Some general external forces and critical mild solutions for the fractional Navier-Stokes equations
Diego Chamorro, Maxence Mansais
TL;DR
This work studies mild solutions to the forced, incompressible fractional Navier–Stokes equations on $\mathbb{R}^d$ with $1<\alpha<2$, formulating the problem in a fixed-point framework within critical spaces. It develops three complementary resolution spaces—$L^{\infty}_\alpha$, parabolic Morrey spaces, and multiplier spaces $\mathcal{V}_\alpha$—and derives sharp estimates for the linear propagator, the bilinear nonlinearity, and the external force to obtain global solutions under small data. The main contributions are (i) a first existence result for the forced fractional NS in $L^{\infty}_\alpha$, (ii) a Morrey-space generalization with forcing in $\dot{\mathcal{W}}^{-{\gamma,\mathfrak{p},\mathfrak{q}}}$, and (iii) a multiplier-space framework yielding global solutions in $\mathcal{V}_\alpha$. The paper also clarifies the relationships and limitations among these spaces, including counterexamples showing the nontriviality of embedding forcing spaces, which informs optimal functional-analytic settings for forced fractional NS. Overall, the results extend the analysis of mild solutions to the forced fractional NS and map the landscape of critical spaces capable of sustaining global mild solutions.
Abstract
In this article we study mild solutions for the forced, incompressible fractional Navier-Stokes equations. These solutions are classically obtained via a fixed-point argument which relies on suitable estimates for the initial data, the nonlinearity and the external forces. Many functional spaces can be considered, however we are mainly interested here in a critical setting which ensures the existence of global solutions. We give some examples of such critical functional spaces and we discuss their relationship with generic external forces.