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

In this note, for 0⩽m<∞0⩽m<∞ and index vectors q=(q1,q2,…,qd)q=(q1,q2,…,qd), r=(r1,r2,…,rd)r=(r1,r2,…,rd), where 12p>2, T>0T>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=∞T=∞) and critical indexes 2p+∑i=1d1qi−m=1, the space II 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].