User jack - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:56:18Z http://mathoverflow.net/feeds/user/8901 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58227/how-to-solve-the-linearized-navier-stokes-equations-in-lp How to solve the linearized Navier-Stokes equations in L^P? jack 2011-03-12T03:33:17Z 2012-06-06T00:37:00Z <p>Let $\Omega\subset \mathbb{R}^3$ be an open set with smooth boundary $\partial \Omega$. Consider the following linearized Navier-Stokes equations in $Q_T=\Omega\times (0,T)$ for an arbitrarily fixed $T\in (0,\infty)$, $$u_t-\Delta u+a(x,t)u+b\cdot \nabla u+\nabla p=f(x,t),\text{div } u=0$$ with the initial and boundary conditions $u(x,0)=0, \left.u(x,t)\right|_{\partial \Omega\times (0,T)}=0$. Here $u(x,t)=(u^1(x,t),u^2(x,t),u^3(x,t))$ and $p(x,t)$ denote the unknown velocity and pressure respectively, $a(x,t)$ and $b(x,t)$ denote the given coefficients.</p> <p>Question: Suppose that $$a\in L^r(0,T; L^s(\Omega)), b\in L^{r_1}(0,T; L^{s_1}(\Omega)),$$ where $2/r+3/s&lt;2$, $2/r_1+3/s_1&lt;1$, and $f(x,t)\in C_0^\infty(\Omega\times (0,T))$, can we solve the above equations in arbitrary $L^p$? Can we get the estimates such as $$\|u_t\|_{L^p(Q_T)}+\|D^2 u\|_{L^p(Q_T)}+\|u\|_{L^p(Q_T)}\leq \|f\|_{L^p(Q_T)}?$$</p> <p>Solonnikov dealed with this problem in his paper "Estimates for solution of nonstationary Navier-Stokes equations" (http://www.springerlink.com/index/N8374858XNT22P11.pdf). However, I can not verify his proof (Page 487 to Page 489).</p> <p>Who can help me? Any comment will be deeply appreciated.</p> http://mathoverflow.net/questions/58533/is-this-interpolation-inequality-right Is this (interpolation) inequality right? jack 2011-03-15T13:39:50Z 2011-03-19T06:38:43Z <p>Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap (\cap_{k=1}^\infty L^{5/3}(0,T;C^k(\bar{\Omega})))$.<br> <strong>Question</strong>: Can we say that $F$ is bounded in $L^q(0,T;C^2(\bar{\Omega}))$ for any $q\in [1,\infty)$?</p> http://mathoverflow.net/questions/58533/is-this-interpolation-inequality-right/58628#58628 Comment by jack jack 2011-03-18T16:54:18Z 2011-03-18T16:54:18Z Thanks. The only problem is the proof of &quot;Proposition 1&quot;. How to prove? Or any reference? http://mathoverflow.net/questions/58533/is-this-interpolation-inequality-right Comment by jack jack 2011-03-16T01:30:08Z 2011-03-16T01:30:08Z Thanks for you all. $F$ is bounded in $L^\infty (0,T;\Omega)$ and $L^{5/3}(0,T;C^k(\bar{\Omega}))$ for all $k\in\mathbb{N}_0$. I have edit the statement.