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...

It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain ${\Omega }_h$ is localized either at the whole lateral surface ${\Gamma }_h$ of the domain, or at a point of ${\Gamma }_h$, while the eigenfunction decays exponentially inside ${\Omega }_h$. Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and Neumann problems, too.

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $\Omega \subset {ℝ}^n$. Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization $P_{m,\Theta }$ of the principal part $P_m$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_{m,\Theta }$. Under additional assumptions $P_m$ must be locally hyperbolic.