THE NAVIER-STOKES EQUATIONS WITH INITIAL VALUES IN BESOV SPACES OF TYPE B-1q,∞+3/q
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We consider weak solutions of the instationary Navier-Stokes system in a smooth bounded domain $\Om \subset {\mathbb R}^3$ with initial value $u_0\in L^{2}_{\sigma}(\Om)$. It is known that a weak solution is a local strong solution in the sense of Serrin if $u_0$ satisfies the optimal initial value condition $u_0\in B^{-1+3/q}_{q,s_q}$ with Serrin exponents $s_q>2, q>3$ such that $\frac2{s_q} + \frac3q =1$. This result has recently been generalized by the authors to weighted Serrin conditions such that $u$ is contained in the weighted Serrin class $\int_0^T (\tau^\alpha \|u(\tau)\|_q)^s\dd\tau<\infty$ with $\frac{2}{s} + \frac{3}{q} =1-2\alpha$, $0<\alpha<\frac12$. This regularity is guaranteed if and only if $u_0$ is contained in the Besov space $B^{-1+3/q}_{q,s}$. In this article we consider the limit case of initial values in the Besov space $B^{-1+3/q}_{q,\infty}$ and in its subspace ${\stackrel{\circ}{B}}{}^{-1+3/q}_{q,\infty}$ based on the continuous interpolation functor. Special emphasis is put on questions of uniqueness within the class of weak solutions.
Abstract
We consider weak solutions of the instationary Navier-Stokes system in a smooth bounded domain $\Om \subset {\mathbb R}^3$ with initial value $u_0\in L^{2}_{\sigma}(\Om)$. It is known that a weak solution is a local strong solution in the sense of Serrin if $u_0$ satisfies the optimal initial value condition $u_0\in B^{-1+3/q}_{q,s_q}$ with Serrin exponents $s_q>2, q>3$ such that $\frac2{s_q} + \frac3q =1$. This result has recently been generalized by the authors to weighted Serrin conditions such that $u$ is contained in the weighted Serrin class $\int_0^T (\tau^\alpha \|u(\tau)\|_q)^s\dd\tau<\infty$ with $\frac{2}{s} + \frac{3}{q} =1-2\alpha$, $0<\alpha<\frac12$. This regularity is guaranteed if and only if $u_0$ is contained in the Besov space $B^{-1+3/q}_{q,s}$. In this article we consider the limit case of initial values in the Besov space $B^{-1+3/q}_{q,\infty}$ and in its subspace ${\stackrel{\circ}{B}}{}^{-1+3/q}_{q,\infty}$ based on the continuous interpolation functor. Special emphasis is put on questions of uniqueness within the class of weak solutions.
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