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

In this note, for 0 ⩽ m < ∞ and index vectors q = ( q 1 , q 2 , … , q d ) , r = ( r 1 , r 2 , … , r d ) , where 1 < q i < ∞ , 1 ⩽ r i ⩽ ∞ , and 1 ⩽ i ⩽ d , we introduce and study mixed-norm Sobolev–Lorentz spaces H ˙ L q , r m , which are more general than the classical Sobolev spaces H ˙ q m . Then we investigate the existence and uniqueness of solutions to the Navier–Stokes equations (NSE) in the spaces L p ( [ 0 , T ] ; H ˙ L q , r m ) where p > 2 , T > 0 , and the initial datum is taken in the space I = { u 0 ∈ ( S ′ ( R d ) ) d : div ( u 0 ) = 0 , ‖ e t Δ u 0 ‖ L p ( [ 0 , T ] ; H ˙ L q , r m ) < ∞ } . 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 2 p + ∑ i = 1 d 1 q i − m = 1 , the space I coincides with the homogeneous Besov space B ˙ L q , r m − 2 p , p . In the case when m = 0 , q 1 = q 2 = ⋯ = q d = r 1 = r 2 = ⋯ = r d , our results recover those of Fabes, Jones and Riviere [10].