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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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##### References:
  Adams, A. R., Sobolev spaces, (1979), Academic Press Boston, MA  Adams, A. R., Reduced Sobolev inequalities, Canad. Math. Bull., 31, 2, 159-167, (1988) · Zbl 0662.46036  Benedek, A.; Panzone, R., The space $$L^p$$ with mixed norm, Duke Math. J., 28, 301-324, (1961) · Zbl 0107.08902  Bergh, J.; Lofstrom, J., Interpolation spaces, (1976), Springer-Verlag · Zbl 0344.46071  Bourgain, J.; Pavloviéc, N., Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255, 9, 2233-2247, (2008) · Zbl 1161.35037  Cannone, M., Ondelettes, paraproduits et Navier-Stokes, (1995), Diderot Editeur Paris · Zbl 1049.35517  Chemin, J.-F.; Lerner, N., Flot de champs de vecteurs non lipschitziens et equations de Navier-Stokes, J. Differential Equations, 121, 2, 314-328, (1995) · Zbl 0878.35089  Chemin, J.-F.; Desjardin, B.; Gallagher, I.; Grenier, E., Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., 34, 2, 315-335, (2000), special issue for R. Temam’s 60th birthday · Zbl 0954.76012  Chemin, J.-F.; Paicu, M.; Zhang, P., Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable, J. Differential Equations, 256, 223-252, (2014) · Zbl 1317.35171  Fabes, E.; Jones, B.; Riviere, N., The initial value problem for the Navier-Stokes equations with data in $$L^p$$, Arch. Ration. Mech. Anal., 45, 222-240, (1972) · Zbl 0254.35097  Federbush, P., Navier and Stokes meet the wavelet, Comm. Math. Phys., 155, 219-248, (1993) · Zbl 0795.35080  Furioli, G.; Lemarie-Rieusset, P. G.; Terraneo, E., Unicité dans $$L^3(\mathbb{R}^3)$$ et d’autres espaces limites pour Navier-Stokes, Rev. Mat. Iberoam., 16, 605-667, (2000) · Zbl 0970.35101  Ferreira, L. C.F.; Medeiros, E. S.; Montenegro, M., A class of elliptic equations in anisotropic spaces, Ann. Mat. Pura Appl. (4), 192, 539-552, (2013) · Zbl 1276.35093  Fujita, H.; Kato, T., On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16, 269-315, (1964) · Zbl 0126.42301  Giga, Y., Solutions of semilinear parabolic equations in $$L^p$$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62, 186-212, (1986) · Zbl 0577.35058  Iftimie, D., The resolution of the Navier-Stokes equations in anisotropic spaces, Rev. Mat. Iberoam., 15, 1, 1-36, (1999) · Zbl 0923.35119  Iftimie, D., A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity, SIAM J. Math. Anal., 33, 6, 1483-1493, (2002) · Zbl 1011.35105  Kato, T., Strong $$L^p$$ solutions of the Navier-Stokes equations in $$\mathbb{R}^m$$ with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073  Kato, T., Strong solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat., 22, 127-155, (1992) · Zbl 0781.35052  Kenig, C. E.; Koch, G. S., An alternative approach to regularity for the Navier-Stokes equations in critical spaces, Ann. Inst. H. Poincare Anal. Non Lineaire, 28, 159-187, (2011) · Zbl 1220.35119  Koch, H.; Tataru, D., Well-posedness for the Navier-Stokes equations, Adv. Math., 157, 1, 22-35, (2001) · Zbl 0972.35084  Kozono, H.; Yamazaki, Y., Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data, Comm. Partial Differential Equations, 19, 959-1014, (1994) · Zbl 0803.35068  Lemarie-Rieusset, P. G., Recent developments in the Navier-Stokes problem, Chapman and Hall/CRC Research Notes in Mathematics, vol. 431, (2002), Chapman and Hall/CRC Boca Raton, FL · Zbl 1034.35093  Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9, 187-195, (1962) · Zbl 0106.18302  Taylor, M. E., Analysis on Morrey spaces and applications to Navier-Stokes equations and other evolution equations, Comm. Partial Differential Equations, 17, 1407-1456, (1992) · Zbl 0771.35047
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