Related papers: Growth estimates for axisymmetric Euler equations …
We consider the semiclassical Schr\"odinger equation on $\mathbb R^d$ given by $$\mathrm{i} \hbar \partial_t \psi = \left(-\frac{\hbar^2}{2} \Delta + W_l(x) \right)\psi + V(t,x)\psi ,$$ where $W_l$ is an anharmonic trapping of the form…
Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow up in the…
We consider stable solutions to the equation $ -\Delta_p u =f(u) $ in a smooth bounded domain $\Omega\subset\mathbb{R}^n $ for a $ C^1 $ nonlinearity $f$. Either in the radial case, or for some model nonlinearities $f$ in a general domain,…
We consider the wave equation in space dimension $3$, with an energy-supercritical nonlinearity which can be either focusing or defocusing. For any radial solution of the equation, with positive maximal time of existence $T$, we prove that…
Analysis of non-compact manifolds almost always requires some controlled behavior at infinity. Without such, one neither can show, nor expect, strong properties. On the other hand, such assumptions restrict the possible applications and…
This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on $\mathbb{R}^3$ within the function space…
We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\Omega\subset\R^3$. We assume that $v_r$, $v_\varphi$, $\omega_\varphi$ vanish on the lateral part of boundary $\partial\Omega$ of the cylinder, and that $v_z$,…
We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions.…
We study the asymptotic behavior of large data radial solutions to the focusing Schr\"odinger equation $i u_t + \Delta u = -|u|^2 u$ in $\R^3$, assuming globally bounded $H^1(\R^3)$ norm (i.e. no blowup in the energy space). We show that as…
In this paper, we consider the compressible Euler equations with time-dependent damping \frac{\a}{(1+t)^\lambda}u in one space dimension. By constructing 'decoupled' Riccati type equations for smooth solutions, we provide some sufficient…
We prove optimal lower bounds for the growth of the energy over balls of minimizers to the vectorial Allen-Cahn energy in two spatial dimensions, as the radius tends to infinity. In the case of radially symmetric solutions, we can prove a…
We study vanishing viscosity solutions to the axisymmetric Euler equations with (relative) vorticity in $L^p$ with $p>1$. We show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions.…
In this paper we prove the uniform-in-time $L^p$ convergence in the inviscid limit of a family $\omega^\nu$ of solutions of the $2D$ Navier-Stokes equations towards a renormalized/Lagrangian solution $\omega$ of the Euler equations. We also…
A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ \omega_t + \nabla^\perp \Psi\cdot \nabla \omega = 0 , \quad \Psi = (-\Delta)^{-1} \omega, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a…
In this paper, we study 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. Our main result is that, if this manifold has $k$ ends…
We prove an abstract theorem giving a $\langle t\rangle^\epsilon$ bound ($\forall \epsilon>0$) on the growth of the Sobolev norms in linear Schr\"odinger equations of the form $i \dot \psi = H_0 \psi + V(t) \psi $ when the time $t \to…
The search of finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by…
We examine the large-time behaviour of solutions to the compressible Navier-Stokes equations under the assumption of radial symmetry. In particular, we calculate a fast time-decay estimate of the norm of the nonlinear part of the solution.…
Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v…
We introduce a symmetric (log-)epiperimetric inequality, generalizing the standard epiperimetric inequality, and we show that it implies a growth-decay for the associated energy: as the radius increases energy decays while negative and…