Stability of solutions of the Navier-Stokes equations backward in time (Q1111853)

The authors use a weighted logarithmic convexity argument to study the stability of solutions of the Navier-Stokes equations, backward in time, \(v_{i,t}=v_ jv_{i,j}-\Delta v_ i+P_ i\), on \(\Omega\) \(\times (0,T]\), \(v_{i,i}=0\) on \(\Omega\) \(\times (0,T]\), \(v_ i(x,0)=b_ i(x)\) on \(\Omega\), \(v_ i(x,t)=h_ i(x,t)\) on \(\Gamma\). Here \(\Omega\) is an exterior region in \({\mathbb{R}}^ 3\) with interior boundary \(\Gamma\), \(\underset \tilde{} b\) and \(\underset \tilde{} h\) are given, P is the pressure, \(\underset \tilde{} v\) is the velocity field and \(T<\infty\) is constant. In particular, they obtain pointwise continuous dependence results assuming that the solution \(u_ i\) satisfes \(u_ i\in L^{6-\epsilon}(\Omega)\) and \(u_{i,t}\in L^{6-\epsilon}(\Omega)\), for any \(\epsilon >0\), with \(P_ i=O(r^{1-\epsilon})\) and \(\nabla P=O(r^{1/2-\epsilon})\).