doi: 10.3934/dcdsb.2021305
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On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

Received  February 2021 Revised  October 2021 Early access January 2022

We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data $ u_{01} \in L^2 $ and $ u_{02} \in H^{-1 + \eta} $ for $ \eta > 0 $.

Citation: David Massatt. On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021305
References:
[1]

D. M. Ambrose and A. L. Mazzucato, Global existence and analyticity for the 2D Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations, 31 (2019), 1525-1547.  doi: 10.1007/s10884-018-9656-0.  Google Scholar

[2]

J. D. Avrin, Large-eigenvalue global existence and regularity results for the Navier-Stokes equation, J. Differential Equations, 127 (1996), 365-390.  doi: 10.1006/jdeq.1996.0074.  Google Scholar

[3]

S. BenachourI. KukavicaW. Rusin and M. Ziane, Anisotropic estimates for the two-dimensional Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations, 26 (2014), 461-476.  doi: 10.1007/s10884-014-9372-3.  Google Scholar

[4]

A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $\mathbb{R}^n$, J. Differential Equations, 240 (2007), 145-163.  doi: 10.1016/j.jde.2007.05.022.  Google Scholar

[5]

P. ColletJ.-P. EckmannH. Epstein and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Comm. Math. Phys., 152 (1993), 203-214.  doi: 10.1007/BF02097064.  Google Scholar

[6]

L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation, Comm. Pure Appl. Math., 58 (2005), 297-318.  doi: 10.1002/cpa.20031.  Google Scholar

[7]

M. GoldmanM. Josien and F. Otto, New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations, Comm. Partial Differential Equations, 40 (2015), 2237-2265.  doi: 10.1080/03605302.2015.1076003.  Google Scholar

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J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Comm. Pure Appl. Math., 47 (1994), 293-306.  doi: 10.1002/cpa.3160470304.  Google Scholar

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Z. Grujić and I. Kukavica, A remark on time-analyticity for the Kuramoto-Sivashinsky equation, Nonlinear Anal., 52 (2003), 69-78.  doi: 10.1016/S0362-546X(01)00910-5.  Google Scholar

[10]

L. T. Hoang, A basic inequality for the Stokes operator related to the Navier boundary condition, J. Differential Equations, 245 (2008), 2585-2594.  doi: 10.1016/j.jde.2008.01.024.  Google Scholar

[11]

L. T. Hoang, Incompressible fluids in thin domains with Navier friction boundary conditions (II), J. Math. Fluid Mech., 15 (2013), 361-395.  doi: 10.1007/s00021-012-0123-0.  Google Scholar

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L. T. Hoang and G. R. Sell, Navier-Stokes equations with Navier boundary conditions for an oceanic model, J. Dynam. Differential Equations, 22 (2010), 563-616.  doi: 10.1007/s10884-010-9189-7.  Google Scholar

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Ju. S. Il'yashenko, Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations, 4 (1992), 585-615.   Google Scholar

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I. Kukavica and D. Massatt, On the global existence of the Kuramoto-Sivashinsky equation, submitted. Google Scholar

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I. KukavicaW. Rusin and M. Ziane, A class of solutions of the Navier-Stokes equations with large data, J. Differential Equations, 255 (2013), 1492-1514.  doi: 10.1016/j.jde.2013.05.009.  Google Scholar

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I. Kukavica and M. Ziane, Regularity of the Navier-Stokes equation in a thin periodic domain with large data, Discrete Contin. Dyn. Syst., 16 (2006), 67-86.  doi: 10.3934/dcds.2006.16.67.  Google Scholar

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Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

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A. Larios and K. Yamazaki, On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto-Sivashinsky equation, Phys. D, 411 (2020), 132560, 14 pp. doi: 10.1016/j.physd.2020.132560.  Google Scholar

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L. Molinet, Local dissipativity in $L^2$ for the Kuramoto-Sivashinsky equation in spatial dimension 2, J. Dynam. Differential Equations, 12 (2000), 533-556.  doi: 10.1023/A:1026459527446.  Google Scholar

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L. Molinet, A bounded global absorbing set for the Burgers-Sivashinsky equation in space dimension two, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 635-640.  doi: 10.1016/S0764-4442(00)00224-X.  Google Scholar

[21]

B. NicolaenkoB. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors, Phys. D, 16 (1985), 155-183.  doi: 10.1016/0167-2789(85)90056-9.  Google Scholar

[22]

F. Otto, Optimal bounds on the Kuramoto-Sivashinsky equation, J. Funct. Anal., 257 (2009), 2188-2245.  doi: 10.1016/j.jfa.2009.01.034.  Google Scholar

[23]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.2307/2152776.  Google Scholar

[24]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions, Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. XI (Paris, 1989–1991), Pitman Res. Notes Math. Ser., vol. 299, Longman Sci. Tech., Harlow, 1994, pp. 205–247.  Google Scholar

[25]

G. Raugel and G. R. Sell, Navier-Stokes equations in thin 3D domains. III. Existence of a global attractor, Turbulence in fluid flows, IMA Vol. Math. Appl., vol. 55, Springer, New York, 1993,137–163. doi: 10.1007/978-1-4612-4346-5_9.  Google Scholar

[26]

M. Rost and J. Krug, Anisotropic Kuramoto–Sivashinsky equation for surface growth erosion, Physical Review Letters, 75 (1995), 3894-3897.  doi: 10.1103/PhysRevLett.75.3894.  Google Scholar

[27]

G. R. Sell and M. Taboada, Local dissipativity and attractors for the Kuramoto-Sivashinsky equation in thin 2D domains, Nonlinear Anal., 18 (1992), 671-687.  doi: 10.1016/0362-546X(92)90006-Z.  Google Scholar

[28]

G. I. Sivashinsky, On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math., 39 (1980), 67-82.  doi: 10.1137/0139007.  Google Scholar

[29]

M. Stanislavova and A. Stefanov, Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation, Discrete Contin. Dyn. Syst., (2009), Dynamical systems, differential equations and applications. 7th AIMS Conference, suppl., 729–738.  Google Scholar

[30]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[31]

R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations, 1 (1996), 499-546.   Google Scholar

[32]

D. Tseluiko and D. T. Papageorgiou, A global attracting set for nonlocal Kuramoto-Sivashinsky equations arising in interfacial electrohydrodynamics, European J. Appl. Math., 17 (2006), 677-703.  doi: 10.1017/S0956792506006760.  Google Scholar

[33]

F. B. Weissler, Local existence and non-existence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102.  doi: 10.1512/iumj.1980.29.29007.  Google Scholar

show all references

References:
[1]

D. M. Ambrose and A. L. Mazzucato, Global existence and analyticity for the 2D Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations, 31 (2019), 1525-1547.  doi: 10.1007/s10884-018-9656-0.  Google Scholar

[2]

J. D. Avrin, Large-eigenvalue global existence and regularity results for the Navier-Stokes equation, J. Differential Equations, 127 (1996), 365-390.  doi: 10.1006/jdeq.1996.0074.  Google Scholar

[3]

S. BenachourI. KukavicaW. Rusin and M. Ziane, Anisotropic estimates for the two-dimensional Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations, 26 (2014), 461-476.  doi: 10.1007/s10884-014-9372-3.  Google Scholar

[4]

A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $\mathbb{R}^n$, J. Differential Equations, 240 (2007), 145-163.  doi: 10.1016/j.jde.2007.05.022.  Google Scholar

[5]

P. ColletJ.-P. EckmannH. Epstein and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Comm. Math. Phys., 152 (1993), 203-214.  doi: 10.1007/BF02097064.  Google Scholar

[6]

L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation, Comm. Pure Appl. Math., 58 (2005), 297-318.  doi: 10.1002/cpa.20031.  Google Scholar

[7]

M. GoldmanM. Josien and F. Otto, New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations, Comm. Partial Differential Equations, 40 (2015), 2237-2265.  doi: 10.1080/03605302.2015.1076003.  Google Scholar

[8]

J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Comm. Pure Appl. Math., 47 (1994), 293-306.  doi: 10.1002/cpa.3160470304.  Google Scholar

[9]

Z. Grujić and I. Kukavica, A remark on time-analyticity for the Kuramoto-Sivashinsky equation, Nonlinear Anal., 52 (2003), 69-78.  doi: 10.1016/S0362-546X(01)00910-5.  Google Scholar

[10]

L. T. Hoang, A basic inequality for the Stokes operator related to the Navier boundary condition, J. Differential Equations, 245 (2008), 2585-2594.  doi: 10.1016/j.jde.2008.01.024.  Google Scholar

[11]

L. T. Hoang, Incompressible fluids in thin domains with Navier friction boundary conditions (II), J. Math. Fluid Mech., 15 (2013), 361-395.  doi: 10.1007/s00021-012-0123-0.  Google Scholar

[12]

L. T. Hoang and G. R. Sell, Navier-Stokes equations with Navier boundary conditions for an oceanic model, J. Dynam. Differential Equations, 22 (2010), 563-616.  doi: 10.1007/s10884-010-9189-7.  Google Scholar

[13]

Ju. S. Il'yashenko, Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations, 4 (1992), 585-615.   Google Scholar

[14]

I. Kukavica and D. Massatt, On the global existence of the Kuramoto-Sivashinsky equation, submitted. Google Scholar

[15]

I. KukavicaW. Rusin and M. Ziane, A class of solutions of the Navier-Stokes equations with large data, J. Differential Equations, 255 (2013), 1492-1514.  doi: 10.1016/j.jde.2013.05.009.  Google Scholar

[16]

I. Kukavica and M. Ziane, Regularity of the Navier-Stokes equation in a thin periodic domain with large data, Discrete Contin. Dyn. Syst., 16 (2006), 67-86.  doi: 10.3934/dcds.2006.16.67.  Google Scholar

[17]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

[18]

A. Larios and K. Yamazaki, On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto-Sivashinsky equation, Phys. D, 411 (2020), 132560, 14 pp. doi: 10.1016/j.physd.2020.132560.  Google Scholar

[19]

L. Molinet, Local dissipativity in $L^2$ for the Kuramoto-Sivashinsky equation in spatial dimension 2, J. Dynam. Differential Equations, 12 (2000), 533-556.  doi: 10.1023/A:1026459527446.  Google Scholar

[20]

L. Molinet, A bounded global absorbing set for the Burgers-Sivashinsky equation in space dimension two, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 635-640.  doi: 10.1016/S0764-4442(00)00224-X.  Google Scholar

[21]

B. NicolaenkoB. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors, Phys. D, 16 (1985), 155-183.  doi: 10.1016/0167-2789(85)90056-9.  Google Scholar

[22]

F. Otto, Optimal bounds on the Kuramoto-Sivashinsky equation, J. Funct. Anal., 257 (2009), 2188-2245.  doi: 10.1016/j.jfa.2009.01.034.  Google Scholar

[23]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.2307/2152776.  Google Scholar

[24]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions, Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. XI (Paris, 1989–1991), Pitman Res. Notes Math. Ser., vol. 299, Longman Sci. Tech., Harlow, 1994, pp. 205–247.  Google Scholar

[25]

G. Raugel and G. R. Sell, Navier-Stokes equations in thin 3D domains. III. Existence of a global attractor, Turbulence in fluid flows, IMA Vol. Math. Appl., vol. 55, Springer, New York, 1993,137–163. doi: 10.1007/978-1-4612-4346-5_9.  Google Scholar

[26]

M. Rost and J. Krug, Anisotropic Kuramoto–Sivashinsky equation for surface growth erosion, Physical Review Letters, 75 (1995), 3894-3897.  doi: 10.1103/PhysRevLett.75.3894.  Google Scholar

[27]

G. R. Sell and M. Taboada, Local dissipativity and attractors for the Kuramoto-Sivashinsky equation in thin 2D domains, Nonlinear Anal., 18 (1992), 671-687.  doi: 10.1016/0362-546X(92)90006-Z.  Google Scholar

[28]

G. I. Sivashinsky, On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math., 39 (1980), 67-82.  doi: 10.1137/0139007.  Google Scholar

[29]

M. Stanislavova and A. Stefanov, Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation, Discrete Contin. Dyn. Syst., (2009), Dynamical systems, differential equations and applications. 7th AIMS Conference, suppl., 729–738.  Google Scholar

[30]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[31]

R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations, 1 (1996), 499-546.   Google Scholar

[32]

D. Tseluiko and D. T. Papageorgiou, A global attracting set for nonlocal Kuramoto-Sivashinsky equations arising in interfacial electrohydrodynamics, European J. Appl. Math., 17 (2006), 677-703.  doi: 10.1017/S0956792506006760.  Google Scholar

[33]

F. B. Weissler, Local existence and non-existence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102.  doi: 10.1512/iumj.1980.29.29007.  Google Scholar

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