In this paper we investigate the singular limiting behavior of slow invariant manifolds for a system of singularly perturbed evolution equations in Banach spaces. The aim is to prove the C${}^1$ stability of invariant manifolds with respect to small values of the singular parameter.

We consider non-linear elliptic equations having a measure in the right-hand side, of the type $diva(x,Du)=\mu ,$ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.

The existence of solutions to the Cauchy problem for a nonlinear parabolic equation describing the gravitational interaction of particles is studied under minimal regularity assumptions on the initial conditions. Self-similar solutions are constructed for some homogeneous initial data.

We consider the homogeneous time-dependent Oseen system in the whole space ${ℝ}^3$. The initial data is assumed to behave as $O\left(\right|x{|}^{-1-ϵ})$, and its gradient as $O\left(\right|x{|}^{-3/2-ϵ})$, when $\left|x\right|$ tends to infinity, where $ϵ$ is a fixed positive number. Then we show that the velocity $u$ decays according to the equation $\left|u(x,t)\right|=O\left(\right|x{|}^{-1})$, and its spatial gradient ${\nabla }_{x}u$ decreases with the rate ${\left|x\right|}^{-3/2}$, for $\left|x\right|$ tending to infinity, uniformly with respect to the time variable $t$. Since these decay rates are optimal even in the stationary case, they should also be the best possible...

The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p,1}^{n/p-1}\left({ℝ}^n\right)\times {\dot{B}}_{p,1}^{n/p}\left({ℝ}^n\right)$ with $n<p<2n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. In the 3D case Guo [7] first constructed a global smooth irrotational solution by using the dispersive Klein-Gordon effect. It has been conjectured that same results should hold in the two-dimensional case. In our recent work [13], we proved the existence of a family of smooth solutions by constructing the wave operators for the 2D system....

Extending our previous work, we show that the Cauchy problem for wave equations with critical exponential nonlinearities in 2 space dimensions is globally well-posed for arbitrary smooth initial data.

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle,$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$where $\Pi$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating...

We study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution $u_{tt}-\Delta u+u+{\left(\right|x|}^{-4}\ast \left|u\right|²)u=0$ in spatial dimension d ≥ 5. We utilize the strategy of Ibrahim et al. (2011) derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering can be reduced to disproving the existence of a soliton-like solution. Employing the technique of Pausader (2010), we consider a virial-type identity in the direction orthogonal to the momentum vector...

We establish absolute continuity of the elliptic measure associated to certain second order elliptic equations in either divergence or nondivergence form, with drift terms, under minimal smoothness assumptions on the coefficients.

In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator ${\Delta }^2$, on an arbitrary bounded Lipschitz domain $D$ in ${\mathbf{R}}^n$. We establish existence and uniqueness results when the boundary values have first derivatives in $L^2\left(\partial D\right)$, and the normal derivative is in $L^2\left(\partial D\right)$. The resulting solution $u$ takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of $\nabla u$ is shown to be in $L^2\left(\partial D\right)$.