Related papers: Global existence for rough differential equations …

We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric H\"older continuous rough paths can be approximated…

The goal of these notes is to provide an introduction to rough partial differential equations. For this purpose, we will present the theory of rough paths to the extend as it is required. Applications to stochastic partial differential…

In the paper [H. Kubo, Global existence for exterior problems of semilinear wave equations with the null condition in 2D, Evol. Equ. Control Theory 2 (2013), no. 2, 319-335], for the 2-D semilinear wave equation system…

We prove the existence of nonradial solutions for the H\'enon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $\alpha$. For sign-changing solutions, the case $\alpha=0$ -- Lane-Emden equation…

We study strong existence and pathwise uniqueness for stochastic differential equations in $\RR^d$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative…

In this paper, we establish a result for existence and uniqueness of stochastic differential equations on Riemannian manifolds, for regular inhomogeneous tensor coefficients with stochastic drift, under geometrical hypothesis on the…

The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential…

We consider the pressureless Euler-Poisson equations with quadratic confinement. For spatial dimension $d\ge 2,\,d\ne 4$, we give a necessary and sufficient condition for the existence of radial global smooth solutions, which is formulated…

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such…

In this paper we are concerned with the global existence of smooth solutions to the turbulent flow equations for compressible flows in $\mathbb{R}^3$. The global well-posedness is proved under the condition that the initial data are close…

As a general rule, differential equations driven by a multi-dimensional irregular path $\Gamma$ are solved by constructing a rough path over $\Gamma$. The domain of definition ? and also estimates ? of the solutions depend on upper bounds…

In this paper we establish local and global existence and uniqueness of solutions for general nonlinear evolution equations with coefficients satisfying some local monotonicity and generalized coercivity conditions. An analogous result is…

In this paper we study rough differential equations driven by Gaussian rough paths from the viewpoint of Malliavin calculus. Under mild assumptions on coefficient vector fields and underlying Gaussian processes, we prove that solutions at a…

We describe a functional framework suitable to the analysis of the Cahn-Hilliard equation on an evolving surface whose evolution is assumed to be given \textit{a priori}. The model is derived from balance laws for an order parameter with an…

This paper studies a class of nonlinear Dirac equations with cubic terms in $R^{1+1}$, which include the equations for the massive Thirring model and the massive Gross-Neveu model. Under the assumptions that the initial data has small…

Here we show a hidden regularity result for nonlinear wave equations with an integral term of convolution type and Dirichlet boundary conditions. Under general assumptions on the nonlinear term and on the integral kernel we are able to…

We consider the Cauchy problem for a system of fully nonlinear parabolic equations. In this paper, we shall show the existence of global-in-time solutions to the problem. Our condition to ensure the global existence is specific to the fully…

We prove the global existence of solutions with small and smooth initial data of a nonlinear dispersive equation for the motion of generalized surface quasi-geostrophic (GSQG) fronts in a parameter regime $1<\alpha<2$, where $\alpha=1$…

This article deals with the second order linear differential equations with entire coefficients. We prove some results involving conditions on coefficients so that the order of growth of every non-trivial solution is infinite.

We study the Taylor expansion for the solutions of differential equations driven by $p$-rough paths with $p>2$. We prove a general theorem concerning the convergence of the Taylor expansion on a nonempty interval provided that the vector…