Related papers: Regularity theory for nonlinear systems of SPDEs

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by L\'evy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity…

We prove maximal regularity results in H\"older and Zygmund spaces for linear stationary and evolution equations driven by a large class of differential and pseudo-differential operators L, both in finite and in infinite dimension. The…

We study the surface quasi-geostrophic equation with an irregular spatial perturbation $$ \partial_{t }\theta+ u\cdot\nabla\theta = -\nu(-\Delta)^{\gamma/2}\theta+ \zeta,\qquad u=\nabla^{\perp}(-\Delta)^{-1}\theta, $$ on…

Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) for $u: \R^+ \times \R^d \to \R^d$ and $p: \R^+…

In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f$$ in the plane (i.e. in…

Let $\Omega\subset R^n$ be a bounded convex domain with $n\ge2$. Suppose that $A$ is uniformly elliptic and belongs to $W^{1,n}$ when $n\ge 3$ or $W^{1,q}$ for some $q>2$ when $n=2$. For $1<p<\infty$, we build up a global second order…

A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^\gamma X(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $\gamma \in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert…

In this article, using DiPerna-Lions theory \cite{Di-Li}, we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and…

We examine the fourth order problem $\Delta^2 u = \lambda f(u) $ in $ \Omega$ with $ \Delta u = u =0 $ on $ \partial \Omega$, where $ \lambda > 0$ is a parameter, $ \Omega$ is a bounded domain in $ R^N$ and where $f$ is one of the following…

In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $\Omega=\mathcal O\times\mathbb R$…

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…

We consider stochastic equations of the prototype $du(t,x) =(\Delta u(t,x)+u(t,x)^{1+\beta})dt+\kappa u(t,x) dW_{t}$ on a smooth domain $D\subset \mathord{\rm I\mkern-3.6mu R\:}^d$, with Dirichlet boundary condition, where $\beta$, $\kappa$…

We prove $L^2$-maximal regularity of linear non-autonomous evolutionary Cauchy problem \begin{equation}\label{eq00}\nonumber \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator…

In this paper we consider nonlinear problems with an operator depending only on the deformation tensor. We consider the class of operators derived from a potential and with $(p,\delta)$ structure, for $1<p\leq 2$ and for all $\delta\geq0$.…

This paper studies the numerical approximation of evolution equations by nonlinear parametrizations $u(t)=\Phi(\param(t))$ with time-dependent parameters $\param(t)$, which are to be determined in the computation. The motivation comes from…

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced…

We study global regularity of nonlinear systems of partial differential equations depending on the symmetric part of the gradient with Dirichlet boundary conditions. These systems arise from variational problems in plasticity with power…

In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of $p$-Laplacian type ($2 \leq p< \infty$) under a strong absorption condition: $ \Delta_p u - \frac{\partial u}{\partial t} = \lambda_0 u_{+}^q…

In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla u)+\lambda_{+}…

In this article, we reproduce results of classical regularity theory of quasilinear elliptic equations in the divergence form, in the setting of Heisenberg Group. The considered cases encompass a very wide class of equations with isotropic…