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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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[1] Ahn, C.; Cho, Y., Lorentz space extension of Strichartz estimates, Proc. Amer. Math. Soc., 133, 12, 3497-3503, (2005) · Zbl 1131.35007
[2] Bergh, J.; Lofstrom, J., Interpolation spaces, (1976), Springer-Verlag, 264 pp · Zbl 0344.46071
[3] 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
[4] Cannone, M., Ondelettes, paraproduits et Navier-Stokes, (1995), Diderot Editeur Paris, 191 pp · Zbl 1049.35517
[5] Cannone, M., A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 13, 3, 515-541, (1997) · Zbl 0897.35061
[6] Cannone, M.; Meyer, Y., Littlewood-Paley decomposition and the Navier-Stokes equations, Methods Appl. Anal., 2, 307-319, (1995) · Zbl 0842.35074
[7] Cannone, M.; Planchon, F., On the nonstationary Navier-Stokes equations with an external force, Adv. Differential Equations, 4, 5, 697-730, (1999) · Zbl 0952.35089
[8] Chemin, J. M., Remarques sur l’existence globale pour le système de Navier-Stokes incompressible, SIAM J. Math. Anal., 23, 20-28, (1992) · Zbl 0762.35063
[9] 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
[10] Fujita, H.; Kato, T., On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16, 269-315, (1964) · Zbl 0126.42301
[11] 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
[12] Giga, Y.; Miyakawa, T., Solutions in \(L^r\) of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 89, 267-281, (1985) · Zbl 0587.35078
[13] Hajaiej, H.; Yu, X.; Zhai, Z., Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms, J. Math. Anal. Appl., 396, 2, 569-577, (2012) · Zbl 1254.26010
[14] Hörmander, L., Linear partial differential operators, (1976), Springer Verlag Berlin, Heidelberg, New York
[15] 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
[16] Kato, T., Strong solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat., 22, 127-155, (1992) · Zbl 0781.35052
[17] Kato, T.; Fujita, H., On the non-stationary Navier-Stokes system, Rend. Semin. Mat. Univ. Padova, 32, 243-260, (1962) · Zbl 0114.05002
[18] Kato, T.; Ponce, G., The Navier-Stokes equations with weak initial data, Int. Math. Res. Not. IMRN, 10, 435-444, (1994) · Zbl 0837.35116
[19] Khai, D. Q.; Tri, N. M., Solutions in mixed-norm Sobolev-Lorentz spaces to the initial value problem for the Navier-Stokes equations, J. Math. Anal. Appl., 417, 819-833, (2014) · Zbl 1308.35166
[20] Khai, D. Q.; Tri, N. M., On the Hausdorff dimension of the singular set in time for weak solutions to the nonstationary Navier-Stokes equation on torus, Vietnam J. Math., 43, 2, 283-295, (2015) · Zbl 1326.35242
[21] Khai, D. Q.; Tri, N. M., On the initial value problem for the Navier-Stokes equations with the initial datum in critical Sobolev and Besov spaces, J. Math. Sci. Univ. Tokyo, (2016), in press; preprint · Zbl 1342.35220
[22] Khai, D. Q.; Tri, N. M., Well-posedness for the Navier-Stokes equations with data in Sobolev-Lorentz spaces, preprint · Zbl 1358.35093
[23] Koch, H.; Tataru, D., Well-posedness for the Navier-Stokes equations, Adv. Math., 157, 1, 22-35, (2001) · Zbl 0972.35084
[24] Le Jan, Y.; Sznitman, A. S., Cascades aléatoires et équations de Navier-Stokes, C. R. Acad. Sci. Paris, Sér. I, 324, 823-826, (1997) · Zbl 0876.35083
[25] Lei, Zh.; Lin, F. H., Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64, 1297-1304, (2011) · Zbl 1225.35165
[26] Lemarie-Rieusset, P. G., Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Res. Notes Math., vol. 431, (2002), Chapman and Hall/CRC Boca Raton, FL · Zbl 1034.35093
[27] Mazzucato, A. L., Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355, 4, 1297-1364, (2003) · Zbl 1022.35039
[28] 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
[29] Wang, B., Ill-posedness for the Navier-Stokes equations in critical Besov spaces \(\dot{B}_{\infty, q}^{- 1}\), Adv. Math., 268, 350-372, (2015) · Zbl 1316.35232
[30] Weissler, F. B., The Navier-Stokes initial value problem in \(L^p\), Arch. Ration. Mech. Anal., 74, 219-230, (1981) · Zbl 0454.35072
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