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Well-posedness for the Navier-Stokes equations with datum in Sobolev-Fourier-Lorentz spaces. (English) Zbl 1339.35214
Summary: In this note, for $$s \in \mathbb{R}$$ and $$1 \leq p$$, $$r \leq \infty$$, we introduce and study Sobolev-Fourier-Lorentz spaces $$\dot{H}_{\mathcal{L}^{p, r}}^s(\mathbb{R}^d)$$. In the family spaces $$\dot{H}_{\mathcal{L}^{p, r}}^s(\mathbb{R}^d)$$, the critical invariant spaces for the Navier-Stokes equations correspond to the value $$s = \frac{d}{p} - 1$$. When the initial datum belongs to the critical spaces $$\dot{H}_{\mathcal{L}^{p, r}}^{\frac{d}{p} - 1}(\mathbb{R}^d)$$ with $$d \geq 2$$, $$1 \leq p < \infty$$, and $$1 \leq r < \infty$$, we establish the existence of local mild solutions to the Cauchy problem for the Navier-Stokes equations in spaces $$L^\infty([0, T]; \dot{H}_{\mathcal{L}^{p, r}}^{\frac{d}{p} - 1}(\mathbb{R}^d))$$ with arbitrary initial value, and existence of global mild solutions in spaces $$L^\infty([0, \infty); \dot{H}_{\mathcal{L}^{p, r}}^{\frac{d}{p} - 1}(\mathbb{R}^d))$$ when the norm of the initial value in the Besov spaces $$\dot{B}_{\mathcal{L}^{\widetilde{p}, \infty}}^{\frac{d}{\widetilde{p}} - 1, \infty}(\mathbb{R}^d)$$ is small enough, where $$\widetilde{p}$$ may take some suitable values.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids
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