Exponential Growth of the Vorticity Gradient for the Euler Equation on the Torus
نویسنده
چکیده
We prove that there are solutions to the Euler equation on the torus with C vorticity and smooth except at one point such that the vorticity gradient grows in L∞ at least exponentially as t → ∞. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [5].
منابع مشابه
Double Exponential Growth of the Vorticity Gradient for the Two-dimensional Euler Equation
For the two-dimensional Euler equation on the torus, we prove that the L∞–norm of the vorticity gradient can grow as double exponential over arbitrary long but finite time provided that at time zero it is already sufficiently large. The method is based on the perturbative analysis around the singular stationary solution studied by Bahouri and Chemin in [1]. Our result on the growth of the vorti...
متن کاملInfinite Superlinear Growth of the Gradient for the Two-dimensional Euler Equation
For two-dimensional Euler equation on the torus, we prove that the L∞ norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of the gradient for finite time.
متن کاملPrimary resonance of an Euler-Bernoulli nano-beam modelled with second strain gradient
In the present manuscript, the second strain gradient (SSG) is utilized to investigate the primary resonance of a nonlinear Euler-Bernoulli nanobeam is analyzed in this paper...
متن کاملDynamic Growth Estimates of Maximum Vorticity for 3d Incompressible Euler Equations and the Sqg Model
By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an applicat...
متن کاملIll-posedness for the incompressible Euler equations in critical Sobolev spaces
For the 2D Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to L∞ ∩ H but escapes H immediately for t > 0. Our main observation is that a localized chunk of vorticity bounded in L∞ ∩H with odd-odd symmetry is able to generate a hyperbolic flow with large velo...
متن کامل