×

zbMATH — the first resource for mathematics

An application of weighted Hardy spaces to the Navier-Stokes equations. (English) Zbl 1306.46040
Summary: In this article, we consider the mapping properties of convolution operators with smooth functions on weighted Hardy spaces \(H^p(w)\) with \(w\) belonging to Muckenhoupt class \(A_\infty\). As a corollary, one obtains decay estimates for the heat semigroup on weighted Hardy spaces. After a weighted version of the div-curl lemma is established, these estimates on weighted Hardy spaces are applied to the investigation of the decay property of global mild solutions to Navier-Stokes equations with the initial data belonging to weighted Hardy spaces.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
35Q30 Navier-Stokes equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Auscher, P.; Russ, E.; Tchamitchian, P., Hardy Sobolev spaces on strongly Lipschitz domains of \(\mathbb{R}^n\), J. Funct. Anal., 218, 1, 54-109, (2005) · Zbl 1073.46022
[2] Bui, H. Q., Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J., 12, 3, 581-605, (1982) · Zbl 0525.46023
[3] Coifman, R.; Lions, P. L.; Meyer, Y.; Semmes, S., Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72, 247-286, (1993) · Zbl 0864.42009
[4] Fefferman, C.; Stein, E., \(H^p\) spaces of several variables, Acta Math., 129, 137-193, (1972) · Zbl 0257.46078
[5] García-Cuerva, J., Weighted \(H^p\)-spaces, Disertationes Math., 162, (1979) · Zbl 0434.42023
[6] Gatto, A. E.; Gutiérrez, C. E.; Wheeden, R. L., Fractional integrals on weighted \(H^p\) spaces, Trans. Amer. Math. Soc., 289, 2, 575-589, (1985) · Zbl 0573.42013
[7] Giga, Y.; Miyakawa, T., Solutions in \(L^r\) of the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 89, 3, 267-281, (1985) · Zbl 0587.35078
[8] Grafakos, L., Modern Fourier analysis, Grad. Texts in Math., vol. 250, (2008), Springer-Verlag
[9] Hytönen, T.; Pérez, C., Sharp weighted bounds involving \(A_\infty\), Anal. PDE, 6, 4, 777-818, (2013) · Zbl 1283.42032
[10] Kato, T., Strong \(L^p\)-solutions of the Navier-Stokes equation in \(\mathbb{R}^n\), with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073
[11] Lee, M.-Y.; Lin, C.-C., The molecule characterization of weighted Hardy spaces, J. Funct. Anal., 188, 442-460, (2002) · Zbl 0998.42013
[12] Lemarié-Rieusset, P. G., Recent development in the Navier-Stokes problem, Chapman & Hall/CRC Res. Notes Math., vol. 431, (2002), Chapman & Hall/CRC Boca Raton, FL · Zbl 1034.35093
[13] Lerner, A. K.; Ombrosi, S.; Pérez, C., Sharp \(A_1\) bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt-Wheeden, Int. Math. Res. Not., 2008, (2008), 11 pp · Zbl 1237.42012
[14] Miyachi, A., Weighted Hardy spaces on a domain, (Proceedings of the Second ISAAC Congress, vol. 1, Fukuoka, 1999, Int. Soc. Anal. Appl. Comput., vol. 7, (2000), Kluwer Acad. Publ. Dordrecht), 59-64 · Zbl 1055.46503
[15] Miyachi, A., Hardy space estimate for the product of singular integrals, Canad. J. Math., 52, 2, 381-411, (2000) · Zbl 0952.42007
[16] Miyakawa, T., Hardy spaces of solenoidal vector fields, with applications to the Navier-Stokes equations, Kyushu J. Math., 50, 1, 1-64, (1996) · Zbl 0883.35088
[17] Miyakawa, T., Application of Hardy space techniques to the time-decay problem for incompressible Navier-Stokes flows in \(\mathbb{R}^n\), Funkcial. Ekvac., 41, 3, 383-434, (1998) · Zbl 1142.35544
[18] Strömberg, J.-O.; Torchinsky, A., Weighted Hardy spaces, Lecture Notes in Math., vol. 1381, (1989), Springer-Verlag Berlin · Zbl 0676.42021
[19] Taibleson, M. H.; Weiss, G., The molecular characterization of certain Hardy spaces. representation theorems for Hardy spaces, Astérisque, vol. 77, 67-149, (1980), Soc. Math. France Paris · Zbl 0472.46041
[20] Tsutsui, Y., The Navier-Stokes equations and weak Herz spaces, Adv. Differential Equations, 16, 11-12, 1049-1085, (2011) · Zbl 1236.35114
[21] Wiegner, M., Decay results for weak solutions of the Navier-Stokes equations on \(\mathbb{R}^n\), J. London Math. Soc. (2), 35, 2, 303-313, (1987) · Zbl 0652.35095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.