CRM: Centro De Giorgi
logo sns
New trends in Partial Differential Equations

Almost optimal initial value conditions for the Navier-Stokes equations: existence, uniqueness, continuity, and stability

speaker: Reinhard Farwig (TU, Darmstadt)

abstract: Consider weak solutions of the instationary Navier-Stokes system in a three-dimensional bounded domain \(\Omega\). It is well-known that an initial value \(u_0\in \mathcal D(A^{1/4})\) or even \(u_0\in L^3_\sigma(\Omega)\), where \(A=-P\Delta\) denotes the Stokes operator, admits a unique regular solution in Serrin's class \(L^{s_q}(0,T;L^q(\Omega))\), \(\frac2{s_q}+\frac3q = 1\), \(2e^{-\tau A}u_0\
_q\big)^{s_q}\, d\tau <\infty. \] This optimal condition can be weakened to an almost optimal condition \(u_0 \in L^2_{\sigma}(\Omega)\) with weighted finite integral \[ \int_0^\infty \big( \tau^\alpha\
e^{-\tau A}u_0\
_q\big)^s\, d\tau <\infty, \] \(s>s_q\), \(q>3,\) satisfy \(\frac2s+\frac3q = 1-2\alpha\), \(0<\alpha<\frac12\). In the case \(s=\infty\) the integral norm has to be replaced by the essential sup-norm. These conditions can be described by the scaling invariant Besov space \(\mathbb B^{-1+3/q}_{q,s}\), \(q>3\), \(s_q

In this talk we present recent results obtained by R. Farwig, Y. Giga and Pen-Yuan Hsu (Tokyo University) on existence, uniqueness and continuity as well as stability in the space \(C^0([0,T);\mathbb B^{-1+3/q}_{q,s})\).

Tue 4 Oct, 15:00 - 15:45, Aula Dini
<< Go back