Local-in-Time Solvability and Space Analyticity for the Navier–Stokes Equations with BMO-Type Initial Data

Abstract

It is proved that there exists a local-in-time solution u ∈ C([0, T ), bmo(R d ) d ) of the Navier-Stokes equations such that every u(t) has an analytic extension on a complex domain whose size only depends on t (and increases with t) and the external force f , assuming only that the initial velocity u 0 is a local BM O function. Our method for proving is a combination and refinement of the work by Grujić and Kukavica [13], Guberović [15] and Kozono et al. [18]. One challenging step is the estimation of the heat and Stokes semigroups from BM O-type spaces to L ∞ ; a result itself of independent interest. We also apply the idea to the analyticity of vorticity with the assistance of Calderón-Zygmund theory.

On the local pressure expansion for the Navier-Stokes equations

“…Note that ũ ∈ C wk ([t 0 , T 0 ); L ∞ ), which is a uniqueness class for mild solutions, see [13] and also [25,33]. Therefore ũ agrees with the strong solutions constructed in [13,25,24,39] for initial data u(t 0 ) on R 3 × (t 0 , T 1 ), for some T 1 ∈ (t 0 , T 0 ] with T 1 − t 0 ≥ C( u L ∞ (t 0 , 1 2 (t 0 +T 0 ); L ∞ ) ) > 0 if T 1 ≤ 1 2 (t 0 + T 0 ). This implies (t − t 0 ) 1/2 ∇ũ ∈ L ∞ (t 0 , T 1 (t 0 ); L ∞ ) (see [24]) and, since this is true for all t 0 , it follows that ∇u ∈ L ∞ loc ((0, T 0 ); L ∞ ).…”

Joint space-time analyticity of mild solutions to the Navier-Stokes equations

“…The above estimates yield the time analyticity without the bounds on the growth of radius of time analyticity. One can also find space analyticity results in [1] for initial data in critical Besov space in Ḃ 3 p −1 p,q (R 3 ) with 1 ≤ p < ∞ and 1 ≤ q ≤ ∞, and in [28]…”

“…Here, u is the R 3 -valued velocity field, p stands for the scalar pressure, and u 0 is an initial datum in the critial space L 3 (R 3 ). The space analyticity of the classical solutions to the Navier-Stokes equations is usually expected as a consequence of parabolic regularity; see, e.g., [17,14,1,28]. The time analyticity of the solutions to the Navier-Stokes equations can be obtained via analytic semigroup properties and complex variables [11,13].…”

Time analyticity for inhomogeneous parabolic equations and the Navier-Stokes equations in the half space

“…For the Navier-Stokes equations, the space analyticity of solutions has been studied extensively in the literature. See, for example, [17,15,14,7,1,3,27] and the references therein. There are also many work regarding the time analyticity for the Navier-Stokes equations.…”