Searcharxiv⌕ Search

arXiv subjects

Walter Rusin

Publications and source records attributed to Walter Rusin.

6 recordsLinked to original sources
arXiv2016

On the local existence for an active scalar equation in critical regularity setting

In this note, we address the local well-posedness for the active scalar equation $\partial_t θ+ u\cdot \nabla θ=0$, where $u = - \nabla^\perp(-Δ)^{-1+β/2}θ$. The local existence of solutions in the Sobolev class $H^{1+β+ε}$, where $ε>0$ and $β\in (1,2)$, has been recently addressed in \cite{HKZ}. The critical case $ε=0$ has remained open. Using a different technique, we prove the local well-posedness in the Besov space $B^{1+β}_{2,1}$, where $β\in (1,2)$. The proof is based on log-Lipschitz estimates for the transport equation.

math.AP↗
arXiv2013

The magneto-geostrophic equations: a survey

We discuss recent results obtained by the authors, regarding the analysis of the magneto-geostrophic equation: a model proposed by Moffat and Loper to study the geodynamo and turbulence in the Earth's fluid core. We conclude this review by indicating some open problems around the MG equation, that remain to be addressed in the future.

math.AP↗
arXiv2012

On the second iterate for active scalar equations

We consider an iterative resolution scheme for a broad class of active scalar equations with a fractional power γof the Laplacian and focus our attention on the second iterate. The main objective of our work is to analyze boundedness properties of the resulting bilinear operator, especially in the super-critical regime. Our results are two-fold: we prove continuity of the bilinear operator in BMO^{1-2γ} - a fractional analogue of the Koch-Tataru space; for equations with an even symbol we show that the B^{-γ}_{\infty,q} -regularity, where q > 2, is in a sense a minimal necessary requirement on the solution.

math.AP↗
arXiv2011

On the supercritically diffusive magneto-geostrophic equations

We address the well-posedness theory for the magento-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (-Δ)^γ, where 0<γ<1, we discover that for γ>1/2 the equations are locally well-posed, while for γ<1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when γgoes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar is given by an unbounded Fourier multiplier which is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocity may be explored in order to obtain an improvement in the regularity of the solutions when the initial data and the force have thin Fourier support, i.e. they are supported on a plane in frequency space. In particular, for such well-prepared data one may prove the local existence and uniqueness of solutions for all values of γ\in (0,1).

math.AP↗
arXiv2011

Logarithmic spikes of gradients and uniqueness of weak solutions to a class of active scalar equations

We study the question weather weak solutions to a class of active scalar equations, with the drift velocity and the active scalar related via a Fourier multiplier of order zero, are unique. Due to some recent results we cannot expect weak solutions to be unique without additional conditions. We analyze the case of some integrability conditions on the gradient of the solutions. The condition is weaker than simply imposing $\nabla θ\in L^\infty$. Lastly, we consider the inviscid limit for the studied class of equations.

math.AP↗