On the Global Existence for the Kuramoto-Sivashinsky Equation

We address the global existence of solutions for the 2D Kuramoto-Sivashinsky equations in a periodic domain $$[0,L_1]\times [0,L_2]$$ with initial data satisfying $$\Vert u_0\Vert _{L^2}\le C^{-1}L_2^{-2}$$ , where C is a constant. We prove that the global solution exists under the condition $$L_2\le 1/C L_1^{3/5}$$ , improving earlier results. The solutions are smooth and decrease energy until they are dominated by $$C L_1^{3/2}L_2^{1/2}$$ , implying the existence of an absorbing ball in $$L^2$$ .

[1]  Luan T. Hoang,et al.  Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (II) , 2010 .

[2]  G. Sell,et al.  Navier-Stokes Equations in Thin 3D Domains III: Existence of a Global Attractor , 1993 .

[3]  G. Sell,et al.  Local dissipativity and attractors for the Kuramoto-Sivashinsky equation in thin 2D domains , 1992 .

[4]  Global Existence and Analyticity for the 2D Kuramoto–Sivashinsky Equation , 2017, 1708.08752.

[5]  G. Sivashinsky.  On Flame Propagation Under Conditions of Stoichiometry , 1980 .

[6]  Ju. S. Il'yashenko.  Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation , 1992 .

[7]  Roger Temam,et al.  Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attr , 1985 .

[8]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[9]  Jonathan Goodman,et al.  Stability of the kuramoto-sivashinsky and related systems† , 1994 .

[10]  Lorenzo Giacomelli,et al.  New bounds for the Kuramoto‐Sivashinsky equation , 2005 .

[11]  L. Hoang.  A basic inequality for the Stokes operator related to the Navier boundary condition , 2008 .

[12]  George R. Sell,et al.  Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions , 1993 .

[13]  Felix Otto,et al.  Optimal bounds on the Kuramoto–Sivashinsky equation , 2009 .

[14]  I. Kukavica,et al.  A remark on time-analyticity for the Kuramoto-Sivashinsky equation , 2003 .

[15]  I. Kukavica,et al.  A class of solutions of the Navier–Stokes equations with large data , 2013 .

[16]  G. Sell,et al.  Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model , 2010 .

[17]  Philip Holmes,et al.  Scale and space localization in the Kuramoto-Sivashinsky equation. , 1999, Chaos.

[18]  D Michelson,et al.  Steady solutions of the Kuramoto-Sivashinsky equation , 1986 .

[19]  D. Papageorgiou,et al.  A global attracting set for nonlocal Kuramoto–Sivashinsky equations arising in interfacial electrohydrodynamics , 2006, European Journal of Applied Mathematics.

[20]  J. Eckmann,et al.  A global attracting set for the Kuramoto-Sivashinsky equation , 1993 .

[21]  Tom Gambill,et al.  Uncertainty estimates and L2 bounds for the Kuramoto-Sivashinsky equation , 2005, math/0508481.

[22]  L. Molinet.  Local Dissipativity in L2 for the Kuramoto–Sivashinsky Equation in Spatial Dimension 2 , 2000 .

[23]  Igor Kukavica,et al.  Regularity of the Navier-Stokes equation in a thin periodic domain with large data , 2006 .

[24]  I. Kukavica,et al.  Anisotropic Estimates for the Two-Dimensional Kuramoto–Sivashinsky Equation , 2014 .

[25]  Roger Temam,et al.  Navier-Stokes equations in three-dimensional thin domains with various boundary conditions , 1996, Advances in Differential Equations.

[26]  David Swanson,et al.  Existence and generalized Gevrey regularity of solutions to the Kuramoto–Sivashinsky equation in Rn , 2007 .

[27]  F. Otto,et al.  New Bounds for the Inhomogenous Burgers and the Kuramoto-Sivashinsky Equations , 2015, 1503.06059.

[28]  Krug,et al.  Anisotropic Kuramoto-Sivashinsky equation for surface growth and erosion. , 1995, Physical review letters.

[29]  J. Avrin.  Large-Eigenvalue Global Existence and Regularity Results for the Navier–Stokes Equation , 1996 .