1 May 2018
Communications in Partial Differential Equations
© 2018 Taylor & Francis We consider the Cauchy problem for the Navier–Stokes equation in ℝ3×]0,∞[ with the initial datum (Formula presented.), a critical space containing nontrivial (−1)−homogeneous fields. For small (Formula presented.) one can get global well-posedness by perturbation theory. When (Formula presented.) is not small, the perturbation theory no longer applies and, very likely, the local-in-time well-posedness and uniqueness fails. One can still develop a good theory of weak solutions with the following stability property: If u(n)are weak solutions corresponding the the initial datum (Formula presented.), and (Formula presented.) converge weakly* in (Formula presented.) to u0, then a suitable subsequence of u(n)converges to a weak solution u corresponding to the initial condition u0. This is of interest even in the special case u0≡0.
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