# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
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.