Search results for " 76B03"
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Solution properties of the incompressible Euler system with rough path advection
2021
The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport velocity possesses an additional rough-in-time, divergence-free vector field. In recent work, we have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles that possess a perturbative geometric rough path Lie-advection constraint. In this paper, we prove the local well-posedness of the system in $L^2$-Sobolev spaces $H^m$ with integer regularity $m\ge \lfloor d/2\rfloor+2$ and establish a Beale-Kato-Majda (BKM)…
On the interior regularity of weak solutions to the 2-D incompressible Euler equations
2016
We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in developing an interior regularity method in the spirit of De Giorgi–Nash–Moser, showing that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result $$\begin{aligned} u \in L_\mathrm{loc}^{2+\varepsilon }(\Omega _T) \implies \mathrm{local\ regularity} \end{aligned}$$ for weak solutions in the energy space $$L_t^\infty L_x^2$$ , satisfying appropriate vorticity estimates. We also obtain impr…