In this paper, we study the well-posedness of the pressureless Euler-Navier-Stokes system in $\mathbb{R}^d$ (with $d\geq 2$) in the critical regularity setting for a density close to $0$. We prove a global existence result for small data for this system, and then give optimal time decay estimates.
We use the Lusternik-Schnirelman theory to prove the existence of a nondecreasing sequence of variational eigenvalues for the subelliptic $p$-Laplacian subject to the Dirichlet boundary condition.
This contribution, built on the companion paper [1], is focused on the different mathematical approaches available for the analysis of the quasilinear approximation in plasma physics.
In this note we prove global well-posedness for the defocusing, cubic nonlinear Schr{ö}dinger equation with initial data lying in a critical Sobolev space.
Using recent work of Bourgain-Dyatlov we show that for any convex co-compact hyperbolic surface Strichartz estimates for the Schrödinger equation hold with an arbitrarily small loss of regularity.
The concept of a fundamental solution to the time-periodic Stokes equations in dimension $n\geq 2$ is introduced. A fundamental solution is then identified and analyzed. Integrability and pointwise estimates are established.
We obtain uniqueness and nondegeneracy results for ground states of Choquard equations $-Δu+u=\left(|x|^{-1}\ast|u|^{p}\right)|u|^{p-2}u$ in $\mathbb{R}^{3}$, provided that $p>2$ and $p$ is sufficiently close to 2.
We study two principle minimizing problems, subject of different constraints. Our open sets are assumed bounded, except mentioning otherwise;precisely $Ω=]0,1[^n \in {\mathbb{R}}^n , n=1 $ or $n=2$.
We obtain a unique continuation result for fractional Schrödinger operators with potential in Morrey spaces. This is based on Carleman inequalities for fractional Laplacians.
We show that a real eigenfunction of the Schrödinger operator changes sign near some point in $\mathbb{R}^n$ under a suitable assumption on the potential.
In this paper, we construct invariant measures for the Ostrovsky equation associated with conservation laws. On the other hand, we prove the local well- posedness of the initial value problem for the periodic Ostrovsky equation with initial data in $H^{s}(\mathbb{T})$ for $s>-1/2$.
In this paper,we will study the boundedness properties of commutator \[ C_{f}=[ f,Δ] \] acting from $\overset{.}{H}^{1}(\mathbb{R}^{d}) $ to $\overset{.}{H}^{-1}(\mathbb{R}^{d}) .$