Baaske, Franka Heat and Navier-Stokes equations in supercritical function spaces. (English) Zbl 1326.35232 Rev. Mat. Complut. 28, No. 2, 281-301 (2015). Let \[ \partial_t \mathbf{u} - \Delta \mathbf{u} + \mathbb{P} \operatorname{div} (\mathbf{u} \otimes \mathbf{u}) =0 \] be the usual Navier-Stokes equations, where \(\mathbb{P}\) stands for the Leray projector, subject to the initial data \(u(\cdot,0) = u_0\). The paper deals with unique strong solutions of these equations and, as a forerunner, non-linear heat equations, in \(\mathbb{R}^n \times (0,T)\) in terms of distinguished inhomogeneous spaces \(A^s_{p,q} (\mathbb{R}^n)\), \(A\in \{B,F \}\), of Besov-Triebel-Lizorkin type. Reviewer: Hans Triebel (Jena) Cited in 4 Documents MSC: 35Q30 Navier-Stokes equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35K05 Heat equation 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35D35 Strong solutions to PDEs Keywords:heat equations; Navier-Stokes equations; strong solutions; function spaces of Besov-Triebel-Lizorkin type PDFBibTeX XMLCite \textit{F. Baaske}, Rev. Mat. Complut. 28, No. 2, 281--301 (2015; Zbl 1326.35232) Full Text: DOI References: [1] Browder, F.E.: Nonlinear equations of evolution. Ann. Math. 80, 485-523 (1964) · Zbl 0127.33602 · doi:10.2307/1970660 [2] Cannone, M.: Harmonic analysis tool for solving the incompressible Navier-Stokes equations. In: Friedlander, S.J., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. III, pp. 161-244. North Holland, Amsterdam (2004) · Zbl 1222.35139 [3] Fujita, H., Kato, T.: On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16, 269-315 (1964) · Zbl 0126.42301 · doi:10.1007/BF00276188 [4] Kato, T.: Nonlinear evolution equations in Banach spaces. In: Proceedings of the Symposium on Applied Mathematics, vol. 17, pp. 50-67. Am. Math. Soc. (1965) · Zbl 0173.17104 [5] Kato, T.: Strong \[L^p\] Lp-solutions of the Navier-Stokes equation in \[\mathbb{R}^nRn\], with applications to weak solutions. Math. Z. 187, 471-480 (1984) · Zbl 0545.35073 · doi:10.1007/BF01174182 [6] Lemarié-Rieusset, P.G.: Recent developements in the Navier-Stokes problem. In: CRC Research Notes in Math., vol. 431, pp. 269-315. Chapman & Hall, Boca Raton (2002) · Zbl 1034.35093 [7] Miao, C., Yuan, B., Zhang, B.: Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Anal. Theory Methods Appl. 68, 461-484 (2008) · Zbl 1132.35047 · doi:10.1016/j.na.2006.11.011 [8] Sickel, W., Triebel, H.: Hölder inequalities and sharp embeddings in function spaces of \[B_{pq}^s\] Bpqs and \[F_{pq}^s\] Fpqs type. Z. Anal. Anwend. 14, 105-140 (1995) · Zbl 0820.46030 · doi:10.4171/ZAA/666 [9] Triebel, H.: Theory of Functions Spaces. Birkhäuser, Basel (1983) · Zbl 1235.46002 · doi:10.1007/978-3-0346-0416-1 [10] Triebel, H.: Theory of Functions Spaces II. Birkhäuser, Basel (1992) · Zbl 1235.46003 · doi:10.1007/978-3-0346-0419-2 [11] Triebel, H.: Higher Analysis. Barth, Leipzig (1992) · Zbl 0783.46001 [12] Triebel, H.: Theory of Functions Spaces III. Birkhäuser, Basel (2006) · Zbl 1104.46001 [13] Triebel, H.: Local functions spaces, heat and Navier-Stokes equations. In: EMS Tracts in Mathematics, vol. 20 (2013) · Zbl 1280.46002 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.