×

zbMATH — the first resource for mathematics

Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in \(\mathbb{R}^ 3\). (English) Zbl 0865.35101
The author investigates the global solvability of the incompressible Navier-Stokes equation in \(\mathbb{R}^3\). For that purpose he introduces the homogeneous Besov spaces \(\dot B^\alpha_{p,q}\). The author proves that a global solution with initial data \(u_0\) exists and is unique in the space \(H^s(\mathbb{R}^3)\), \(s>1/2\), if \(|u_0|_{\dot B^{-1/4}_{4,\infty}}\) is smaller than some universal constant, and the solution is in \(L^p\), \(p>3/2\), if \(u_0\in L^p\cap\dot B^{-(1-3/2p)}_{2p,\infty}\) and \(|u_0|_{\dot B^{-(1-3/2p)}_{2p,\infty}}\) is smaller than some constant depending on \(p\). In the latter result \(u_0\in L^p\cap\dot B^{-(1-3/2p)}_{2p,\infty}\) may be replaced by \(u_0\in L^p\cap L^3\), and if \(p>3\) by \(L^2\cap L^p\).
Some regularity results of solutions are also established. In the proofs, the author transforms the original problem into an integral equation, and uses the exact expression of the heat kernel instead of the semigroup theory to obtain the necessary estimates.

MSC:
35Q30 Navier-Stokes equations
58D25 Equations in function spaces; evolution equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Beirä o da Vega, H., Existence and asymptotic behaviour for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. Journal, Vol. 36, 1, 149-166, (1987)
[2] Bergh, J.; Löfstrom, J., Interpolation spaces, an introduction, (1976), Springer-Verlag · Zbl 0344.46071
[3] Bony, J. M., Calcul symbolique et propagation des singularités dans LES équations aux dérivées partielles non linéaires, Ann. Sci. Ecole Norm. Sup., Vol. 14, 209-246, (1981) · Zbl 0495.35024
[4] Chemin, J.-Y., Remarques sur l’existence globale pour le système de Navier-Stokes incompressible, SIAM Journal Math. Anal., Vol. 23, 20-28, (1992) · Zbl 0762.35063
[5] Giga, Y., Solutions for semi-linear parabolic equations in L^{p} and regularity of weak solutions of the Navier-Stokes system, Journal of differential equations, Vol. 61, 186-212, (1986) · Zbl 0577.35058
[6] Giga, Y.; Miyakawa, T., Solutions in L^{r} of the Navier-Stokes initial value problem, Arch. Rat. Mech. Anal., Vol. 89, 267-281, (1985) · Zbl 0587.35078
[7] Kajikiya, R.; Miyakawa, T., On L^{2} decay of weak solutions of the Navier-Stokes equations in R^{n}, Math. Zeit., Vol. 192, 135-148, (1986) · Zbl 0607.35072
[8] Kato, T., Strong L^{P} solutions of the Navier-Stokes equations in R^{m} with applications to weak solutions, Math. Zeit., Vol. 187, 471-480, (1984) · Zbl 0545.35073
[9] Kato, T.; Fujita, H., On the non-stationnary Navier-Stokes system, Rend. Sem. Math. Univ. Padova, Vol. 32, 243-260, (1962) · Zbl 0114.05002
[10] Kato, T.; Fujita, H., On the Navier-Stokes initial value problem I, Arch. Rat. Mech. Anal., Vol. 16, 269-315, (1964) · Zbl 0126.42301
[11] Peetre, J., New thoughts on Besov spaces, (1976), Duke Univ. Math. Series, Duke University Durham · Zbl 0356.46038
[12] Serrín, J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., Vol. 9, 187-195, (1962) · Zbl 0106.18302
[13] Stein, E. M., Singular integral and differentiability properties of functions, (1970), Princeton University Press
[14] Taylor, M., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. in POE, Vol. 17, 1407-1456, (1992) · Zbl 0771.35047
[15] Triebel, H., theory of function spaces, volume 78 of monographs in mathematics, (1983), Birkhausen
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.