Related papers: Double exponential growth of the vorticity gradien…
For $1<p<\infty$, we establish the $L_{p}$ boundedness of the maximal Riesz transforms in terms of the Riesz transforms on quantum tori $L_{p}(\mathbb{T}^{d}_{\theta})$, and quantum Euclidean space $L_{p}(\mathbb{R}^{d}_{\theta})$. In…
Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic…
We give an elementary proof of weighted resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V(x) - E$ in dimension $n \neq 2$, where $h, \, E > 0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit…
We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly…
The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The…
A certain model for epitaxial film growth has recently attracted attention, with the existence of small global solutions having being proved in both the case of the n-dimensional torus and free space. We address a regularity question for…
We consider the two-dimensional quasilinear wave equations with quadratic nonlinearities. We introduce a new class of null forms and prove uniform boundedness of the highest order norm of the solution for all time. This class of null forms…
Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the…
Vorticity dynamics of the three-dimensional incompressible Euler equations is cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In…
We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition…
In this paper we study the dynamics of eigenvalues of the deformation tensor for solutions of the 3D incompressible Euler equations. Using the evolution equation of the $L^2$ norm of spectra, we deduce new a priori estimates of the $L^2$…
We consider the Cauchy problem for the full free boundary Euler equations in $3$d with an initial small velocity of size $O(\epsilon_0)$, in a moving domain which is initially an $O(\epsilon_0)$ perturbation of a flat interface. We assume…
We say that the vanishing viscosity limit holds in the classical sense if the velocity for a solution to the Navier-Stokes equations converges in the energy norm uniformly in time to the velocity for a solution to the Euler equations. We…
We study the two-dimensional semigeostrophic system on the flat torus in the small-amplitude scaling and quantify its approximation by incompressible Euler in dual variables. On a natural perturbative bootstrap window for the…
We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocusing nonlinear Schrodinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for…
We develop a quasilinear theory of the 2D Euler equation and derive an integro-differential equation for the evolution of the coarse-grained vorticity. This equation respects all the invariance properties of the Euler equation and conserves…
In this paper, we prove the existence and uniqueness of subsonic solutions to the steady Euler flows past a smooth, axisymmetric obstacle. Specifically, for a broad class of prescribed positive axial velocities in the upstream, the subsonic…
The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the…
This paper addresses the long-time dynamics of solutions to the 2D incompressible Euler equations. We construct solutions with continuous vorticity $\omega_{\varepsilon}(x,t)$ concentrated around points $\xi_{j}(t)$ that converge to a sum…
We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of…