Explicit estimates on the torus for the sup-norm and the crest factor of solutions of the modified Kuramoto-Sivashinky equation in one and two space dimensions

DOI10.1007/s10884-019-09762-1OpenAlexW2964211179WikidataQ115383430 ScholiaQ115383430MaRDI QIDQ2181125

Guido Gentile, Michele V. Bartuccelli, Jonathan H. B. Deane

Publication date: 18 May 2020

Published in: Journal of Dynamics and Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1609.09394

zbMATH Keywords

periodic boundary conditionsbest constantsinterpolation inequalitiesdissipative partial differential equationscrest factor

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Initial-boundary value problems for higher-order parabolic equations (35K35) Sobolev spaces and other spaces of ``smooth functions, embedding theorems, trace theorems (46E35) A priori estimates in context of PDEs (35B45) Hilbert spaces of continuous, differentiable or analytic functions (46E20) Semilinear parabolic equations (35K58)

Related Items (3)

On the crest factor and its relevance in detecting turbulent behaviour in solutions of partial differential equations ⋮ On the crest factor for dissipative partial differential equations ⋮ On the nature of space fluctuations of solutions of dissipative partial differential equations

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Anisotropic estimates for the two-dimensional Kuramoto-Sivashinsky equation
• Sharp constants for the \(L^{\infty }\)-norm on the torus and applications to dissipative partial differential equations.
• Relations between energy and enstrophy on the global attractor of the 2-D Navier-Stokes equations
• Sharp estimates for the number of degrees of freedom for the damped-driven 2-D Navier-Stokes equations
• Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations
• Brézis-Wainger inequality on Riemannian manifolds
• On the Eckhaus instability for spatially periodic patterns
• The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems
• On the dimension of the attractors in two-dimensional turbulence
• Integral manifolds and inertial manifolds for dissipative partial differential equations
• Integral representation and \(L^ \infty\) bounds for solutions of the Helmholtz equation on arbitrary open sets in \(\mathbb{R}^ 2\) and \(\mathbb{R}^ 3\)
• Infinite-dimensional dynamical systems in mechanics and physics.
• Sharp Sobolev interpolation inequalities for the Stokes operator
• The energy method, stability, and nonlinear convection.
• Local dissipativity in \(L^2\) for the Kuramoto-Sivashinsky equation in spatial dimension 2
• A global attracting set for the Kuramoto-Sivashinsky equation
• Pattern selection of solutions of the Swift--Hohenberg equation
• Length scales in solutions of the complex Ginzburg-Landau equation
• Chaotic spatial patterns described by the extended Fisher-Kolmogorov equation
• Explicit estimates on the torus for the sup-norm and the dissipative length scale of solutions of the Swift-Hohenberg equation in one and two space dimensions
• Global existence and analyticity for the 2D Kuramoto-Sivashinsky equation
• Sharp constants in the Sobolev embedding theorem and a derivation of the Brezis-Gallouet interpolation inequality
• Asymptotic expansions and extremals for the critical Sobolev and Gagliardo–Nirenberg inequalities on a torus
• Brezis–Gallouet–Wainger type inequality for irregular domains†
• Length scales in solutions of the Navier-Stokes equations
• The Well-Posedness of the Kuramoto–Sivashinsky Equation
• Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations
• Local dissipativity and attractors for the Kuramoto-Sivashinsky equation in thin 2D domains
• Nonlinear dynamics of a dispersive anisotropic Kuramoto–Sivashinsky equation in two space dimensions
• Positivity and the attractor dimension in a fourth-order reaction-diffusion equation
• Best Constants in Multiplicative Inequalities for Sup-Norms
• Applied Analysis of the Navier-Stokes Equations
• Some Canonical Bifurcations in the Swift–Hohenberg Equation
• Pattern formation outside of equilibrium
• Lieb-Thirring integral inequalities and their applications to attractors of the Navier-Stokes equations

This page was built for publication: Explicit estimates on the torus for the sup-norm and the crest factor of solutions of the modified Kuramoto-Sivashinky equation in one and two space dimensions