Holder Continuous Solutions of Active Scalar Equations

Abstract

We consider active scalar equations ${\partial }_t\theta +\nabla \cdot (u\theta )=0$, where $u=T[\theta ]$ is a divergence-free velocity field, and $T$ is a Fourier multiplier operator with symbol $m$. We prove that when $m$ is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with Hölder regularity $C_{t,x}^{1/9-}$. In fact, every integral conserving scalar field can be approximated in ${\mathcal{D}}^{\prime }$ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when the multiplier $m$ is odd, weak limits of solutions are solutions, so that the $h$-principle for odd active scalars may not be expected.

arXiv comment61 pages

References in corpus (5)

• Energy Spectrum of Quasi-Geostrophic Turbulence
• h-Principles for the incompressible Euler equations
• On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations
• Dissipative continuous Euler flows in two and three dimensions
• Dissipative Euler flows with Onsager-critical spatial regularity

Cited by in corpus (2)

• Asymptotic Stability for some Stratified Stationary Solutions of the Inviscid Incompressible Porous Medium Equation
• Non-uniqueness of Admissible Weak Solutions to Compressible Euler Systems with Source Terms