On the Radius of Analyticity of Solutions to 3D Navier-Stokes System with Initial Data in L ^p

Abstract

Given initial data u ₀ ∈ L ^p (ℝ³) for some p in $\left[ {3,{{18} \over 5}} \right[$, the auhtors first prove that 3D incompressible Navier-Stokes system has a unique solution u = u _L+v with ${u_L}\mathop = \limits^{{\rm{def}}} \,{{\rm{e}}^{t\Delta }}{u_0}$ and $v \in {{\tilde L}^\infty }\left( {\left[ {0,T} \right];{{\dot H}^{{5 \over 2} - {6 \over p}}}} \right) \cap {{\tilde L}^1}\left( {\left] {0,T} \right[;{{\dot H}^{{9 \over 2} - {6 \over p}}}} \right)$ for some positive time T. Then they derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in [Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631–1643, Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033] with initial data in Ḣ ^s(ℝ³) for $s \in \left[ {{1 \over 2},{3 \over 2}} \right[$.

Keywords

Incompressible Navier-Stokes equations / Radius of analyticity / Littlewood-Paley theory

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