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PDE Real Analysis Seminar
Seminar information archive ～04/18｜Next seminar｜Future seminars 04/19～
| Date, time & place | Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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2011/10/31
13:30-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Horst Heck
(Technische Universität Darmstadt)
Stationary Weak Solutions of the Navier-Stokes Equations Past an Obstacle (ENGLISH)
Horst Heck
(Technische Universität Darmstadt)
Stationary Weak Solutions of the Navier-Stokes Equations Past an Obstacle (ENGLISH)
[ Abstract ]
Consider the stationary Navier-Stokes equations in an exterior smooth domain $\\Omega$. If the flow condition $u_\\infty$ for $u$ at infinity is non-zero and the external force $f\\in \\dot H^{-1}_2(\\Omega)$ is given Leray constructed a weak solution $u\\in \\dot H^1_2(\\Omega)$, the homogeneous Sobolev space, with $u-u_\\infty\\in L^6(\\Omega)$.
We show that if in addition $f\\in \\dot H^{-1}_q(\\Omega)$ for some $q\\in (4/3,4)$ then the weak solution has the property $u-u_\\infty\\in L^{4q/(4-q)}(\\Omega)$.
This additional integrability implies that $u$ satisfies the energy identity. Further consequences are uniqueness results for small $u_\\infty$ and $f$ and continuous dependence of the solution with respect to $u_\\infty$.
The presented results are joint work with Hyunseok Kim and Hideo Kozono.
Consider the stationary Navier-Stokes equations in an exterior smooth domain $\\Omega$. If the flow condition $u_\\infty$ for $u$ at infinity is non-zero and the external force $f\\in \\dot H^{-1}_2(\\Omega)$ is given Leray constructed a weak solution $u\\in \\dot H^1_2(\\Omega)$, the homogeneous Sobolev space, with $u-u_\\infty\\in L^6(\\Omega)$.
We show that if in addition $f\\in \\dot H^{-1}_q(\\Omega)$ for some $q\\in (4/3,4)$ then the weak solution has the property $u-u_\\infty\\in L^{4q/(4-q)}(\\Omega)$.
This additional integrability implies that $u$ satisfies the energy identity. Further consequences are uniqueness results for small $u_\\infty$ and $f$ and continuous dependence of the solution with respect to $u_\\infty$.
The presented results are joint work with Hyunseok Kim and Hideo Kozono.
