Related papers: Non-uniqueness for the transport equation with Sob…

In this paper, we extend the fractional Sobolev spaces with variable exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$, where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We are…

We give a necessary and sufficient condition for transport-entropy inequalities in dimension one. As an application, we construct a new example of a probability distribution verifying Talagrand's T2 inequality and not the logarithmic…

This paper studies quantitative uniqueness properties in $L^p$ spaces for Fokker-Planck and transport-diffusion equations under two new assumptions on their velocity field $b=b(x,t)$. We first prove $L^p$-stability estimates for…

We give an example of a bounded divergence free autonomous vector field in $\mathbb R^3$ (and of a nonautonomous bounded divergence free vector field in $\mathbb R^2$) and of a bounded initial data for which the Cauchy problem for the…

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist…

In this paper we study two basic facts of optimal transportation on Wiener space W. Our first aim is to answer to the Monge Problem on the Wiener space endowed with the Sobolev type norm (k,gamma) to the power of p (cases p = 1 and p > 1…

We study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, for a certain range of exponents $p$ and $q$, we construct a $(W^{1, p}, W^{1, q})$-extension domain which is not an $(L^{1,…

Existence of a generalized solution to a strongly singular convective elliptic equation in the whole space is established. The differential operator, patterned after the (p,q)-Laplacian, can be non-homogeneous. The result is obtained by…

We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},\omega)$, where $\omega$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $\alpha\to1^-$ of Marchaud fractional derivatives of order $\alpha$.…

In this paper we study a one-dimensional space-discrete transport equation subject to additive Levy forcing. The explicit form of the solutions allows their analytic study. In particular we discuss the invariance of the covariance structure…

We will study variations in Sobolev spaces of optimal transport maps with the standard Gaussian measure as the reference measure. Some dimension free inequalities will be obtained. As application, we construct solutions to Monge-Ampere…

In this work, we extend existing well-posedness by noise results for the stochastic transport and continuity equations by treating them as special cases of the linear advection equation of $k$-forms, which arises naturally in geometric…

We prove compactness and hence existence for solutions to a class of non linear transport equations. The corresponding models combine the features of linear transport equations and scalar conservation laws. We introduce a new method which…

H-distributions associated to weakly convergent sequences in Sobolev spaces are determined. It is shown that a weakly convergent sequence $(u_n)$ in $W^{-k,p}( \R^d)$ has the property that $\theta u_n$ converges strongly in $W^{-k,p}(\R^d)$…

Using a generalization of complexes, called 2-complexes, this paper defines and analyzes new Sobolev spaces of matrix fields and their interrelationships within a commuting diagram. These spaces have very weak second-order derivatives. An…

We introduce the notion of weighted singular vectors and weighted uniform exponent with respect to a set of weights. We prove invariance of these exponents for affine subspaces and submanifolds inside those affine subspaces. For certain…

In the Euclidean setting the Sobolev spaces $W^{\alpha,p}\cap L^\infty$ are algebras for the pointwise product when $\alpha>0$ and $p\in(1,\infty)$. This property has recently been extended to a variety of geometric settings. We produce a…

In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problem for the Dirichlet fractional $(p,q)$-Laplacian. The nonlinearity considered…

We study linear and nonlinear PDEs defined on the space of $\mathcal{P}(\mathbb{T}^d)$ over the flat torus $\mathbb{T}^d$, equipped with the Dirichlet-Ferguson measure $\mathcal{D}$. We first develop an analytic framework based on the…

We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to…