We prove the existence and uniqueness of global strong solutions to the Cauchy problem for 3D incompressible MHD equations with nonlinear damping terms. Moreover, the preliminary L² decay for weak solutions is also established.

Using as a main tool the time-regularizing convolution operator introduced by R. Landes, we obtain regularity results for entropy solutions of a class of parabolic equations with irregular data. The results are obtained in a very general setting and include known previous results.

Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A\left(t\right)$-caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.

We study regularity results for solutions $u\in H{W}^{1,p}\left(\Omega \right)$ to the obstacle problem $${\int }_{\Omega }𝒜(x,{\nabla }_{ℍ}u){\nabla }_{ℍ}(v-u)\mathrm{d}x\ge 0\forall v\in {𝒦}_{\psi ,u}\left(\Omega \right)$$ such that $u\ge \psi$ a.e. in $\Omega$, where ${𝒦}_{\psi ,u}\left(\Omega \right)=\{v\in H{W}^{1,p}\left(\Omega \right):v-u\in H{W}_0^{1,p}\left(\Omega \right)v\ge \psi \text{a.e.}\text{in}\Omega \}$, in Heisenberg groups ${ℍ}^n$. In particular, we obtain weak differentiability in the $T$-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{array}{ccc}\hfill dT\psi \in H{W}_{\mathrm{loc}}^{1,p}\left(\Omega \right)& \Rightarrow Tu\in L_{\mathrm{loc}}^p\left(\Omega \right),{\left|{\nabla }_{ℍ}\psi \right|}^p\in L_{\mathrm{loc}}^q\left(\Omega \right)\hfill & \hfill \Rightarrow |{\nabla }_{ℍ}{u|}^p\in L_{\mathrm{loc}}^q\left(\Omega \right),\end{array}d$$ where $2<p<4$, $q>1$.

It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T4$ configurations. We then use this to construct weak solutions to the unstable interface problem (the...

In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), $$u_{tt}+f\left(x\right)u_t+{(-\Delta )}^{\alpha /2}\left(u^m\right)=h(t,x){\left|u\right|}^p,$$ posed in $(0,T)\times {ℝ}^N,$ where ${(-\Delta )}^{\alpha /2},0<\alpha \le 2$ is $\alpha /2$-fractional power of $-\Delta .$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times 2$ system of the same type.