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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Journal of Differential Equations
( Vol. 149 NO.2 / (1998))
Boundary Regularity of Weak Solutions of the Navier-Stokes Equations

Pages. 211-247
Abstract We prove that a solution to Navier-Stokes equations is in $L^2(0,infty: H^2(Omega))$ under the critical assumption that $u in L^{r,r^prime},3/r + 2/r^prime leqslant 1$ with $r geqslant 3$. A boundary $L^infty$ estimate for the solution is derived if the pressure on the boundary is bounded. Here our estimate is local. Indeed, employing Moser type iteration and the reverse Holder inequality, we find an integral estimate for $L^infty$-norm of $u$. Moreover the solution is $C^{1,alpha}$ continuous up to boundary if the tangential derivatives of the pressure on the boundary are bounded. Then, from the bootstrap argument a local higher regularity theorem follows, that is, the velocity is as regular as the boundary data of the pressure. 1. Introduction and statement of the result 2. $L^\infty(0,\infty : H^1(\Omega)) \cap L^2(0,\infty : H^2(\Omega))$ regularity 3. Estimate of pressure 4. $L^\infty$ estimate of velocity 5. Higher regularity