A class of $q$-nonlinear parabolic systems with a nondiagonal principal matrix and strong nonlinearities in the gradient is considered.We discuss the global in time solvability results of the classical initial boundary value problems in the case of two spatial variables. The systems with nonlinearities $q\in (1,2)$, $q=2$, $q>2$, are analyzed.

Let $Q_T$ be a cylinder in ${\mathbb{R}}^{n+1}$ and $x=\left(x^{\prime },t\right)\in {\mathbb{R}}^n\times \mathbb{R}$. It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator $$\left\{\begin{array}{cc}{u}_t-\stackrel{n}{\sum _{i,j=1}}{a}^{ij}\left(x\right)D_{ij}u=f\left(x\right)\hfill & \text{q.o. in }{Q}_T,\hfill \\ u\left(x\right)=0\hfill & \text{su }\partial Q_T,\hfill \end{array}\right.$$ in the Morrey spaces $W_{p,\lambda }^{2,1}\left(Q_T\right)$, $p\in \left(1,\mathrm{\infty }\right)$, $\lambda \in \left(0,n+2\right)$, supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.

Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{\mathbb{R}}$, the real underlying manifold to $X$. Let $\mathrm{\Omega }$ be an open subset of $S$ with $\partial \mathrm{\Omega }$ analytic, $Y$ a complexification of $S$. We first recall the notion of $\mathrm{\Omega }$-tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\overline{\partial }}_b$ to the extendability of $CR$ functions on $\mathrm{\Omega }$ to $\mathrm{\Omega }$-tuboids of $X$. Next, if $X$ has complex dimension 2, we give results...

This paper is devoted to the study of smooth flows of density-dependent fluids in ${ℝ}^N$ or in the torus ${𝕋}^N$. We aim at extending several classical results for the standard Euler or Navier-Stokes equations, to this new framework. Existence and uniqueness is stated on a time interval independent of the viscosity $\mu$ when $\mu$ goes to $0$. A blow-up criterion involving the norm of vorticity in $L^1(0,T;L^{\infty })$ is also proved. Besides, we show that if the density-dependent Euler equations have a smooth...

Using a calibration method we prove that, if $\Gamma \subset \Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\Gamma$ and regular outside, then the function $u_{\beta }$ which solves $$\left\{\begin{array}{cc}\Delta u_{\beta }=\beta (u_{\beta }-g)\hfill & \text{in}\Omega \setminus \Gamma \hfill \\ {\partial }_{\nu }{u}_{\beta }=0\hfill & \text{on}\partial \Omega \cup \Gamma \hfill \end{array}\right.$$ is in turn discontinuous along $\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $${\int }_{\Omega \setminus S_u}{\left|\nabla u\right|}^{2}dx+{\mathscr{H}}^{n-1}\left(S_u\right)+\beta {\int }_{\Omega \setminus S_u}{(u-g)}^{2}dx,$$ over $SBV\left(\Omega \right)$, for $\beta$ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown. ...