TY - JOUR

T1 - On stability of weak Navier-Stokes solutions with large L3,∞ initial data.

AU - Barker, Tobias

AU - Seregin, Gregory

AU - Šverák, Vladimir

PY - 2018/5/7

Y1 - 2018/5/7

N2 - We consider the Cauchy problem for the Navier–Stokes equation in ℝ3×]0,∞[ with the initial datum u0∈L3weak, a critical space containing nontrivial (−1)−homogeneous fields. For small ||u0||L3weak one can get global well-posedness by perturbation theory. When ||u0||L3weak 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 u(n)0, and u(n)0 converge weakly* in L3weak 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.

AB - We consider the Cauchy problem for the Navier–Stokes equation in ℝ3×]0,∞[ with the initial datum u0∈L3weak, a critical space containing nontrivial (−1)−homogeneous fields. For small ||u0||L3weak one can get global well-posedness by perturbation theory. When ||u0||L3weak 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 u(n)0, and u(n)0 converge weakly* in L3weak 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.

U2 - 10.1080/03605302.2018.1449219

DO - 10.1080/03605302.2018.1449219

M3 - Article

SN - 0360-5302

VL - 43

SP - 628

EP - 651

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 4

ER -