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Solutions in mixed-norm Sobolev-Lorentz spaces to the initial value problem for the Navier-Stokes equations. (English) Zbl 1308.35166
Summary: In this note, for \(0\leq m<\infty\) and index vectors \(\mathbf q=(q_1,q_2,\dots,q_d)\), \(\mathbf r=(r_1,r_2,\dots,r_d)\), where \(1<q_i<\infty\), \(1\leq r_i\leq\infty\), and \(1\leq i\leq d\), we introduce and study mixed-norm Sobolev-Lorentz spaces \(\dot H_{L^{q,r}}^m\), which are more general than the classical Sobolev spaces \(\dot H_q^m\). Then we investigate the existence and uniqueness of solutions to the Navier-Stokes equations (NSE) in the spaces \(L^p([0,T];\dot H_{L^{q,r}}^m)\) where \(p>2\), \(T>0\), and the initial datum is taken in the space \[ \mathcal I=\{u_0\in(\mathcal S'(\mathbb R^d))^d:\operatorname{div}(u_0)=0,\quad ||e^{t\Delta}u_0||_{L^p([0,T];\dot H_{L^{q,r}}^m)}<\infty\}. \] 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 \((T=\infty)\) and critical indexes \(\frac{2}{p}+\sum_{i=1}^d\frac{1}{q_i}-m=1\), the space \(\mathcal I\) coincides with the homogeneous Besov space \(\dot B_{L^{q,r}}^{m-\frac{2}{p},p}\). In the case when \[ m=0,\quad q_1=q_2=\cdots=q_d=r_1=r_2=\cdots=r_d, \] our results recover those of E. B. Fabes et al. [Arch. Ration. Mech. Anal. 45, 222–240 (1972; Zbl 0254.35097)].

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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