Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection

We study the Kuramoto-Sivashinsky equation (KSE) in scalar form on the two-dimensional torus with and without advection by an incompressible vector field. We prove local existence of mild solutions for arbitrary data in L2. We then study the issue of global existence. We prove global existence for the KSE in the presence of advection for arbitrary data, provided the advecting velocity field v satisfies certain conditions that ensure the dissipation time of the associated hyperdiffusion-advection equation is sufficiently small. In the absence of advection, global existence can be shown only if the linearized operator does not admit any growing mode and for sufficiently small initial data.

[1]  J. Hyman,et al.  The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems , 1986 .

[2]  Igor Kukavica,et al.  On the Global Existence for the Kuramoto-Sivashinsky Equation , 2021, Journal of Dynamics and Differential Equations.

[3]  Marco Cannone,et al.  Chapter 3 - Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations , 2005 .

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

[5]  L. Collins,et al.  Effect of nonunity lewis number on premixed flame propagation through isotropic turbulence , 1995 .

[6]  E. Titi,et al.  Large dispersion, averaging and attractors: three 1D paradigms , 2016, Nonlinearity.

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

[8]  J. Bedrossian,et al.  Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations , 2019, The Annals of Probability.

[9]  T. Gallay.  Enhanced Dissipation and Axisymmetrization of Two-Dimensional Viscous Vortices , 2017, Archive for Rational Mechanics and Analysis.

[10]  G. Alberti,et al.  Exponential self-similar mixing by incompressible flows , 2016, 1605.02090.

[11]  Giovanni Fantuzzi,et al.  Bounds on mean energy in the Kuramoto–Sivashinsky equation computed using semidefinite programming , 2018, Nonlinearity.

[12]  E. Tadmor,et al.  Suppressing Chemotactic Blow-Up Through a Fast Splitting Scenario on the Plane , 2017, Archive for Rational Mechanics and Analysis.

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

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

[15]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[16]  J. Bedrossian,et al.  Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows , 2015, 1510.08098.

[17]  Yuanyuan Feng,et al.  Phase Separation in the Advective Cahn–Hilliard Equation , 2019, J. Nonlinear Sci..

[18]  Yuanyuan Feng,et al.  Dissipation enhancement by mixing , 2018, Nonlinearity.

[19]  Spatial Analyticity on the Global Attractor for the Kuramoto–Sivashinsky Equation , 2000 .

[20]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[21]  Gautam Iyer,et al.  Convection-induced singularity suppression in the Keller-Segel and other non-linear PDEs , 2019, Transactions of the American Mathematical Society.

[22]  Eitan Tadmor,et al.  The well-posedness of the Kuramoto-Sivashinsky equation , 1986 .

[23]  P. Constantin,et al.  Diffusion and mixing in fluid flow , 2005 .

[24]  M. Cannone.  Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations , 2022 .

[25]  Si-ming He.  Suppression of blow-up in parabolic–parabolic Patlak–Keller–Segel via strictly monotone shear flows , 2017, Nonlinearity.

[26]  A. Kiselev,et al.  Suppression of Chemotactic Explosion by Mixing , 2015, 1508.05333.

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

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

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

[30]  T. Elgindi,et al.  Universal mixers in all dimensions , 2018, Advances in Mathematics.

[31]  M. Stanislavova,et al.  The Kuramoto-Sivashinsky equation in R^1 and R^2: effective estimates of the high-frequency tails and higher Sobolev norms , 2007, 0711.4005.

[32]  P. Constantin,et al.  Relaxation in Reactive Flows , 2008 .

[33]  Pierre Collet,et al.  Analyticity for the Kuramoto-Sivashinsky equation , 1993 .

[34]  Michele Coti Zelati,et al.  On the Relation between Enhanced Dissipation Timescales and Mixing Rates , 2018, Communications on Pure and Applied Mathematics.

[35]  Y. Smyrlis,et al.  Analyticity for Kuramoto–Sivashinsky‐type equations in two spatial dimensions , 2016 .

[36]  Toan T. Nguyen,et al.  Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method , 2018, Journal of Functional Analysis.

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

[38]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

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

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