. We study the Boltzmann equation without Grad's angular cut-off assumption. We introduce a suitable renormalized formulation, which allows the cross-section to be singular in both the angular and the relative velocity variables. This situation occurs as soon as one is interested in long-range interactions. Together with several new estimates, this enables us to prove existence of weak solutions, and to give a proof of a conjecture by Lions (appearance of strong compactness) under general, fully realistic assumptions. Contents 1. Introduction 1 2. Assumptions on the cross-section and main results 12 3. Renormalized formulation 17 4. Strong compactness and passage to the limit 27 5. Does a borderline kinetic singularity induce compactness ? 34 Appendix : On the defect measure 36 References 41 1. Introduction Since the work of DiPerna and Lions [23] on the Cauchy problem for the Boltzmann equation, ten years ago, it has been a well-known open problem to extend their theory to physicall...

We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.

We prove the well posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L¹ data. This paper complements the work by Chen and Perthame [19], who developed a pure L¹ theory based on the notion of kinetic solutions.

We first introduce, using a functional app'oach, the notion of capacity related to the parabolic p laplace operator. Then we prove a decomposition theorem for measures (in space and time) that do not charge the sets of null capacity. We apply this result to prove existence and uniqueness of renofinalized solutions ibr nonlinem' pm'abolic initial boundary value problems with such measures as right hand side.

We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk 2 , which corresponds with the time-dependent minimal surface equation. We also study the asimptotic behavoiur of the solutions.

In this paper we prove some existence and regularity results for weak solutions to a class of nonlinear parabolic equations whose prototype is > < > u(x; 0) = 0 in u(x; t) = 0 on ; where cylinder ; r 1, q 1. 1

A three-dimensional thermoviscoelastic system derived from the balance laws of momentum and energy is considered. To describe structural phase transitions in solids, the stored energy function is not assumed to be convex as a function of the deformation gradient. A novel feature for multi-dimensional, nonconvex, and non-isothermal problems is that no regularizing higher order terms are introduced. The mechanical dissipation is not linearized. We prove existence global in time. The approach is based on a fixed-point argument using an implicit time discretization and the theory of renormalized solutions for parabolic equations with L1 data. 1.

nonlinear equations