Generalized kinetic equations

We study the abstract differential equation on a Hilbert space H, which represents a variety of different kinetic equations. T is assumed bounded and self-adjoint on H, and A (unbounded) positive self-adjoint and Fredholm. For partial range boundary conditions and 0x<, we prove existence and (non-) uniqueness theorems and give representations of the solution. Various examples from neutron transport, radiative transfer of polarized and unpolarized light, and electron transport are given.This paper is dedicated to K.M. Case on the occasion of his sixtieth birthday     

Abstract time-dependent transport equations

We consider the abstract time-dependent linear transport equation as an initial-boundary value evolution problem in the Banach spaces L_p, 1 ⩽ p < ∞, or on a space of measures on a (possibly time-dependent) kinetic phase space. Existence, uniqueness, dissipativity, and positivity results are proved for very general, possibly time-dependent, transport operators and boundary conditions. When the phase space, boundary conditions, and transport operator are independent of time, corresponding results are obtained for the associated semigroup.     

A constructive approach to steady nonlinear kinetic equations

We study steady boundary value problems of nonlinear kinetic theory. Using a continuation argument based on the variation of the Knudsen number we derive a method for the construction of steady solutions of discrete velocity models in a slab. This method is readily transformed into a numerical code. In a preliminary numerical test case the numerical scheme turns out to yield solutions even for Knudsen numbers small enough to calculate with high precision the asymptotic flow field adjacent to a kinetic boundary layer. Thus, we are able to numerically simulate in a simplified situation the transition from a (mesoscopic) kinetic boundary layer to the (macroscopic) far field.     

Chemotaxis: from kinetic equations to aggregate dynamics

The hydrodynamic limit for a kinetic model of chemotaxis is investigated. The limit equation is a non local conservation law, for which finite time blow-up occurs, giving rise to measure-valued solutions and discontinuous velocities. An adaptation of the notion of duality solutions, introduced for linear equations with discontinuous coefficients, leads to an existence result. Uniqueness is obtained through a precise definition of the nonlinear flux as well as the complete dynamics of aggregates, i.e. combinations of Dirac masses. Finally a particle method is used to build an adapted numerical scheme.     

Weak solutions to stochastic porous media equations

A stochastic version of the porous medium equation is studied. The corresponding Kolmogorov equation is solved in a space where is an invariant measure. Then a weak solution, that is a solution in the sense of the corresponding martingale problem, is constructed.     

Abstract green formula and applications to boundary integral equations

The resolution of boundary value problems by integral equations is usually based on isomorphisms between the solution of the boundary value problem and boundary data. Using an abstract Green formula in a Hilbert space framework, we prove these isomorphisms. Many applications are given, like the Dirichlet and Neumann problems for the Laplace operator, as well as the clamped and free plate problems in the plane.     

On wave equations with supercritical nonlinearities

We prove that the solution operators e_t (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]¹ ?L_r+1) ×L₂(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]^s_q￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here e_t(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of {

Hypoelliptic regularity in kinetic equations

We establish new regularity estimates, in terms of Sobolev spaces, of the solution f to a kinetic equation. The right-hand side can contain partial derivatives in time, space and velocity, as in classical averaging, and f is assumed to have a certain amount of regularity in velocity. The result is that f is also regular in time and space, and this is related to a commutator identity introduced by Hörmander for hypoelliptic operators. In contrast with averaging, the number of derivatives does not depend on the L^p space considered. Three type of proofs are provided: one relies on the Fourier transform, another one uses Hörmander's commutators, and the last uses a characteristics commutator. Regularity of averages in velocity are deduced. We apply our method to the linear Fokker–Planck operator and recover the known optimal regularity, by direct estimates using Hörmander's commutator.     

Uncertainty principle and kinetic equations

A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation.     

Uniform quasi-differentiability of semigroup to nonlinear reaction-diffusion equations with supercritical exponent

A new approach is established to show that the semigroup {S(t)}_(t≤0) generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L~q(?)(2 ≤q ∞) with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.     

Abstract differential equations with almost-periodic solutions

In this paper we present some quite simple results concerning almost-periodic solutions of abstract differential equations. We start with a general proposition about linear equations, then we establish some new facts about bounded or relatively compact solutions which become almost-periodic; finally, we study “separation from 0 properties” of non-trivial almost-periodic solutions for equations with bounded or unbounded operator coefficients.     

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂R^N(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.     

Radial entire solutions for supercritical biharmonic equations

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions n ≤ 12 and n ≥ 13. Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.