On the behavior of the solutions of the Navier-Stokes equations lying on invariant manifolds
The authors consider the Navier-Stokes equations \[ du/dt+(u\cdot \nabla)u=-\nabla p+\nu \Delta u,\quad \nabla \cdot u=0 \] with a periodicity condition \(u(x+Le_ j)=u(x)\), \(\forall x\in R^ N\), \(j=1,...,N\); and the initial condition \(u(0)=u_ 0\). Their main result is, that the function \(t\to \| S(t)u_ 0\|\) decreases on (0,\(\infty)\), if \(u_ 0\) belongs to the connected component of 0 in \(B_ R\cap M_{k-1}\). Here \(\| u\|^ 2=\sum^{N}_{i=1}\int_{\Omega}| \nabla u_ i|^ 2 dx\), \(B_ R=\{u\in V:\) \(\| u\| <R\}\). R is such that the eigenvalue \(\Lambda =\Lambda_ k\) of the Stokes operator fulfils the inequality \(\Lambda^{1/2}\geq 2((2c/\nu)R)\), \(M_ k\) is the spectral manifold in the set of initial data invariant under the action of the nonlinear semigroup \((S(t))_{t\geq 0}.\) Further it is shown that for large \(\Lambda\), for any initial data \(u_ 0\in M_{\Lambda}\) with \(\| u_ 0\| \geq O(\Lambda^{1/4})\) there exists a time interval with a length \(O(\Lambda^{-2/3} \log \Lambda)\) such that the solution u of the three dimensional problem arising from \(u_ 0\) decreases in V with an amplitude \(O(\Lambda^{1/4})\).
scientific article; zbMATH DE number 3962331
scientific article