Abstract. We derive new models for gravity driven shallow water flows in several space dimensions over a general topography. A first model is valid for small slope variation, i.e. small curvature, and a second model is valid for arbitrary topography. In both cases no particular assumption is made on the velocity profile in the material layer. The models are written for an arbitrary coordinate system, and several formulations are provided. A Coulomb friction term is derived within the same framework, relevant in particular for debris avalanches. All our models are invariant under rotation, admit a conservative energy equation, and preserve the steady state of a lake at rest.

We consider the porous medium equation on a compact Riemannian manifold and give a new proof of the contraction of its semigroup in the Wasserstein distance. This proof is based on the insight that the porous medium equation does not increase the size of infinitesimal perturbations along gradient flow trajectories and on an Eulerian formulation for the Wasserstein distance using smooth curves. Our approach avoids the existence result for optimal transport maps on Riemannian manifolds.

Abstract. An entropy solution u of a multi-dimensional scalar conservation law is not necessarily in BV , even if the conservation law is genuinely nonlinear. We show that u nevertheless has the structure of a BV -function in the sense that the shock location is codimension-one rectifiable. This result highlights the regularizing effect of genuine nonlinearity in a qualitative way; it is based on the locally finite rate of entropy dissipation. The proof relies on the geometric classification of blow-ups in the framework of the kinetic formulation.

We consider compressible pressureless fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid itself. We explain how this flow can be described by a differential inclusion on the space of transport maps, in particular when a sticky particle dynamics is assumed. We study a discrete particle approximation and we prove global existence and stability results for solutions of this system. In the particular case of the Euler-Poisson system in the attractive regime our approach yields an explicit representation formula for the solutions

Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical symmetry when the origin r = 0 is included. These partial differential equations are hyperbolic, but fail to be strictly hyperbolic when the fluid mass density vanishes and vacuum is reached. Furthermore, when geometric effects are taken into account, the sup-norm of solutions can not be controlled since there exist no invariant regions. To overcome these difficulties and to establish an existence theory for solutions with arbitrarily large amplitude, we search for solutions with finite mass and total energy. Our strategy of proof takes advantage of the particular structure of the Euler equations, and leads to a versatile framework covering general compressible fluid problems. We establish first higherintegrability estimates for the mass density and the total energy. Next, we use arguments from the theory of compensated compactness and Young measures, extended here to sequences of solutions with finite mass and total energy. The third ingredient of the proof is a characterization of the unbounded support of entropy admissible Young measures. This requires the study of singular products involving measures and principal values.The unknowns of this system are the density ρ 0 and the velocity u, which are functions of the independent variables (t, x) ∈ [0, ∞) × R. The pressure P (ρ) is related to the internal energy U (ρ) by the relationfor all ρ 0. We restrict ourselves to polytropic perfect gases, for which U (ρ) = κ γ−1 ρ γ and P (ρ) = κρ γ .Here γ > 1 is the adiabatic coefficient, and κ := θ 2 /γ with θ := (γ − 1)/2 are constants. The case of general pressure laws will be addressed in future work. R ψ(s) χ(s|ρ, u), σ(s|ρ, u) ds, (1.10) and we impose the entropy inequalitiesin the distribution sense. We use the notation g ,ρ := ∂ ρ g for all functions g. Definition 1.1. Let (ρ, u) be given initial data with finite mass and total energy. A pair of measurable functions (ρ, u) : [0, ∞) × Ω −→ [0, ∞) × R is called an entropy solution with finite mass and energy (or a finite energy solution, for short) to the Cauchy problem (1.1) & (1.7) if the following is true: 1. The total mass is conserved in time: for almost every (a.e.) t M [ ρ ](t) = M . 2. The total energy is bounded in time: for a.e. t E[ ρ, u ](t) E.