Event Detail

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Friday, August 12, 2022 - 11:00 to 13:00
Valley Library Willamette West Seminar Room

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In this dissertation, we will consider some localization methods for estimating solutions to partial differential equations related to fluids models. In particular, we focus on two non-local systems, the Navier-Stokes (and Euler) equations describing incompressible fluid behavior, and the aggregation equations. We introduce compactly supported weights and algebraic weights, and use them to inspect solutions of these models. Via compactly supported weights, we derive short-time existence of solutions to the Euler equations in $H^s_{ul}(\R^d)$ for sufficiently nice initial data. We also look at the concept of localized stability--the stability of solutions in a fixed region. We derive a localized stability result for the Navier-Stokes equations with compactly supported weights and a localized stability result for the aggregation equations with algebraic weights.\bigskip

The latter portion of this dissertation is part of joint work with David Ambrose, Elaine Cozzi, and Jim Kelliher. Applying one of the estimates derived earlier, we prove local existence of the 3D Euler equations with solenoidal initial data in the uniformly local Sobolev space $H^{s}_{ul}$ with regularity index $s\geq 3.$ Given solenoidal $u^0\in H^s_{ul}$, we construct a sequence of solutions to Euler equations and show it converges in an appropriate sense to a solution to the Euler equations with initial data $u^0$. This joint work also shows local existence of solutions to 2D SQG in the H\"{o}lder space $C^r$ for $r>1$ and in $H^s_{ul}$ for $s\geq3$. We briefly outline these latter arguments.