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| The classical theory of viscous incompressible fluid flow is governed by the celebrated Navier-Stokes equations.
There are two main approaches for the construction of solutions to the Navier-Stokes equations. In the 1934 pioneering paper by Leray, weak solutions are obtained for all divergence free initial data $u_0 \in L^2 (\bbfR^3)^3$ and $F=0$. The second approach leads to mild solutions. These solutions are given by an integral formulation using the Duhamel principle and they are obtained by means of the Banach contraction principle. \\
I will describe a link between these two approaches. That is I will show that the initial value problem with data $V(x, 0)$ perturbed by an arbitrarily large divergence free $L^2$-vector field has a global-in-time weak solution in the sense of Leray and this weak solution
converges as $t\rightarrow \infty $ in the energy $L^2$-norm towards the mild solution $V = V (x, t)$. That is sufficiently small mild solutions of the Navier-Stokes equations are asymptotically stable weak solutions under all divergence free initial perturbations from $L^2 (\bbfR^3)^3$. |
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