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On global solutions and blow-up for Kuramoto-Sivashinsky-type models, and well-posed Burnett equations. (English) Zbl 1176.35094
Author’s abstract: The initial-boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto-Sivashinsky equation
\[ v_t+ v_{xxxx}+v_{xx}=\tfrac 12 (v^2)_x \]
and other related \(2m\)th-order semilinear parabolic partial differential equations in one dimension and in \(\mathbb{R}^N\) are considered. Global existence and blow-up as well as \(L^\infty\)-bounds are reviewed by using:
(i)
classic tools of interpolation theory and Galerkin methods,
(ii)
eigenfunction and nonlinear capacity methods,
(iii)
Henry’s version of weighted Gronwall’s inequalities,
(iv)
two types of scaling (blow-up) arguments.
For the IBVPs, existence of global solutions is proved for both Dirichlet and “Navier” boundary conditions. For some related \(2m\)th-order PDEs in \(\mathbb{R}^N\times R_+\), uniform boundedness of global solutions of the Cauchy problem is established.
As another related application, the well-posed Burnett-type equations,
\[ v_t+(v\cdot \nabla)v=-\nabla p-(-\Delta)^mv, \quad \operatorname{div} v=0\text{ in }\mathbb{R}^N \times \mathbb{R}_+,\;m\geq 1, \] are considered. For \(m=1\), these are the classic Navier-Stokes equations. As a simple illustration, it is shown that a uniform \(L^p(\mathbb{R}^N)\)-bound on locally sufficiently smooth \(v(x,t)\) for \(p>N/(2m-1)\) implies a uniform \(L^\infty(\mathbb{R}^N)\)-bound, and hence the solutions do not blow up. For \(m=1\) and \(N=3\), this gives \(p>3\), which reflects the famous Leray-Prodi-Serrin-Ladyzhenskaya regularity results \((L^{p\cdot q}\) criteria), and re-derives Kato’s class of unique mild solutions in \(\mathbb{R}^N\). Truly bounded classic \(L^2\)-solutions are shown to exist in dimensions \(N<2(2m-1)\).

MSC:
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35K30 Initial value problems for higher-order parabolic equations
35K35 Initial-boundary value problems for higher-order parabolic equations
35B45 A priori estimates in context of PDEs
35Q30 Navier-Stokes equations
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References:
[1] Tseluko, D.; Papageorgiou, D.T., A global contracting set for nonlinear kuramoto – sivashinsky equations arising in interfacial electrohydrodynamics, European J. appl. math., 17, 677-703, (2006) · Zbl 1132.37322
[2] Lin, S.P., Finite amplitude side-band of a viscous film, J. fluid mech., 63, 417-429, (1974) · Zbl 0283.76035
[3] Bellout, H.; Benachour, S.; Titi, E.S., Finite time singularity versus global regularity for hyper-viscous hamilton – jacobi-like equations, Nonlineraity, 16, 1967-1989, (2003) · Zbl 1038.35089
[4] Biagoni, H.A.; Bona, J.L.; Iorio, R.J.; Scialom, R.J., On the Korteweg-de Vries-kuramoto – sivashinsky equations, Adv. differential equations, 1, 1-20, (1996) · Zbl 0844.35103
[5] Bronski, J.C.; Gambill, T.N., Uncertainty estimates an \(L_2\) bounds for the kuramoto – sivashinsky equation, Nonlinearity, 19, 2023-2039, (2006) · Zbl 1110.37062
[6] Cao, Y.; Titi, E.S., Trivial stationary solutions to the kuramoto – sivashinsky and certain nonlinear elliptic equations, J. differential equations, 231, 755-767, (2006) · Zbl 1113.35008
[7] Elgin, J.N.; Wu, X., Stability of cellular states of the kuramoto – sivashinsky equation, SIAM J. appl. math., 56, 1621-1638, (1996) · Zbl 0868.35049
[8] Giacomelli, L.; Otto, F., New bounds for the kuramoto – sivashinsky equation, Comm. pure appl. math., LVIII, 297-318, (2005) · Zbl 1062.35113
[9] Grujić, Z.; Kukavica, I., A remark on time-analyticity for the kuramoto – sivashinsky equations, Nonlinear anal., 52, 69-78, (2003) · Zbl 1020.35095
[10] Kaikina, E.I., Subcritical kuramoto – sivashinsky-type equations in a half-line, J. differential equations, 220, 279-321, (2006) · Zbl 1090.35087
[11] Kent, Ph.; Elgin, J., Travelling waves of the kuramoto – sivashinsky equation: period-multiplying bifurcation, Nonlinearity, 5, 899-919, (1992) · Zbl 0771.35006
[12] Kukavica, I.; Malcok, M., Backward behaviour of solutions of the kuramoto – sivashinsky equation, J. math. anal. appl., 307, 455-464, (2005) · Zbl 1080.35121
[13] Larkin, N.A., Korteweg – de Vries and kuramoto – sivashinsky equations in bounded domains, J. math. anal. appl., 297, 169-185, (2004) · Zbl 1075.35070
[14] Sell, G.; Taboada, M., Local dissipativity and attractors for the kuramoto – sivashinsky equation in thin 2D domains, Nonlinear anal., 18, 671-687, (1992) · Zbl 0784.35046
[15] Tadmor, E., The well-posedness of the kuramoto – sivashinsky equation, SIAM J. math. anal., 17, 884-893, (1986) · Zbl 0606.35073
[16] Yang, D., Dynamics for the stochastic nonlinear kuramoto – sivashinsky equation, J. math. anal. appl., 330, 550-570, (2007) · Zbl 1120.60061
[17] Mitidieri, E.; Pohozaev, S.I., ()
[18] von Kármán, Th., Über laminare und turbulente reibung, Zamm, 1, 233-252, (1921) · JFM 48.0968.01
[19] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, C. R. acad. sci. Paris, 196, 527, (1933) · JFM 59.0763.02
[20] Leray, J., Sur le mouvement d’un liquide vosqueus emplissant l’espace, Acta math., 63, 193-248, (1934) · JFM 60.0726.05
[21] Chae, D., Nonexistence of asymptotically self-similar singularities in the Euler and the navier – stokes equations, Math. ann., 338, 435-449, (2007) · Zbl 1147.35068
[22] Nečas, J.; Ružička, M.; Šverák, V., On larey’s self-similar solutions of the navier – stokes equations, Acta math., 176, 283-294, (1996) · Zbl 0884.35115
[23] 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
[24] Hou, T.Y.; Li, R., Nonexistence of locally self-similar blow-up for the 3D incompressible navier – stokes equations, Discrete contin. dynam. syst., 18, 637-642, (2007), Full text in: arXiv:math/0603126v1[math.AP] · Zbl 1194.35307
[25] Dong, H.; Du, D., Partial regularity of solutions to the four-dimensional navier – stokes equations at the first blow-up time, Comm. math. phys., 273, 785-801, (2007) · Zbl 1156.35442
[26] Galaktionov, V.A.; Vazquez, J.L., A stability technique for evolution partial differential equations. A dynamical systems approach, (2004), Birkhäuser Boston, Berlin
[27] Andreev, V.K.; Kaptsov, O.V.; Pukhnachov, V.V.; Rodionov, A.A., Applications of group-theoretical methods in hydrodynamics, (1998), Kluwer Acad. Publ. Dordrecht · Zbl 0912.35001
[28] Ohkitani, K., A blow-up problem of a class of axisymmetric navier – stokes equations with infinite energy, J. math. phys., 48, 065205, (2007), 1-13 · Zbl 1144.81396
[29] Galaktionov, V.A., On blow-up space jets for the navier – stokes equations in \(\mathbb{R}^3\) with convergence to Euler equations, J. math. phys., 49, 113101, (2008) · Zbl 1159.81322
[30] Escauriaza, L.; Seregin, G.; S˘verák, V., \(L_{3, \infty}\)-solutions of the navier – stokes equations and backward uniqueness, Russian math. surveys, 58, 211-250, (2003) · Zbl 1064.35134
[31] 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
[32] Neustupa, J.; Penel, P., On regularity of a weak solutions to the navier – stokes equation with generilized impermeability boundary conditions, Nonlinear anal., 66, 1753-1769, (2007) · Zbl 1119.35057
[33] Seregin, G., Navier – stokes equations: almost \(L_{3, \infty}\)-case, J. math. fluid mech., 9, 34-43, (2007) · Zbl 1128.35085
[34] Frank-Kamenetskii, D.A., Towards temperature distributions in a reaction vessel and the stationary theory of thermal explosion, Doklady acad. nauk SSSR, 18, 411-412, (1938)
[35] Velazquez, J.J.L., Estimates on \((N - 1)\)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana univ. math. J., 42, 445-476, (1993) · Zbl 0802.35073
[36] Velazquez, J.J.L.; Galaktionov, V.A.; Herrero, M.A., The space structure near a blow-up point for semilinear heat equations: A formal approach, Comput. math. math. phys., 31, 46-55, (1991) · Zbl 0747.35014
[37] Galaktionov, V.A., On a spectrum of blow-up patterns for a higher-order semilinear parabolic equations, Proc. roy. soc. lond. A, 457, 1-21, (2001)
[38] 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
[39] Budd, C.; Galaktionov, V., Stability and spectra of blow-up in problems with quasi-linear gradient diffusivity, Proc. roy. soc. lond. A, 454, 2371-2407, (1998) · Zbl 1023.35053
[40] Galaktionov, V.A., Geometric Sturmian theory of nonlinear parabolic equations and applications, (2004), Chapman & Hall/CRC Boca Raton, Florida · Zbl 1075.35017
[41] Galaktionov, V.A., Evolution completeness of separable solutions of non-linear diffusion equations in bounded domains, Math. meth. appl. sci., 27, 1755-1770, (2004) · Zbl 1065.35157
[42] Qionglei, C.; Zhifei, Z., Regularity criterion via the pressure on weak solutions to the 3D navier – stokes equations, Proc. amer. math. soc., 135, 1829-1837, (2007) · Zbl 1126.35047
[43] Gala, S., A note on the uniqueness of mild solutions to the navier – stokes equations, Arch. math., 88, 448-454, (2007) · Zbl 1119.35047
[44] Kato, T., Strong \(L^p\) solutions of the navier – stokes equations in \(\mathbf{R}^m\) with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073
[45] Waymire, E.C., Probability & incompressible navier – stokes equations: an overview of some recent developments, Probab. surveys, 2, 1-32, (2005) · Zbl 1189.76424
[46] Leray, J., Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, J. math. pures appl., 12, 1-82, (1933) · Zbl 0006.16702
[47] Ladyzhenskaya, O.A., Mathematical problems of the dynamics of viscous incompressible fluids, (1970), Nauka Moscow · Zbl 0215.29004
[48] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéaires, () · Zbl 0189.40603
[49] Eidelman, S.D., Parabolic systems, (1969), North-Holland Publ. Comp. Amsterdam/London
[50] Friedman, A., Partial differential equations, (1983), Robert E. Krieger Publ. Comp. Malabar
[51] Taylor, M.E., Partial differential equations III. nonlinear equations, (1996), Springer New York, Tokyo
[52] Weissler, F.B., Semilinear evolution equations in Banach spaces, J. funct. anal., 32, 277-296, (1979) · Zbl 0419.47031
[53] 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
[54] Baoxiang, W., The Cauchy problem for critical and subcritical semilinear parabolic equations in \(L^r\) (I), Nonliner anal. TMA, 48, 747-764, (2002) · Zbl 1002.35057
[55] Cui, S., Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear anal. TMA, 43, 293-323, (2001) · Zbl 0963.35075
[56] Galaktionov, V.A.; Pohozaev, S.I., Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana univ. math. J., 51, 1321-1338, (2002) · Zbl 1082.35079
[57] Henry, D., ()
[58] Galaktionov, V.A.; Williams, J.F., On very singular similarity solutions of a higher-order semilinear parabolic equation, Nonlinearity, 17, 1075-1099, (2004) · Zbl 1063.35077
[59] Chaves, M.; Galaktionov, V.A., \(L^\infty\) and decay estimates in higher-order semilinear diffusion-adsorption equations, J. math. anal. appl., 341, 575-587, (2008) · Zbl 1141.35010
[60] Hamilton, R., The formation of singularities in the riccu flow, Surveys in differ. geom., Vol. II, 7-136, (1995), Cambridge, MA, 1993
[61] Samarskii, A.A.; Galaktionov, V.A.; Kurdyumov, S.P.; Mikhailov, A.P., Blow-up in quasilinear parabolic equations, (1995), Walter de Gruyter Berlin, New York · Zbl 1020.35001
[62] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Basel, Berlin · Zbl 0816.35001
[63] Smoller, J., Shock waves and reaction-diffusion equations, (1983), Springer-Verlag New York, Berlin · Zbl 0508.35002
[64] Evans, J.D.; Galaktionov, V.A.; Williams, J.F., Blow-up and global asymptotics of the limit unstable cahn – hilliard equation, SIAM J. math. anal., 38, 64-102, (2006) · Zbl 1110.35023
[65] Bernoff, A.J.; Bertozzi, A.L., Singularities in a modified kuramoto – sivashinsky equation describing interface motion for phase transition, Physica D, 85, 375-404, (1995) · Zbl 0899.76190
[66] Quittner, P.; Souplet, P., Superlinear parabolic problems and their equilibria, (2007), Birkhäuser
[67] Majda, A.J.; Bertozzi, A.L., Vorticity and incompressible flow, (2002), Cambridge Univ. Press Cambridge · Zbl 0983.76001
[68] Cannone, M.; Karch, G., About the regularized navier – stokes equations, J. math. fluid mech., 7, 1-28, (2005) · Zbl 1096.35099
[69] Gustafson, S.; Kang, K.; Tsai, T.-P., Interior regularity criteria for suitable weak solutions of the navier – stokes equations, Comm. math. phys., 273, 161-176, (2007) · Zbl 1126.35042
[70] Hopf, E., Ueber die anfangswertaufgbe für die hydrodynamischen grundgleichungen, Math. nachr., 4, 213-231, (1951) · Zbl 0042.10604
[71] Kolmogorov, A.N.; Fomin, S.V., Elements of the theory of functions and functional analysis, (1976), Nauka Moscow · Zbl 0235.46001
[72] Gallay, T.; Wayne, C.E., Invariant manifolds and long-time asymptotics of the navier – stokes and vorticity equations on \(\mathbb{R}^2\), Arch. ration. mech. anal., 163, 209-258, (2002) · Zbl 1042.37058
[73] V.A. Galaktionov, On blow-up twisters” for the Navier-Stokes equations in \(\mathbb{R}^3\): A view from reaction-diffusion theory, Adv. Differential Equations (submitted for publication)
[74] Bardos, C.; Titi, E.S., Euler equations for incompressible ideal fluids, Russian. math. surveys, 62, 409-451, (2007) · Zbl 1139.76010
[75] Constantin, P., On the Euler equations of incompressible fluids, Bull. amer. math. soc., 44, 603-621, (2007) · Zbl 1132.76009
[76] Ladyzhenskaya, O.A., Solutions in the large” to the boundary-values problem for the navier – stokes equations in two space variables, Soviet phys. dokl., 123, 1128-1131, (1958) · Zbl 0090.41502
[77] Ladyzhenskaya, O.A., Mathematical problems of the dynamics of viscous incompressible flow, (1961), Gos. Izdat. Fiz.-Mat. Moscow, second ed., Nauka, Moscow, 1970; Gordon and Breach, 2nd ed., 1969 · Zbl 0184.52603
[78] Mattingly, J.C.; Sinai, Ya.G., An elementary proof of the existence and uniqueness theorem for the navier – stokes equations, Comm. contemp. math., 1, 497-516, (1999) · Zbl 0961.35112
[79] Rosenau, P., Extending hydrodynamics via the regularization of the Chapman-Enskog expansions, Phys. lett. A, 40, 7193-7196, (1989)
[80] C. Fefferman, Existence & smoothness of the Navier-Stokes equation, The Clay Math. Inst., http://www.esi2.us.es/ mbilbao/claymath.htm · Zbl 1194.35002
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