We study the local behaviour of solutions of the following type of equation,-Δu - V(x)u + g(u) = 0 when V is singular at some points and g is a non-decreasing function. Emphasis is put on the case when V(x) = c|x|-2 and g has a power-like growth.

We demonstrate that there exist no self-similar solutions of the incompressible magnetohydrodynamics (MHD) equations in the space $L^3\left({\mathbf{R}}^3\right)$. This is a consequence of proving the local smoothness of weak solutions via blowup methods for weak solutions which are locally $L^3$. We present the extension of the Escauriaza-Seregin-Sverak method to MHD systems.

In this article we consider local solutions for stochastic Navier Stokes equations, based on the approach of Von Wahl, for the deterministic case. We present several approaches of the concept, depending on the smoothness available. When smoothness is available, we can in someway reduce the stochastic equation to a deterministic one with a random parameter. In the general case, we mimic the concept of local solution for stochastic differential equations.

This paper proves the local well-posedness of strong solutions to a two-phase model with magnetic field and vacuum in a bounded domain $\Omega \subset {ℝ}^3$ without the standard compatibility conditions.

In this paper we establish the existence and uniqueness of the local solutions to the incompressible Euler equations in $ℝ$${}^N$, $N\ge 3$, with any given initial data belonging to the critical Besov spaces $B_{p,1}^{N/p+1}$. Moreover, a blowup criterion is given in terms of the vorticity field....

We consider the 2D magnetic Prandtl equation in the Prandtl-Hartmann regime in a periodic domain and prove the local existence and uniqueness of solutions by energy methods in a polynomial weighted Sobolev space. On the one hand, we have noted that the $x$-derivative of the pressure $P$ plays a key role in all known results on the existence and uniqueness of solutions to the Prandtl-Hartmann regime equations, in which the case of favorable $P$$...$

We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation ${\partial }_{t}u-\partial {³}_{txx}u+2\kappa {\partial }_{x}u+{\partial }_x[g\left(u\right)/2]=\gamma (2{\partial }_{x}u\partial {²}_{xx}u+u\partial {³}_{xxx}u)$ for the initial data u₀(x) in the Besov space $B_{p,r}^s\left(ℝ\right)$ with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given $C^m$-function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_0\in H^{s-1+\epsilon }$, $w_0\in H^{s-1}$ and $b_0\in H^s$ for $s>\frac{3}{2}$ and any $0<\epsilon <1$. The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$.

Dans $L^2({ℝ}^d)$, nous démontrons un résultat de localisation exponentielle pour un opérateur de Schrödinger semi-classique à potentiel périodique perturbé par de petites perturbations aléatoires indépendantes identiquement distribuées placées au fond de chaque puits. Pour ce faire, on montre que notre opérateur, restreint à un intervalle d’énergie convenable, est unitairement équivalent à une matrice aléatoire infinie dont on contrôle bien les coefficients. Puis, pour ce type de matrices, on prouve un résultat...

We are concerned with a transmission problem for the Kirchhoff plate equation where one small part of the domain is made of a viscoelastic material with the Kelvin-Voigt constitutive relation. We obtain the logarithmic stabilization result (explicit energy decay rate), as well as the wellposedness, for the transmission system. The method is based on a new Carleman estimate to obtain information on the resolvent for high frequency. The main ingredient of the proof is some careful analysis for the...

We discuss the 3D incompressible micropolar fluid equations, and give logarithmically improved regularity criteria in terms of both the velocity field and the pressure in Morrey-Campanato spaces, BMO spaces and Besov spaces.