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PDE Real Analysis Seminar
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Date, time & place | Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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2016/08/29
10:30-11:30 Room #268 (Graduate School of Math. Sci. Bldg.)
Nguyen Cong Phuc (Louisiana State University)
The Navier-Stokes equations: stationary existence, conditional regularity, and self-similar singularities (English)
https://www.math.lsu.edu/~pcnguyen/
Nguyen Cong Phuc (Louisiana State University)
The Navier-Stokes equations: stationary existence, conditional regularity, and self-similar singularities (English)
[ Abstract ]
In this talk, both stationary and time-dependent Navier-Stokes equations are discussed. The common theme is that the quadratic nonlinearity and the pressure are both treated as weights generally belonging to a Sobolev space of negative order. We obtain the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces that belong to a critical space. This result can be viewed as the stationary counterpart of an existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with small initial data in $BMO^{-1}$. In another direction, some new local energy bounds are obtained for the time-dependent Navier-Stokes equations which imply the regularity condition $L_{t}^{\infty}(X)$, where $X$ is a non-endpoint borderline Lorentz space $X=L_{x}^{3, q}, q\not=\infty$. The analysis also allows us to rule out the existence of Leray's backward self-similar solutions to the Navier–Stokes equations with profiles in $L^{12/5}(\mathbb{R}^3)$ or in the Marcinkiewicz space $L^{q, \infty}(\mathbb{R}^{3})$ for any $q \in (12/5, 6)$.
This talk is based on joint work with Tuoc Van Phan and Cristi Guevara.
[ Reference URL ]In this talk, both stationary and time-dependent Navier-Stokes equations are discussed. The common theme is that the quadratic nonlinearity and the pressure are both treated as weights generally belonging to a Sobolev space of negative order. We obtain the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces that belong to a critical space. This result can be viewed as the stationary counterpart of an existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with small initial data in $BMO^{-1}$. In another direction, some new local energy bounds are obtained for the time-dependent Navier-Stokes equations which imply the regularity condition $L_{t}^{\infty}(X)$, where $X$ is a non-endpoint borderline Lorentz space $X=L_{x}^{3, q}, q\not=\infty$. The analysis also allows us to rule out the existence of Leray's backward self-similar solutions to the Navier–Stokes equations with profiles in $L^{12/5}(\mathbb{R}^3)$ or in the Marcinkiewicz space $L^{q, \infty}(\mathbb{R}^{3})$ for any $q \in (12/5, 6)$.
This talk is based on joint work with Tuoc Van Phan and Cristi Guevara.
https://www.math.lsu.edu/~pcnguyen/
