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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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2007/09/05
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Reinhard Farwig (Darmstadt University of Technology)
Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Reinhard Farwig (Darmstadt University of Technology)
Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion
[ Abstract ]
Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.
The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.
[ Reference URL ]Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.
The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
