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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[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. 

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[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. 

[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.

[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.

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