Related papers: An energy method for rough partial differential eq…
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…
In this article, we study a non-Newtonian Stokes-Transport system. This set of PDEs was introduced as a model for describing the behavior of a cloud of particles in suspension in a Stokes fluid, and is a nonlinear coupling between a…
We prove the existence of probabilistically strong solutions for large classes of possibly degenerate stochastic differential equations with locally Sobolev-regular coefficients, using the restricted Yamada-Watanabe theorem. Our approach…
We prove existence and uniqueness results for (mild) solutions to some non-linear parabolic evolution equations with a rough forcing term. Our method of proof relies on a careful exploitation of the interplay between the spatial and time…
Via abstract results on maximal monotone operators and compactness property of Nemickii operator, existence of a weak solution for a class of nonlinear parabolic systems of partial differential equations is proven.
We consider local weak solutions to the widely degenerate parabolic PDE \[ \partial_{t}u-\mathrm{div}\left((\vert Du\vert-\lambda)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\qquad\mathrm{in}\ \ \Omega_{T}=\Omega\times(0,T), \] where…
Unlike many deterministic PDEs, stochastic equations are not amenable to the classical variational theory of Euler-Lagrange. In this paper, we show how self-dual variational calculus leads to solutions of various stochastic partial…
This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the…
In this paper, we showed that for some given suitable density and pressure, there exist infinitely many compactly supported solutions with prescribed energy profile. The proof is mainly based on the convex integration scheme. We construct…
Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-\nu)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in…
We show pathwise uniqueness for a class of degenerate It\^{o}-SDE among all of its weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. Consequently, by the Yamada-Watanabe Theorem and a weak existence…
This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the…
In this paper, we study the existence and uniqueness of weak solution of a nonlinear poroelasticity model. To better describe the proccess of deformation and diffusion underlying in the original model, we firstly reformulate the nonlinear…
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric…
We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on…
We prove weak uniqueness of mild solutions for general classes of SPDEs on a Hilbert space. The main novelty is that the drift is only defined on a Sobolev-type subspace and no H\"older-continuity assumptions are required. This framework…
Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove…
We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic…
In this paper, a class of systems of pseudo-parabolic PDEs is considered. These systems (S)$_\varepsilon$ are derived as a pseudo-parabolic dissipation system of Kobayashi--Warren--Carter energy, proposed by [Kobayashi et al., Physica D,…
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…