×

zbMATH — the first resource for mathematics

Abstract \(L^ p\) estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. (English) Zbl 0739.35067
The authors first generalize the result of G. Dore and A. Venni [Math. Z. 196, 189-201 (1987; Zbl 0615.47002)] on abstract Cauchy- problems \(u'+Au=f\) to operators \(A\), for which one has the estimate \[ \| A^{iy}\|\leq K\exp(\theta| y|)\quad\hbox { with } \theta<\pi/2 (*) \] for the pure imaginary powers of \(A\), and show furthermore that the constant \(C\) in the estimate \[ \int^ T_ 0(\| u'\|^ s+\| Au\|^ s)dt\leq C\int^ T_ 0\| f\|^ sdt \] is independent of \(T\), which is important for ”global in time” results. The estimate (*) for the Stokes operator in \(L_{q,\sigma}\) on exterior domains was proved by the authors in J. Fac. Sci., Univ. Tokyo, Sect. I A 36, 103-130 (1989; Zbl 0689.76012). Hence their abstract results imply then new \(L_ p-L_ q\) estimates of integral type (global in time) for the solutions of the Navier-Stokes system \[ u'-\Delta u+(u\nabla u)+\nabla p=f,\quad\hbox { div} u=0\quad\hbox {and } u=0\hbox { on } \partial \Omega. \] If \(\|\centerdot\|_{q,s}\) denotes the norm in \(L_ s(\mathbb{R}^ +,L_ q)\), they get \[ \| u'\|_{q,s}+\| D^ 2u\|_{q,s}+\|\nabla p\|_{q,s}+\| p\|_{r,s}\leq C(\| f\|_{q,s}+\| f\|_{2,2}+\| u(0)\|_ Y) \] (for certain \(Y\)), if \(n+1={n\over q}+{2\over s}\), \({n\over r}={n\over q}-1\). For \(n=3\) this implies the crucial result \(p\in L_{5/3}(\mathbb{R}_ +\times\Omega)\) for proving regularity for large \(t\).

MSC:
35Q30 Navier-Stokes equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agmon, S; Douglis, A; Nirenberg, L; Agmon, S; Douglis, A; Nirenberg, L, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II, Comm. pure appl. math., Comm. pure appl. math., 17, 35-92, (1964) · Zbl 0123.28706
[2] Benedek, A; Calderón, A.P; Panzone, R, Convolution operators on Banach space valued functions, (), 356-365 · Zbl 0103.33402
[3] Borchers, W; Miyakawa, T, L2-decay for the Navier-Stokes flow in halfspaces, Math. ann., 282, 139-155, (1988) · Zbl 0627.35076
[4] Borchers, W; Miyakawa, T, Algebraic L2-decay for Navier-Stokes flows in exterior domains, Acta math., 165, 189-227, (1990) · Zbl 0722.35014
[5] Borchers, W; Sohr, H, On the semigroup of the Stokes operator for exterior domains, Math. Z., 196, 415-425, (1987) · Zbl 0636.76027
[6] Bourgain, J, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. mat., 21, 163-168, (1983) · Zbl 0533.46008
[7] Burkholder, D.L, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, (), 270-286
[8] Butzer, P; Berens, H, Semi-groups of operators and approximations, (1967), Springer Berlin/Heidelberg/New York · Zbl 0164.43702
[9] 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
[10] Cannarsa, P; Vespri, V, On maximal Lp regularity for the abstract Cauchy problem, Boll. un. mat. ital. (6), 5-13, 165-175, (1986) · Zbl 0608.35027
[11] Cattabriga, L, Su un problema al contorno relativo al sistema di equazioni di Stokes, (), 308-340 · Zbl 0116.18002
[12] \scP. Constantin, Remarks on the Navier-Stokes equations, preprint.
[13] Dore, G; Venni, A, On the closedness of the sum of two closed operators, Math. Z., 196, 189-201, (1987) · Zbl 0615.47002
[14] Duff, G.F.D, Derivative estimates for the Navier-Stokes equations in a three dimensional region, Acta math., 164, 145-210, (1990) · Zbl 0728.35083
[15] Foias, C; Guillopé, C; Temam, R, New a priori estimates for Navier-Stokes equations in dimension 3, Comm. partial differential equations, 6, 329-359, (1981) · Zbl 0472.35070
[16] Fujita, H; Kato, T, On the Navier-Stokes initial value problem, I, Arch. rational mech. anal., 16, 269-315, (1964) · Zbl 0126.42301
[17] Fujiwara, D; Morimoto, H, An L_r-theorem of the Helmholtz decomposition of vector fields, J. fac. sci. univ. Tokyo sect. IA math., 24, 685-700, (1977) · Zbl 0386.35038
[18] Galdi, G; Maremonti, P, Monotic decreasing and asymptotic behavior of the kinematic energy for weak solutions of the Navier-Stokes equations in exterior domains, Arch. rational mech. anal., 94, 253-266, (1986) · Zbl 0617.35108
[19] Giga, Y, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Math. Z., 178, 297-329, (1981) · Zbl 0473.35064
[20] Giga, Y, Domains of fractional powers of the Stokes operators in Lr spaces, Arch. rational mech. anal., 89, 251-265, (1985) · Zbl 0584.76037
[21] Giga, Y, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. differential equations, 62, 186-212, (1986) · Zbl 0577.35058
[22] Giga, Y; Miyakawa, T, Solutions in Lr to the Navier-Stokes initial value problem, Arch. rational mech. anal., 89, 267-281, (1985) · Zbl 0587.35078
[23] Giga, Y; Sohr, H, On the Stokes operator in exterior domains, J. fac. sci. univ. Tokyo sect. IA math., 36, 103-130, (1989)
[24] Giga, Y; Sohr, H, Note on the Cauchy problem in Banach spaces with applications to the Navier-Stokes equation in exterior domain, (1988), unpublished preprint
[25] Iwashita, H, Lq − lr estimates for solutions of non-stationary Stokes equations in exterior domains and the Navier-Stokes initial value problems in lq spaces, Math. ann., 285, 265-288, (1989) · Zbl 0659.35081
[26] Komatsu, H, Fractional powers of operators, Pacific J. math., 19, 285-346, (1966) · Zbl 0154.16104
[27] Ladyzhenskaya, O.A, The mathematical theory of viscous incompressible flow, (1969), Gordon & Breach New York · Zbl 0184.52603
[28] Maremonti, P, Partial regularity of a generalized solution to the Navier-Stokes equations in exterior domains, Comm. math. phys., 110, 75-87, (1987) · Zbl 0617.35021
[29] Masuda, K, Weak solutions of Navier-Stokes equations, Tôhoku math. J., 36, 623-646, (1984) · Zbl 0568.35077
[30] McCracken, M, The resolvent problem for the Stokes equations on halfspaces in Lp, SIAM J. math. anal., 12, 201-228, (1981) · Zbl 0475.35073
[31] Miyakawa, T, On nonstationary solutions of the Navier-Stokes equations in exterior domains, Hiroshima math. J., 12, 115-140, (1982) · Zbl 0486.35067
[32] Miyakawa, T; Sohr, H, On energy inequality, smoothness and large time behavior in L2 for weak solutions of the Navier-Stokes equations in exterior domains, Math. Z., 199, 455-478, (1988) · Zbl 0642.35067
[33] Nirenberg, L, On elliptic partial differential equations, Ann. scuola norm. sup. Pisa cl. sci. (3), 13, 115-162, (1959) · Zbl 0088.07601
[34] Prüss, J; Sohr, H, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203, 429-452, (1990) · Zbl 0665.47015
[35] de Francia, J.L.Rubio, Martingale and integral transforms of Banach space valued functions, (), 195-222
[36] Serrin, J, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. rational mech. anal., 9, 187-195, (1962) · Zbl 0106.18302
[37] Sobolevskii, P.E, Coerciveness inequalities for abstract parabolic equations, Dokl. akad. nauk SSSR, 157, 52-55, (1964), [In Russian]
[38] Sohr, H, Zur regularitätstheorie der instationären gleichungen von Navier-Stokes, Math. Z., 184, 359-375, (1983) · Zbl 0506.35084
[39] Sohr, H; von Wahl, W, A new proof of Leray’s structure theorem and the smoothness of weak solutions of Navier-Stokes equations for large \(¦x¦\), Bayreuth. math. schr., 20, 153-204, (1985) · Zbl 0681.35073
[40] Sohr, H; von Wahl, W, On the regularity of the pressure of weak solutions of Navier-Stokes equations, Arch. math., 46, 428-439, (1986) · Zbl 0574.35070
[41] Sohr, H; von Wahl, W; Wiegner, M, Zur asymptotik der gleichungen von Navier-Stokes, Nachr. akad. wiss. Göttingen, 3, 1-15, (1986)
[42] Solonnikov, V.A, Estimates for solutions of nonstationary Navier-Stokes equations, J. soviet math., 8, 467-523, (1977) · Zbl 0404.35081
[43] Triebel, H, Interpolation theory, function spaces, differential operators, (1978), North-Holland Amsterdam/New York/Oxford · Zbl 0387.46032
[44] von Wahl, W, The equations of Navier-Stokes and abstract parabolic equations, () · Zbl 0643.35083
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.