Related papers: A priori bounds for rough differential equations w…
We define in this work a notion of Young differential inclusion $$ dz_t \in F(z_t)dx_t, $$ for an $\alpha$-Holder control $x$, with $\alpha>1/2$, and give an existence result for such a differential system. As a by-product of our proof, we…
Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the…
We prove a priori and a posteriori H\"older bounds and Schauder $C^{1,\alpha}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$ \mathrm{div}\left(|u|^a A\nabla…
Standard finite element discretizations of the Richards equation may violate the discrete minimum principle, producing unphysical negative saturations. While existing bound-preserving methods typically rely on computationally expensive…
We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp…
We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path…
We consider the following problem $ -\Delta_{p}u= h(x,u) \mbox{ in }\Omega$, $u\in W^{1,p}_{0}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $1<p<N$, with a smooth boundary. In this paper we assume that…
We study an initial-boundary-value problem of a nonlinear Korteweg-de Vries equation posed on a finite interval (0,2pi). The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous…
We consider a rather general class of evolutionary PDEs involving dissipation (of possibly fractional order), which competes with quadratic nonlinearities on the regularity of the overall equation. This includes as prototype models,…
We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form $du_t- L_tu_tdt= N(u_t)dt + \sum_{i = 1}^dF_i(u_t)d\mathbf X^i_t$ where $(L_t)_{t\in[0,T]}$ is a…
In a recent paper, Bouchard, Elie and Reveillac \cite{BER} have studied a new class of Backward Stochastic Differential Equations with weak terminal condition, for which the $T$-terminal value $Y_T$ of the solution $(Y,Z)$ is not fixed as a…
In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity…
Existence and uniqueness of the scattering solutions is proved for a class of bounded rough obstacles which is much larger than the class of Lipschitz obstacles. Integral equations method is not used. The approach is based on the…
Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-posedness and existence of a flow for RDEs with Osgood…
A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the…
In the present paper we shall establish n-dimensional Hardy's inequalities with non-doubling weight functions of the distance to the boundary, where the boundary is a $C^2$ class bounded domain of $R^N$. This work is essentially based on…
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical $p$-laplacian equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method…
This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path with as . Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large…
In this note we consider differential equations driven by a signal $x$ which is $\gamma$-H\"older with $\gamma>1/3$, and is assumed to possess a lift as a rough path. Our main point is to obtain existence of solutions when the coefficients…
We prove in this short report the existence of a fundamental solution (F.S.) for the Cauchy initial boundary problem on the whole space for the parabolic differential equation having at origin the point of non-integrable unbounded…