equations. Our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro-macro decomposition of the Boltzmann equation. The expansion nature of the rarefaction waves and the suitable microscopic version of the H-theorem are essential elements of our analysis.     

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.     

Estimates of Solutions and Asymptotic Symmetry for Parabolic Equations on Bounded Domains

We consider fully nonlinear parabolic equations on bounded domains under Dirichlet boundary conditions. Assuming that the equation and the domain satisfy certain symmetry conditions, we prove that each bounded positive solution of the Dirichlet problem is asymptotically symmetric. Compared with previous results of this type, we do not assume certain crucial hypotheses, such as uniform (with respect to time) positivity of the solution or regularity of the nonlinearity in time. Our method is based on estimates of solutions of linear parabolic problems, in particular on a theorem on asymptotic positivity of such solutions.     

Convergence of Solutions to the Boltzmann Equation in the Incompressible Euler Limit

 We consider here the problem of deriving rigorously, for well-prepared initial data and without any additional assumption, dissipative or smooth solutions of the incompressible Euler equations from renormalized solutions of the Boltzmann equation. This completes the partial results obtained by Golse [B. Perthame and L. Desvillettes eds., Series in Applied Mathematics 4 (2000), Gauthier-Villars, Paris] and Lions & Masmoudi [Arch. Rational Mech. Anal. 158 (2001), 195–211]. (Accepted June 6, 2002) Published online December 3, 2002 Communicated by Y. BRENIER     

Upper Maxwellian Bounds for the Spatially Homogeneous Boltzmann Equation

For the spatially homogeneous Boltzmann equation with cutoff hard potentials, it is shown that solutions remain bounded from above uniformly in time by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.     

Global Nonlinear Stability for Steady Ideal Fluid Flow in Bounded Planar Domains

We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in L^p for p>4/3. A related stability theorem using conservation of angular momentum in a circular domain is also proved.     

Global Well-Posedness in Spatially Critical Besov Space for the Boltzmann Equation

The unique global strong solution in the Chemin–Lerner type space to the Cauchy problem on the Boltzmann equation for hard potentials is constructed in a perturbation framework. Such a solution space is of critical regularity with respect to the spatial variable, and it can capture the intrinsic properties of the Boltzmann equation. For the proof of global well-posedness, we develop some new estimates on the nonlinear collision term through the Littlewood–Paley theory.     

Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces K^m+1_a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.