Solutions in mixed-norm Sobolev–Lorentz spaces to the initial value problem for the Navier–Stokes equations

Abstract

In this note, for 0⩽m<∞ and index vectors q=(q1,q2,…,qd), r=(r1,r2,…,rd), where 1<qi<∞, 1⩽ri⩽∞, and 1⩽i⩽d, we introduce and study mixed-norm Sobolev–Lorentz spaces H˙Lq,rm, which are more general than the classical Sobolev spaces H˙qm. Then we investigate the existence and uniqueness of solutions to the Navier–Stokes equations (NSE) in the spaces Lp([0,T];H˙Lq,rm) where p>2, T>0, and the initial datum is taken in the spaceI={u0∈(S′(Rd))d:div(u0)=0,‖etΔu0‖Lp([0,T];H˙Lq,rm)<∞}. The results have a standard relation between existence time and data size: large time with small datum or large datum with small time. In the case of global solutions (T=∞) and critical indexes 2p+∑i=1d1qi−m=1, the space I coincides with the homogeneous Besov space B˙Lq,rm−2p,p. In the case whenm=0,q1=q2=⋯=qd=r1=r2=⋯=rd, our results recover those of Fabes, Jones and Riviere [10].