×

Solutions for semilinear parabolic equations in \(L^ p\) and regularity of weak solutions of the Navier-Stokes system. (English) Zbl 0577.35058

See the preview in Zbl 0549.35063.

MSC:

35K55 Nonlinear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35Q30 Navier-Stokes equations

Citations:

Zbl 0549.35063
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Baras, P., Non-unicité des solutions d’une équation d’évolution non-linéaire, Ann. Fac. Sci. Toulouse Math., 5, 287-302 (1983) · Zbl 0553.35046
[2] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 771-831 (1982) · Zbl 0509.35067
[3] Fabes, E. B.; Jonesand, B. F.; Riviere, N. M., The initial value problem for the Navier-Stokes equations with data in \(L^p\), Arch. Rational Mech. Anal., 45, 222-240 (1972) · Zbl 0254.35097
[4] Fabes, E. B.; Lewis, J. E.; Riviere, N. M., Boundary value problems for the Navier-Stokes equations, Amer. J. Math., 99, 626-668 (1977) · Zbl 0386.35037
[5] Foias, C.; Temam, R., Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pure Appl., 58, 339-368 (1979) · Zbl 0454.35073
[6] Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t = Δu + u^{1 + α} \), J. Fac. Sci. Univ. Tokyo, 13, 109-124 (1966), Sect. I · Zbl 0163.34002
[7] Fujita, H., On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, (Proc. Symp. Pure Math., 18 (1968)), 131-161, Part I
[8] Giga, Y.; Miyakawa, T., Solutions in \(L_r\) of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89, 267-281 (1985) · Zbl 0587.35078
[9] Giga, Y., Weak and strong solutions of the Navier-Stokes initial value problem, Publ. Res. Inst. Math. Sci., 19, 887-910 (1983) · Zbl 0547.35101
[10] Giga, Y., Regularity criteria for weak solutions of the Navier-Stokes system, (Proceedings, Amer. Math. Soc. Summer Inst. (1983)), in press · Zbl 0598.35094
[11] Giga, Y.; Kohn, R., Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38, 297-319 (1985) · Zbl 0585.35051
[12] Hopf, E., Über die Anfangwertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4, 213-231 (1951) · Zbl 0042.10604
[13] Kaniel, S.; Shinbrot, M., Smoothness of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 24, 302-324 (1967) · Zbl 0152.44902
[14] Kato, T.; Fujita, H., On the nonstationary Navier-Stokes system, (Rend. Sem. Mat. Univ. Padova, 32 (1962)), 243-260 · Zbl 0114.05002
[15] Kato, T., Strong \(L^p\)-solutions of the Navier-Stokes equation in \(R^m\) with applications to weak solutions, Math. Z., 187, 471-480 (1984) · Zbl 0545.35073
[16] Ladyzhenskaya, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and Quasilinear Equations of Parabolic Type, (Translations of Mathematical Monographs, Vol. 23 (1968), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · Zbl 0164.12302
[17] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05
[18] Ni, W. M.; Sacks, P., Singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc., 287, 657-671 (1985) · Zbl 0573.35046
[19] Reed, M.; Simon, B., Fourier Analysis, (Methods of Modern Mathematical Physics, Vol. 2 (1972), Academic Press: Academic Press New York)
[20] Scheffer, V., Turbulence and Hausdorff dimension, (Turbulence and Navier-Stokes Equations. Turbulence and Navier-Stokes Equations, Lecture Notes in Math., Vol. 565 (1976), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg), 94-112
[21] Serrin, J., The initial value problem for the Navier-Stokes equations, (Langer, R. E., Nonlinear Problems (1963), Univ. of Wisconsin Press: Univ. of Wisconsin Press Madison), 69-98
[22] Sohr, H., Zur regularitätstheorie der instationären Gleichungen von Navier-Stokes, Math. Z., 184, 359-375 (1983) · Zbl 0506.35084
[23] Sohr, H.; von Wahl, W., On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations, Manuscripta Math., 49, 27-59 (1984) · Zbl 0567.35069
[24] Solonnikov, V. A., Estimates for solutions of nonstationary Navier-Stokes equations, J. Soviet Math., 8, 467-529 (1977) · Zbl 0404.35081
[25] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J
[26] Temam, R., Navier-Stokes Equations (1977), North-Holland: North-Holland Amsterdam · Zbl 0335.35077
[27] von Wahl, W., Regularity of weak solutions of the Navier-Stokes equations, (Proceedings, Amer. Math. Soc. Summer Inst. (1983)), in press · Zbl 0643.35083
[28] Weissler, F. B., Semilinear evolution equations in Banach spaces, J. Funct. Anal., 32, 277-296 (1979) · Zbl 0419.47031
[29] Weissler, F. B., Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. Math. J., 29, 79-102 (1980) · Zbl 0443.35034
[30] Weissler, F. B., The Navier-Stokes initial value problem in \(L^p\), Arch. Rational Mech. Anal., 74, 219-230 (1981) · Zbl 0454.35072
[31] Weissler, F. B., Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38, 29-40 (1981) · Zbl 0476.35043
[32] Masuda, K., Weak solutions of the Navier-Stokes equations, Tôhoku Math. J., 36, 623-646 (1984) · Zbl 0568.35077
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.