Let $\Omega$ be a bounded simply connected domain in the complex plane, $ℂ$. Let $N$ be a neighborhood of $\partial \Omega$, let $p$ be fixed, $1\<p\<\infty ,$ and let $\widehat{u}$ be a positive weak solution to the $p$ Laplace equation in $\Omega \cap N.$ Assume that $\widehat{u}$ has zero boundary values on $\partial \Omega$ in the Sobolev sense and extend $\widehat{u}$ to $N\setminus \Omega$ by putting $\widehat{u}\equiv 0$ on $N\setminus \Omega .$ Then there exists a positive finite Borel measure $\widehat{\mu }$ on $ℂ$ with support contained in $\partial \Omega$ and such that$$\begin{array}{c}\hfill \int |\nabla \widehat{u}{|}^{p-2}\langle \nabla \widehat{u},\nabla \phi \rangle dA=-\int \phi d\widehat{\mu }\end{array}$$whenever $\phi \in C_0^{\infty }\left(N\right).$ If $p=2$ and if $\widehat{u}$ is the Green function for $\Omega$ with pole at $x\in \Omega \setminus \overline{N}$ then the measure $\widehat{\mu }$ coincides with harmonic measure...

We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called $L^{\infty }$-truncation method, used to obtain the strong convergence of the velocity...

In this paper we consider positive unbounded solutions of second order quasilinear ordinary differential equations. Our objective is to determine the asymptotic forms of unbounded solutions. An application to exterior Dirichlet problems is also given.

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation $Au=b$. We prove that if $A$ is a quasi-uniformly monotone and hemi-continuous operator, then $A^{-1}$ is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence...

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970’s, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution...

On considère l’équation de Vlasov-Poisson en dimension 3. On montre des résultats d’existence et d’unicité de solutions faibles de l’équation de Vlasov-Poisson avec densité bornée pour des données initiales ayant strictement moins de six moments dans $L_{x,\xi }^1$. La preuve est basée sur une nouvelle approche qui consiste à établir des effets de moments a priori pour des équations de transport avec des termes de force peu réguliers.

We study the 2D magnetohydrodynamic (MHD) equations for a viscous incompressible resistive fluid, a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the magnetic fields, in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality with a large class of non-autonomous external forces. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal...

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.

We consider second order semilinear hyperbolic functional differential equations where the lower order terms contain functional dependence on the unknown function. Existence and uniqueness of solutions for t ∈ (0,T), existence for t ∈ (0,∞) and some qualitative properties of the solutions in (0,∞) are shown.