Differential Equations with singular fields

This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow (bounded jacobian) is considered. The main result provides existence under the condition that the vector field belongs to $BV$ in dimension 2 and $SBV$ in higher dimensions

Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients

In this paper we give an affirmative answer to an open question mentioned in [Le Bris and Lions, Comm. Partial Differential Equations 33 (2008), 1272--1317], that is, we prove the well-posedness of the Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients.Comment: 11 pages. The proof has been modifie

Stochastic differential equations with coefficients in Sobolev spaces

We consider It\^o SDE \d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\delta(A_0)$ w.r.t. the Gaussian measure $\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\text{loc}^{1,p'}$ for some $p'>1$. Assume \int_{\R^d} \exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \d\gamma_d0, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X_t)_# \gamma_d admits a density with respect to $\gamma_d$. In particular, if the coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue measure \Leb_d quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps.Comment: 31 page

Tangent Lines and Lipschitz Differentiability Spaces

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces.We show that any tangent space of a Lipschitz differentiability space contains at least n distinct tangent lines, obtained as the blow-up of n Lipschitz curves, where n is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these n distinct tangent lines span an n-dimensional part of the tangent space. \ua9 2016 Fabio Cavalletti and Tapio Rajala

Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements

Motivated by models of fracture mechanics, this paper is devoted to the analysis of unilateral gradient flows of the Ambrosio-Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. In the spirit of gradient flows in metric spaces, such evolutions are defined in terms of curves of maximal unilateral slope, and are constructed by means of implicit Euler schemes. An asymptotic analysis in the Mumford-Shah regime is also carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set.Comment: accepted in Ann. Inst. H. Poincar\'e, Anal. Nonli

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans ans Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum

Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

In this article we study the fractal Navier-Stokes equations by using stochastic Lagrangian particle path approach in Constantin and Iyer \cite{Co-Iy}. More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by L\'evy processes. Basing on this representation, a self-contained proof for the existence of local unique solution for the fractal Navier-Stokes equation with initial data in \mW^{1,p} is provided, and in the case of two dimensions or large viscosity, the existence of global solution is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for L\'evy processes with time dependent and discontinuous drifts is proved.Comment: 19 page

Well-posedness of the transport equation by stochastic perturbation

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.Comment: Addition of new part

Positive solutions of indefinite logistic growth models with flux-saturated diffusion

This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator - div (del u/root 1+vertical bar del u vertical bar(2)) - lambda a(x)f(u) in Omega, u - 0 on partial derivative Omega, with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, Omega is a bounded domain in R-N with a regular boundary partial derivative Omega, lambda > 0 represents a diffusivity parameter, a is a continuous weight which may change sign in Omega, and f: [0, L] -> R, with L > 0 a given constant, is a continuous function satisfying f(0) = f (L) = 0 and f (s) > 0 for every s is an element of [0, L]. Depending on the behavior of f at zero, three qualitatively different bifurcation diagrams appear by varying lambda. Typically, the solutions we find are regular as long as lambda is small, while as a consequence of the saturation of the flux they may develop singularities when A becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, f (s) = s(L - s) and a 1, having no similarity with the case of linear diffusion based on the Fick-Fourier's la