Regularity for fully non linear equations with non local drift

We prove Holder regularity for solutions of non divergence integro-differential equations with non necessarily even kernels. The even/odd decomposition of the kernel can be understood as a sum of a diffusion and a drift term. In our case we assume that such drift have the order smaller than or equal to the diffusion and at least one. For example we can say something about the following equation $\Delta^{1/2}u + |Du| = f$. The main step relies in a localized version of the Aleksandrov-Bakelman-Pucci estimate. Our estimates are also uniform as the order of the equation goes to two.Comment: Worked on the presentation, corrected some typos and added some figure

H\"older estimates for non-local parabolic equations with critical drift

In this paper we extend previous results on the regularity of solutions of integro-differential parabolic equations. The kernels are non necessarily symmetric which could be interpreted as a non-local drift with the same order as the diffusion. We provide an Oscillation Lemma and a Harnack Inequality which can be used to prove higher regularity estimates

Another approach to some rough and stochastic partial differential equations

In this note we introduce a new approach to rough and stochastic partial differential equations (RPDEs and SPDEs): we consider general Banach spaces as state spaces and -- for the sake of simiplicity -- finite dimensional sources of noise, either rough or stochastic. By means of a time-dependent transformation of state space and rough path theory we are able to construct unique solutions of the respective R- and SPDEs. As a consequence of our construction we can apply the pool of results of rough path theory, in particular we obtain strong and weak numerical schemes of high order converging to the solution process

Regularity Properties of Degenerate Diffusion Equations with Drifts

This paper considers a class of nonlinear, degenerate drift- diffusion equations. We study well-posedness and regularity properties of the solutions, with the goal to achieve uniform H\"{o}lder regularity in terms of $L^p$-bound on the drift vector field. A formal scaling argument yields that the threshold for such estimates is $p=d$, while our estimates are for $p>d+\frac{4}{d+2}$. On the other hand we are able to show by a series of examples that one needs $p>d$ for such estimates, even for divergence free drift.Comment: 31 pages, 1 figur

Flows for Singular Stochastic Differential Equations with Unbounded Drifts

In this paper, we are interested in the following singular stochastic differential equation (SDE) $${\rm d} X_t = b(t,X_t) {\rm d} t + {\rm d} B_{t},\ 0\leq t\leq T,\ X_0 = x \in \mathbb{R}^d,$$ where the drift coefficient $b:[0,T]\times \mathbb{R}^{d}\longrightarrow \mathbb{R}^{d}$ is Borel measurable, possibly unbounded and has spatial linear growth. The driving noise $B_{t}$ is a $d-$ dimensional Brownian motion. The main objective of the paper is to establish the existence and uniqueness of a strong solution and a Sobolev differentiable stochastic flow for the above SDE. Malliavin differentiability of the solution is also obtained (cf.\cite{MMNPZ13, MNP2015}). Our results constitute significant extensions to those in \cite{Zvon74, Ver79, KR05, MMNPZ13, MNP2015} by allowing the drift $b$ to be unbounded. We employ methods from white-noise analysis and the Malliavin calculus. As application, we prove existence of a unique strong Malliavin differentiable solution to the following stochastic delay differential equation $${\rm d} X (t) = b (X(t-r), X(t,0,(v,\eta)) {\rm d} t + {\rm d} B(t), \,t \geq 0 ,\textbf{ } (X(0), X_0)= (v, \eta) \in \mathbb{R}^d \times L^2 ([-r,0], \mathbb{R}^d),$$ with the drift coefficient $b: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Borel-measurable function bounded in the first argument and has linear growth in the second argument.Comment: 42 page

Universal statistics of density of inertial particles sedimenting in turbulence

We solve the problem of spatial distribution of inertial particles that sediment in Navier-Stokes turbulence with small ratio $Fr$ of acceleration of fluid particles to acceleration of gravity $g$. The particles are driven by linear drag and have arbitrary inertia. We demonstrate that independently of the particles' size or density the particles distribute over fractal set with log-normal statistics determined completely by the Kaplan-Yorke dimension $D_{KY}$. When inertia is not small $D_{KY}$ is proportional to the ratio of integral of spectrum of turbulence multiplied by wave-number and $g$. This ratio is independent of properties of particles so that the particles concentrate on fractal with universal, particles-independent statistics. We find Lyapunov exponents and confirm predictions numerically. The considered case includes typical situation of water droplets in clouds.Comment: 24 pages, minor correction

Weak differentiability of Wiener functionals and occupation times

In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Le\~ao, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite $p$-variation ($p\ge 2$) regularity is also discussed. Under stronger regularity conditions in the sense of finite $(p,q)$-variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established.Comment: Revised version. To appear in Bulletin des Sciences Math\'ematiques. arXiv admin note: text overlap with arXiv:1707.04972, arXiv:1408.142

On a transport equation with nonlocal drift

In \cite{CordobaCordobaFontelos05}, C\'ordoba, C\'ordoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions $$\partial_t \theta +u \; \partial_x \theta = 0, \qquad u = H \theta,$$ where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible H\"older regularization effects of this equation and its consequences to the equation with diffusion $$\partial_t \theta + u \; \partial_x \theta + \Lambda^\gamma \theta = 0, \qquad u = H \theta,$$ where $\Lambda = (-\Delta)^{1/2}$, and $1/2 \leq \gamma <1$. Our results also apply to the model with velocity field $u = \Lambda^s H \theta$, where $s \in (-1,1)$. We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the H\"older class in $C^{(s+1)/2}$, for all positive time

Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime

We consider a large-scale flexible service system with two large server pools and two types of customers. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible -- they can serve both types 1 and 2. (This is a so-called "N-system." Our results hold for a more general class of systems as well.) The service rate of a customer depends both on its type and the pool where it is served. We study a priority service discipline, where type 2 has priority in pool 2, and type 1 prefers pool 1. We consider the Halfin-Whitt asymptotic regime, where the arrival rate of customers and the number of servers in each pool increase to infinity in proportion to a scaling parameter $n$, while the overall system capacity exceeds its load by $O(\sqrt{n})$. For this system we prove tightness of diffusion-scaled stationary distributions. Our approach relies on a single common Lyapunov function $G^{(n)}(x)$, depending on parameter $n$ and defined on the entire state space as a functional of the {\em drift-based fluid limits} (DFL). Specifically, $G^{(n)}(x)=\int_0^\infty g(y^{(n)}(t)) dt$, where $y^{(n)}(\cdot)$ is the DFL starting at $x$, and $g(\cdot)$ is a "distance" to the origin. ($g(\cdot)$ is same for all $n$). The key part of the analysis is the study of the (first and second) derivatives of the DFLs and function $G^{(n)}(x)$. The approach, as well as many parts of the analysis, are quite generic and may be of independent interest.Comment: 22 pages, 3 figure

On the non-diffusive Magneto-Geostrophic equation

Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular that the active scalar. In \cite{Friedlander-Vicol_3}, the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized in frequency straight lines crossing the origin. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the $H^{5/2^{+}}(\mathbb{T}^3)$ norm of the perturbation.Comment: 19 pages, revised introduction, perturbations localized in straight lines. arXiv admin note: substantial text overlap with arXiv:1110.1129 by other author