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Boundary characteristic point regularity for Navier-Stokes equations: blow-up scaling and petrovskii-type criterion (a formal approach). (English) Zbl 1248.35036

The 3D Navier-Stokes equations \[ \frac{\partial u}{\partial t}+(u\cdot\nabla)u-\Delta u+\nabla p=0,\quad \text{div}\,u=0 \] are considered in a non-cylindrical domain \(Q\subset \mathbb{R}^3\times [-1,0)\). \(Q\) is a smooth domain of a backward parabolic shape with vertex \((0,0)\). The vertex is the only characteristic point. It means that the plane \(\{t=0\}\) is tangent to \(\partial Q\) at the origin and other characteristics for \(t\in[-1,0)\) intersect \(\partial Q\) transversely. The equations are added by initial and boundary conditions \[ \begin{aligned} & u(x,-1)=u_0(x)\;\text{on}\;Q\cap\{t=-1\},\\ & u=0\;\text{on}\;\partial Q. \end{aligned} \] The problem of regularity (in Wiener’s sense, see [the second author, Appl. Anal. 71, No. 1–4, 149–165 (1999; Zbl 1034.35048)]) of the vertex \((0,0)\) is studied. The vertex \((0,0)\) of the backward paraboloid \(Q\) is regular according to Wiener if for any bounded data \(u_0(x)\) the solution \(u\) satisfies to the condition \(u(0,0^-)=0\). The authors use a blow-up scaling and a special matching with a boundary layer near \(\partial Q\). They prove that the regularity of the vertex does not depend of a convection term. The similar regularity analysis of the well-posed Burnett equations \[ \begin{aligned} \frac{\partial u}{\partial t}+(u\cdot\nabla)u+\Delta^2 u+\nabla & p=0,\quad \text{div}\,u=0\;\text{in}\;Q,\\ & u=\nabla u\cdot n=0\;\text{on}\;\partial Q,\\ & u(x,-1)=u_0\end{aligned} \] is presented too.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35Q30 Navier-Stokes equations

Citations:

Zbl 1034.35048
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References:

[1] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Gauthier-Villars: Gauthier-Villars Dunod, Paris · Zbl 0189.40603
[2] Fujita, H.; Sauer, N., Construction of weak solutions of the Navier-Stokes equation in a non cylindrical domain, Bull. Amer. Math. Soc., 75, 465-468 (1969) · Zbl 0194.41401
[3] Fujita, H.; Sauer, N., On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries, J. Fac. Sci., Univ. Tokyo, Sect. IA, 17, 403-420 (1970) · Zbl 0206.39702
[4] J.O. Sather, The initial-boundary value problem for the Navier-Stokes equations in Regions with Moving Boundaries, Ph.D. Thesis, University of Minnesota, 1963, 76 pp.; J.O. Sather, The initial-boundary value problem for the Navier-Stokes equations in Regions with Moving Boundaries, Ph.D. Thesis, University of Minnesota, 1963, 76 pp.
[5] Ladyženskaja, O. A., An initial-boundary value problem for the Navier-Stokes equations in domains with boundary changing in time, Zap. Naučn. Sem. Leningr. Otdel. Steklov. Mat. Inst. (LOMI), 11, 97-128 (1968), (in Russian)
[6] Neustupa, J., Existence of a weak solution of the Navier-Stokes equation in a general time-varying domain by the Rothe method, Math. Methods Appl. Sci., 32, 653-683 (2009) · Zbl 1160.35494
[7] Majda, A. J.; Bertozzi, A. L., Vosticity and Incompressible Flow (2002), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[8] Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis (1985), North-Holland: North-Holland Amsterdam · Zbl 0572.35083
[9] Maz’ya, V., On the Wiener type regularity of a boundary point for the polyharmonic operator, Appl. Anal., 71, 149-165 (1999) · Zbl 1034.35048
[10] Wiener, N., The Dirichlet problem, J. Math. and Phys. Mass. Inst. Tech., 3, 127-146 (1924), Reprinted in: N. Wiener, P. Masani (Ed.), Mathematicians of Our Time 10, in: Collected works with Commentaries, vol. I, MIT Press, Cambridge, Mass., 1976, pp. 394-413 · JFM 51.0361.01
[11] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, C. R. Acad. Sci. Paris, 196, 527 (1933) · JFM 59.0763.02
[12] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05
[13] Kondrat’ev, V. A., Asymptotics of solutions of the Navier-Stokes equation near an angular point on the boundary, J. Appl. Math. Mech., 31, 125-129 (1967)
[14] Deuring, P.; von Wahl, W., Strong solutions of the Navier-Stokes system in Lipschitz bounded domains, Math. Nachr., 171, 111-148 (1995) · Zbl 0835.35109
[15] Kohr, M.; Pintea, C., Stokes-Brinkman transmission problems on Lipschitz and \(C^1\) domains in Riemannian manifolds, Comm. Pure Appl. Anal., 9, 493-537 (2010) · Zbl 1228.35099
[16] Kweon, J. R., Regularity of solutions for the Navier-Stokes system of incompressible flows on a polygon, J. Differential Equations, 235, 166-198 (2007) · Zbl 1119.35055
[17] Maz’ya, V.; Rossmann, J., Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Ration. Mech. Anal., 194, 669-712 (2009) · Zbl 1253.76019
[18] Maz’ya, V.; Rossmann, J., A maximum modulus estimate for solutions of the Navier-Stokes system in domains of polyhedral type, Math. Nachr., 282, 459-469 (2009) · Zbl 1170.35077
[19] Maz’ya, V. G.; Slutskii, A. S., Asymptotic analysis of the Navier-Stokes system in a plane domain with thin channels, Asymptotic Anal., 23, 59-89 (2000) · Zbl 0956.35103
[20] Mitrea, M.; Monniaux, S., The regularity of the Stokes operator and the Fujita-Kato approach to the Navier-Stokes initial value problem in Lipschitz domains, J. Funct. Anal., 254, 1522-1574 (2008) · Zbl 1143.47031
[21] Wolf, J., On the boundary regularity of suitable weak solutions to the Navier-Stokes equations, Ann. Univ. Ferrara, 56, 97-139 (2010) · Zbl 1205.35213
[22] Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J., (Spectral Problems with Corner Singularities of Solutions to Elliptic Equations. Spectral Problems with Corner Singularities of Solutions to Elliptic Equations, Math. Surveys Monogr., vol. 85 (2001), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0965.35003
[23] Mitrea, M.; Monniaux, S., On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds, Trans. Amer. Math. Soc., 361, 3125-3157 (2009) · Zbl 1170.42010
[24] V.A. Galaktionov, On blow-up twistors for the Navier-Stokes equations in \(\mathbb{R}^3\) arXiv:0901.4286; V.A. Galaktionov, On blow-up twistors for the Navier-Stokes equations in \(\mathbb{R}^3\) arXiv:0901.4286
[25] Petrowsky, J., Über die Lösungen der ersten Randwertaufgabe der Wärmeleitungsgleichung, Uc˘enye Zapiski Moscovsk. Gosud. Univ., 2, 55-59 (1934), Moscow, USSR (in German) with Russian summary · Zbl 0012.21001
[26] Petrovsky, I. G., Zur ersten Randwertaufgabe der Wärmeleitungsleichung, Compos. Math., 1, 383-419 (1935) · Zbl 0010.29903
[27] Galaktionov, V. A., On regularity of a boundary point in higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach, NoDEA, 16, 597-655 (2009), (arXiv:0901.3986) · Zbl 1219.35123
[28] Galaktionov, V.; Maz’ya, V., Boundary Characteristic Point Regularity for Semilinear Reaction-Diffusion Equations: Towards an ODE Criterion, J. Math. Sci. (New York, Springer), vol. 175, 3 (2011), (arXiv:1106.4696) · Zbl 1286.35123
[29] Kozlov, V.; Maz’ya, V., An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities, J. Funct. Anal., 217, 448-488 (2004) · Zbl 1068.35180
[30] Galaktionov, V. A.; Vazquez, J. L., A stability technique for evolution partial differential equations, (A Dynamical Systems Approach. A Dynamical Systems Approach, Progr. in Nonl. Differ. Equat. and Their Appl., vol. 56 (2004), Birkhäuser: Birkhäuser Boston/Berlin) · Zbl 0953.35112
[31] Grisvard, P., Elliptic Problems in Non-Smooth Domains (1985), Pitman: Pitman Boston · Zbl 0695.35060
[32] Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J., Elliptic boundary value problems in domains with point singularities, (Math. Surveys Monogr., vol. 52 (1997), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0947.35004
[33] Maz’ya, V.; Rossmann, J., Elliptic Equations in Polyhedral Domains, Mathematical Surveys and Monographs, vol. 162 (2010), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1196.35005
[34] Maz’ya, V. G.; Soloviev, A. A., (Boundary Integral Equations on Contours with Peaks. Boundary Integral Equations on Contours with Peaks, Operator Theory: Advances and Applications, vol. 196 (2010), Birkhäuser Verlag: Birkhäuser Verlag Basel) · Zbl 1179.45001
[35] Kozlov, V.; Maz’ya, V., Asymptotic formula for solutions to elliptic equations near the Lipschitz boundary, Ann. Mat. Pura Appl., 184, 4, 185-213 (2005) · Zbl 1223.35144
[36] Mayboroda, S.; Maz’ya, V., Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation, Invent. Math., 175, 287-334 (2009) · Zbl 1166.35014
[37] Maz’ya, V., The Wiener test for higher order elliptic equations, Duke Math. J., 115, 479-512 (2002) · Zbl 1129.35346
[38] Maz’ya, V.; Slutskii, A. S., Asymptotic solution to the Dirichlet problem for a two-dimensional Riccatti’s type equation near a corner point, Asymptotic Anal., 39, 169-185 (2004) · Zbl 1073.35093
[39] Grisvard, P., (Singularities in Boundary Value Problems. Singularities in Boundary Value Problems, PMA, vol. 22 (1992), Springer-Verlag: Springer-Verlag New York) · Zbl 0766.35001
[40] Landis, E. M., Necessary and sufficient conditions for regularity of a boundary point in the Dirichlet problem for the heat-conduction equation, Dokl. Akad. Nauk SSSR. Dokl. Akad. Nauk SSSR, Soviet Math., 10, 380-384 (1969) · Zbl 0187.03803
[41] Lanconelli, E., Sul problema di Dirichlet per l’equazione del calore, Ann. Mat. Pura Appl., 97, 83-114 (1973) · Zbl 0277.35058
[42] Evans, L. C.; Gariepy, R. F., Wieners criterion for the heat equation, Arch. Ration. Mech. Anal., 78, 293-314 (1982) · Zbl 0508.35038
[43] Watanabe, H., The initial-boundary value problems for the heat operator in non-cylindrical domains, J. Math. Soc. Japan, 49, 399-430 (1997) · Zbl 0911.35056
[44] Sturm, C., Mémoire sur une classe d’équations à différences partielles, J. Math. Pures Appl., 1, 373-444 (1836)
[45] Birman, M. S.; Solomjak, M. Z., Spectral theory of self-adjoint operators in Hilbert space (1987), D. Reidel: D. Reidel Dordrecht, Tokyo
[46] Samarskii, A. A.; Galaktionov, V. A.; Kurdyumov, S. P.; Mikhailov, A. P., Blow-up in Quasilinear Parabolic Equations (1995), Walter de Gruyter: Walter de Gruyter Berlin/New York · Zbl 1020.35001
[47] Vladimirov, V. S., Equations of Mathematical Physics (1971), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0231.35002
[48] Friedman, A., Partial Differential Equations (1983), Robert E. Krieger Publ. Comp: Robert E. Krieger Publ. Comp Malabar
[49] Eidelman, S. D., Parabolic Systems (1969), North-Holland Publ. Comp: North-Holland Publ. Comp Amsterdam, London
[50] S.D. Eidelman, Parabolic equations, in: Modern Problems of Math. Fundam. Achiev., vol. 63 (Part. Differ. Equat.), VINITI AN SSSR: 1990, Moscow, p. 244. English transl.: J. Soviet Math., 1991.; S.D. Eidelman, Parabolic equations, in: Modern Problems of Math. Fundam. Achiev., vol. 63 (Part. Differ. Equat.), VINITI AN SSSR: 1990, Moscow, p. 244. English transl.: J. Soviet Math., 1991.
[51] Hale, J. K., Asymptotic Behavior of Dissipative Systems (1988), AMS: AMS Providence, RI · Zbl 0642.58013
[52] C. Fefferman, Existence & smoothness of the Navier-Stokes equation, The Clay Math. Inst., http://www.esi2.us.es/ mbilbao/claymath.htm; C. Fefferman, Existence & smoothness of the Navier-Stokes equation, The Clay Math. Inst., http://www.esi2.us.es/ mbilbao/claymath.htm · Zbl 1194.35002
[53] Nečas, J.; Ružička, M.; Šverák, V., On Leray’s self-similar solutions of the Navier-Stokes equations, Acta Math., 176, 283-294 (1996) · Zbl 0884.35115
[54] Tsai, T.-P., On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Ration. Mech. Anal., 143, 29-51 (1998) · Zbl 0916.35084
[55] Miller, J. R.; O’Leary, M.; Schonbek, M., Nonexistence of singular pseudo-self-similar solutions of the Navier-Stokes system, Math. Ann., 319, 809-815 (2001) · Zbl 0983.35103
[56] Hou, T. Y.; Li, R., Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations, Discrete Contin. Dyn. Syst., 18, 637-642 (2007), (full text: arXiv:math/0603126v1) · Zbl 1194.35307
[57] Gallay, T.; Wayne, C. E., Long-time asymptotics of the Navier-Stokes and vorticity equations on \(R^3\), Phil. Trans. Roy. Soc. London, 360, 2155-2188 (2002) · Zbl 1048.35055
[58] Gallay, T.; Wayne, C. E., Invariant manifolds and long-time asymptotics of the Navier-Stokes and vorticity equations on \(R^2\), Arch. Ration. Mech. Anal., 163, 209-258 (2002) · Zbl 1042.37058
[59] Gallay, T.; Wayne, C. E., Long-time asymptotics of the Navier-Stokes equations in \(R^2\) and \(R^3\), Z. Angew. Math. Mech., 86, 256-257 (2006) · Zbl 1094.35089
[60] Brandolese, L., Fine properties of self-similar solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 192, 375-401 (2009) · Zbl 1169.76014
[61] Brandolese, L.; Vigneron, F., New asymptotic profiles of nonstationary solutions of the Navier-Stokes system, J. Math. Pures Appl., 88, 64-86 (2007) · Zbl 1127.35033
[62] Angenent, S., Large-time asymptotics of the porous medium equation, (Ni, W.-M.; Peletier, L. A.; Serrin, J., Nonl. Differ. Equat. and Their Equilib. States I (Berkeley, CA, 1986). Nonl. Differ. Equat. and Their Equilib. States I (Berkeley, CA, 1986), MSRI Publ., vol. 12 (1988), Springer Verlag: Springer Verlag New York, Tokyo), 21-34
[63] Kolmogorov, A. N.; Fomin, S. V. (1976), Nauka: Nauka Moscow
[64] Rosenau, P., Extending hydrodynamics via the regularization of the Chapman-Enskog expansions, Phys. Lett. A, 40, 7193-7196 (1989)
[65] Galaktionov, V. A.; Mitidieri, E.; Pohozaev, S. I., On global solutions and blow-up for Kuramoto-Sivashinsky-type models and well-posed Burnett equations, Nonl. Anal., 70, 2930-2952 (2009), (arXiv:0902.0257) · Zbl 1176.35094
[66] Galaktionov, V. A., Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach, Nonlinearity, 22, 1695-1741 (2009), (arXiv:0901.4307) · Zbl 1197.35062
[67] Egorov, Yu. V.; Galaktionov, V. A.; Kondratiev, V. A.; Pohozaev, S. I., Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Differential Equations, 9, 1009-1038 (2004) · Zbl 1122.35040
[68] Fedoryuk, M. V., Singularities of the kernels of Fourier integral operators and the asymptotic behaviour of the solution of the mixed problem, Russian Math. Surveys, 32, 67-120 (1977) · Zbl 0386.35005
[69] Barbatis, G., Explicit estimates on the fundamental solution of higher-order parabolic equations with measurable coefficients, J. Differential Equations, 174, 442-463 (2001) · Zbl 0993.35037
[70] Barbatis, G., Sharp heat-kernel estimates for higher-order operators with singular coefficients, Proc. Edinb. Math. Soc., 47, 2, 53-67 (2004) · Zbl 1058.35082
[71] Galaktionov, V. A.; Svirshchevskii, S. R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (2007), Chapman& Hall/CRC: Chapman& Hall/CRC Boca Raton, Florida · Zbl 1153.35001
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