Pressure and intermittency in passive vector turbulence

We investigate the scaling properties a model of passive vector turbulence with pressure and in the presence of a large-scale anisotropy. The leading scaling exponents of the structure functions are proven to be anomalous. The anisotropic exponents are organized in hierarchical families growing without bound with the degree of anisotropy. Nonlocality produces poles in the inertial-range dynamics corresponding to the dimensional scaling solution. The increase with the P\'{e}clet number of hyperskewness and higher odd-dimensional ratios signals the persistence of anisotropy effects also in the inertial range.Comment: 4 pages, 1 figur

Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations

Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocity field computed through a selfconsistant interaction potential. Blow up of regular solutions is now well established for such system. In Carrillo et al. (Duke Math J (2011)), a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developped. We propose in this work to establish the link between this approach and duality solutions. This latter concept of solutions allows in particular to define a flow associated to the velocity field. Then an existence and uniqueness theory for duality solutions is developped in the spirit of James and Vauchelet (NoDEA (2013)). However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case

Large-time Behavior of the Solutions to Rosenau Type Approximations to the Heat Equation

In this paper we study the large-time behavior of the solution to a general Rosenau type approximation to the heat equation, by showing that the solution to this approximation approaches the fundamental solution of the heat equation at a sub-optimal rate. The result is valid in particular for the central differences scheme approximation of the heat equation, a property which to our knowledge has never been observed before.Comment: 20 page

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow

Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial-boundary value problem. The proof utilizes the kinetic formulation and the compensated compactness method. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial-boundary value problem for the Degasperis-Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.Comment: 24 page