Anisotropic Estimates for the Two-Dimensional Kuramoto–Sivashinsky Equation

We address the global solvability of the Kuramoto–Sivashinsky equation in a rectangular domain \([0,L_1]\times [0,L_2]\) . We give sufficient conditions on the width \(L_2\) of the domain, depending on the length \(L_1\) , so that the obtained solutions are global. Our proofs are based on anisotropic estimates.     

Local Dissipativity in L2 for the Kuramoto–Sivashinsky Equation in Spatial Dimension 2

We show the local dissipativity of the Kuramoto–Sivashinsky equation with periodic boundary conditions in a rectangular thin enough domain. More precisely, we give a sufficient condition on the width L ₂ of the domain, depending on the length L ₁, so that there exists a bounded local attracting set in _per ² which will be estimated, as well as its basin of attraction. We thus improve, and make more transparent, a result due to Sell and Taboada [14]. Finally, in the second part, we test our approach to another model to which it also applies.     

Global Attractor for a Nonlinear Oscillator Coupled to the Klein–Gordon Field

The long-time asymptotics is analyzed for all finite energy solutions to a model\(\mathbf{U}(1)\)-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as t→ ± ∞ to the set of all “nonlinear eigenfunctions” of the form ψ(x)e^{?iω t}. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time spectrum in the spectral gap [ ? m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single harmonic \(\omega\in[-m,m]\).The research is inspired by Bohr’s postulate on quantum transitions and Schrödinger’s identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled\(\mathbf{U}(1)\)-invariant Maxwell–Schrödinger and Maxwell–Dirac equations.     

Bounds on Coarsening Rates for the Lifschitz–Slyozov–Wagner Equation

This paper is concerned with the large time behavior of solutions to the Lifschitz–Slyozov–Wagner (LSW) system of equations. Point-wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data. These bounds complement the time averaged upper bounds obtained by Dai and Pego, and the point-wise in time upper and lower bounds obtained by Niethammer and Velasquez for solutions with initial data close to a self-similar solution.     

Finite Dimensional Global Attractor for a Fractional Schrödinger Type Equation with Mixed Anisotropic Dispersion

Journal of Dynamics and Differential Equations - We study the Cauchy problem for a class of nonlinear damped fractional Schrödinger type equation in a two dimensional unbounded domain. Then,...     

Solitary wave solutions to the Kuramoto–Sivashinsky equation by means of the homotopy analysis method

The homotopy analysis method (HAM) is used to find a family of solitary solutions of the Kuramoto–Sivashinsky equation. This approximate solution, which is obtained as a series of exponentials, has a reasonable residual error. The homotopy analysis method contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of series solution. This method is reliable and manageable.     

Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.     

Global Dynamics Above the Ground State for the Nonlinear Klein–Gordon Equation Without a Radial Assumption

We extend our result Nakanishi and Schlag in J. Differ. Equ. 250(5):2299–2333, 2011) to the non-radial case, giving a complete classification of global dynamics of all solutions with energy that is at most slightly above that of the ground state for the nonlinear Klein–Gordon equation with the focusing cubic nonlinearity in three space dimensions.     

Vortex Dynamics for the Nonlinear Maxwell–Klein–Gordon Equation

We derive the vortex dynamics for the nonlinear Maxwell–Klein–Gordon equation with the Ginzburg–Landau type potential. In particular, we consider the case when the external electric fields are of order \({O( | \log \epsilon |^{\frac{1}{2}})}\). We study the convergence of the space–time Jacobian \({\partial_t \psi \cdot i \nabla \psi}\) as an interaction term between the vortices and electric fields. An explicit form of the limiting vector measure is shown.     

Stability in the Kuramoto–Sakaguchi model for finite networks of identical oscillators

Nonlinear Dynamics - We study the Kuramoto–Sakaguchi model composed by N identical phase oscillators symmetrically coupled. Ranging from local (one-to-one, $$R=1$$ ) to global (all-to-all,...     

Bounded Domain Problem for the Modified Buckley–Leverett Equation

The focus of the present study is the modified Buckley–Leverett (MBL) equation describing two-phase flow in porous media. The MBL equation differs from the classical Buckley–Leverett (BL) equation by including a balanced diffusive–dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers a non-monotone water saturation profile for certain Riemann problems as suggested by the experimental observations. In this paper, we first show that for the MBL equation, the solution of the finite interval \([0,L]\) boundary value problem converges to that of the half line \([0,+\infty )\) boundary value problem exponentially as \(L\rightarrow +\infty \) . This result provides a justification for the use of the finite interval in numerical studies for the half line problem [Y. Wang and C.-Y. Kao, Central schemes for the modified Buckley–Leverett equation, J. Comput. Sci. 4(1–2), 12 – 23, 2013]. Furthermore, we numerically verify that the convergence rate is consistent with the theoretical derivation. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.     

The Visous Cahn–Hilliard Equation as a Limit of the Phase Field Model: Lower Semicontinuity of the Attractor

We consider the one-dimensional viscous Cahn–Hilliard equation with Dirichlet boundary conditions as the limit of a corresponding Dirichlet boundary value problem for the phase field model and we prove the convergence of the attractor. No assumption on the hyperbolicity of the stationary solutions is made.     

Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison

We consider two magnetohydrodynamic-α (MHDα) models with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). More precisely, we consider the Navier–Stokes-α-MHD and the Modified Leray-α-MHD models. Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. In both cases, the global existence of the solution and of a global attractor can be shown. We provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the long-time dynamics of the involved system and gives information about the numerical stability of the model. We get the same bound that holds for the Simplified Bardina-MHD model, considered in a previous paper (this result provides, in some sense, an intermediate bound between the number of degrees of freedom for the Simplified Bardina model and the Navier–Stokes-α equation in the nonmagnetic case). However, the Navier–Stokes-α-MHD system is preferable since, in the ideal case, it conserves more quadratic invariants derived from the standard MHD model.     

Global Existence for the Generalized Two-Component Hunter–Saxton System

We study the global existence of solutions to a two-component generalized Hunter–Saxton system in the periodic setting. We first prove a persistence result for the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter–Saxton system under proper assumptions on the initial data. This significantly improves recent results.     

A Birkhoff–Lewis Type Theorem for the Nonlinear Wave Equation

We give an extension of the celebrated Birkhoff–Lewis theorem to the nonlinear wave equation. Accordingly, we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin, which is an elliptic equilibrium of the associated infinite-dimensional Hamiltonian system.     

Local Exact Controllability for the One-Dimensional Compressible Navier–Stokes Equation

In this paper we deal with the isentropic (compressible) Navier-Stokes equation in one space dimension and we adress the problem of the boundary controllability for this system. We prove that we can drive initial conditions which are sufficiently close to some constant states to those constant states. This is done under some natural hypotheses on the time of control and on the regularity on the initial conditions.     

Stability for Rayleigh–Benard Convective Solutions of the Boltzmann Equation

We consider the Boltzmann equation for a gas in a horizontal slab, subject to a gravitational force. The boundary conditions are of diffusive type, specifying the wall temperatures, so that the top temperature is lower than the bottom one (Benard setup). We consider a 2-dimensional convective stationary solution, which for small Knudsen numbers is close to the convective stationary solution of the Oberbeck–Boussinesq equations, near and above the bifurcation point, and prove its stability under 2-d small perturbations, for Rayleigh numbers above and close to the bifurcation point and for small Knudsen numbers.     

On the Regularity Set and Angular Integrability for the Navier–Stokes Equation

We investigate the size of the regular set for suitable weak solutions of the Navier–Stokes equation, in the sense of Caffarelli–Kohn–Nirenberg (Commun Pure Appl Math 35:771–831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space \({\{t > 0\}}\) in an appropriate limit. In particular, we obtain that if the \({L^{2}}\) norm with weight \({|x|^{-\frac12}}\) of the data tends to 0, the regular set invades \({\{t > 0\}}\); this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).     

On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

In this work we investigate unique continuation properties of solutions to the initial value problem associated to the Benjamin–Ono equation in weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb R )\cap L^2(|x|^{2r}dx)$ for $s\in \mathbb R $ , and $s\ge 1$ , $s\ge r$ . More precisely, we prove that the uniqueness property based on a decay requirement at three times can not be lowered to two times even by imposing stronger decay on the initial data.