We consider Kirchhoff equations for vibrating bodies in any dimension in presence of a time-periodic external forcing with period 2pi/omega and amplitude epsilon, both for Dirichlet and for space-periodic boundary conditions.
We prove existence, regularity and local uniqueness of time-periodic solutions of period 2pi/omega and order epsilon, by means of a Nash-Moser iteration scheme. The results hold for parameters (omega, epsilon) in Cantor sets having measure asymptotically full as epsilon tends to 0.
(What's new in version 2: the case of finite-order Sobolev regularity, the case of space-periodic boundary conditions, a different iteration scheme in the proof, some references).
In this paper we prove the existence of small-amplitude quasi-periodic solutions with Sobolev regularity, for the $d$-dimensional forced Kirchhoff equation with periodic boundary conditions. This is the first result of this type for a quasi-linear equations in high dimension. The proof is based on a Nash-Moser scheme in Sobolev class and a regularization procedure combined with a multiscale analysis in order to solve the linearized problem at any approximate solution.
It is well-known that the classical hyperbolic Kirchhoff equation admits infinitely many simple modes, namely time-periodic solutions with only one Fourier component in the space variables. In this paper we assume that, for a suitable choice of the nonlinearity, there exists a heteroclinic connection between two simple modes with different frequencies. Under this assumption, we cook up a forced Kirchhoff equation that admits a solution that blows-up in finite time, despite the regularity and boundedness of the forcing term. The forcing term can be chosen with the maximal regularity that preven…
We prove the existence of infinitely many nontrivial weak periodic solutions for a class of fractional Kirchhoff problems driven by a relativistic Schrödinger operator with periodic boundary conditions and involving different types of nonlinearities.
We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$ with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound $\varepsilon^{-2}$ given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.
We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig-Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on ${\mathbb T}^\infty$.
We describe the asymptotic behaviour of solutions of unviscid Burgers equation on the circle with time-periodic forcing. These solutions converge to periodic states, but the period of these limit states may be greater than the period of the forcing. We obtain as a corollary a new result on the flow of the associated Hamiltonian system.
We study topological structure of the $ω$-limit sets of the skew-product semiflow generated by the following scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f(t,u,u_x)$ is $C^2$-admissible with time-recurrent structure including almost-periodicity and almost-automorphy. Contrary to the time-periodic cases (for which any $ω$-limit set can be imbedded into a periodically forced circle flow), it is shown that one cannot expect that any $ω$-limit set can be imbedded into an almost-periodically force…
In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost-periodically forced scalar reaction-diffusion equation \begin{equation}\label{eq0} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\, 0<x<L \end{equation} with periodic boundary condition \begin{equation} \label{bdc1} u(t,0)=u(t,L),\quad u_x(t,0)=u_x(t,L), \end{equation} where $f$ is uniformly almost periodic in $t$. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is proved that any compact minimal invariant set (throughout this paper, we…
In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrödinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schrödinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.
We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.
We consider several models of nonlinear wave equations subject to very strong damping and quasi-periodic external forcing. This is a singular perturbation, since the damping is not the highest order term. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under very general non-resonance conditions on the frequency, we show the existence of asymptotic expansions of the response solution; moreover, we prove that the response solution indeed exists and depends analytically on $\varepsilon$ (where $\varepsilon$ is the inverse of t…
We consider the skew-product semiflow which is generated by a scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f$ is uniformly almost periodic in $t$. The structure of the minimal set $M$ is thoroughly investigated under the assumption that the center space $V^c(M)$ associated with $M$ is no more than $2$-dimensional. Such situation naturally occurs while, for instance, $M$ is hyperbolic or uniquely ergodic. It is shown in this paper that $M$ is a $1$-cover of the hull $H(f)$ provided that $M$ is…
We study the non-autonomously forced Burgers equation $$ u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t) $$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \mathbb{R}$. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.
Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2π\mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.
In this paper we consider a class of fully nonlinear forced and reversible Schroedinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such has diffeomorphisms of the torus and pseudo-differential operators. This procedure automtically produces a change of variables, well def…
In this paper we consider the long time behavior of the weakly damped, forced Korteweg-de Vries equation in the Sololev spaces of the negative indices in the periodic case. We prove that the solutions are uniformly bounded in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$. Moreover, we show that the solution-map possesses a global attractor in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$, which is a compact set in $H^{s+3}(\T)$.
In this paper we prove the existence and the stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for the 1-dimensional forced Kirchoff equation with periodic boundary conditions. This is the first KAM result for a quasi-linear wave-type equation. The main difficulties are: $(i)$ the presence of the highest order derivative in the nonlinearity which does not allow to apply the classical KAM scheme, $(ii)$ the presence of double resonances, due to the double multiplicity of the eigenvalues of $- \partial_{xx}$. The proof is based on a Nash-Moser scheme in Sobolev class…
Criteria for the existence of $T$-periodic solutions of nonautonomous parabolic equation $u_t = Δu + f(t,x,u)$, $x\in\mathbb{R}^N$, $t>0$ with asymptotically linear $f$ will be provided. It is expressed in terms of time average function $\hat f$ of the nonlinear term $f$ and the spectrum of the Laplace operator $Δ$ on $\mathbb{R}^N$. One of them says that if the derivative $\hat f_\infty$ of $\hat f$ at infinity does not interact with the spectrum of $Δ$, i.e. $\mathrm{Ker} (-Δ+ \hat f_\infty)=\{0\}$, then the parabolic equation admits a $T$-periodic solution. Another theorem is derived in the…
Stochastically forced Navier-Stokes equations interacting with an elastic structure
University of California at Berkeley（加州大学伯克利分校）；Technische Universität Darmstadt（达姆施塔特工业大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基奎，巴斯克科学基金会）
In this paper we prove the existence of quasi-periodic, small-amplitude, solutions for quasi-linear Hamiltonian perturbations of the non-linear Schroedinger equation on the torus in presence of a quasi-periodic forcing. In particular we prove that such solutions are linearly stable. The proof is based on a Nash-Moser implicit function theorem and on a reducibility result on the linearized operator in a neighborhood of zero. The proof of the reducibility relies on changes of coordinates such as diffeomorphisms of the torus, pseudo-differential operators and a KAM-reducibility arguments. Due to …
We establish the existence and uniqueness of an ergodic invariant measure for 2D fractionally dissipated stochastic Euler equations on the periodic box, for any power of the dissipation term.
We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods inspired to regularity theory of Rabinowitz
In the present paper, we establish the uniqueness and nondegeneracy of positive energy solutions to the Kirchhoff equation \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1<p<5$ are constants. Then, as applications, we derive the existence and local uniqueness of solutions to the perturbed Kirchhoff problem \begin{eqnarray*} -\left(ε^2a+εb\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+V(x)u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3} \end{eqnarray*} for $ε>0$ sufficiently small, under some mild assump…
We consider a conservative system consisting of an elastic plate interacting with a gas filling a semi-infinite tube. The plate is placed on the bottom of the tube. The dynamics of the gas velocity potential is governed by the linear wave equation. The plate displacement satisfies the linear Kirchhoff equation. We show that this system possesses an infinite number of periodic solutions with the frequencies tending to infinity. This means that the well-known property of decaying of local wave energy in tube domains does not hold for the system considered.
We study the asymptotic behavior of the forced linear Euler and nonlinear Navier-Stokes equations close to Couette flow in a periodic channel. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $H^{s}, s>-1$. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier-Stokes equations. Hence, these results provide an indication that nonli…
This is a habilitation self-report describing results on forced periodic solutions of partial differential equations. Here averaging principle in the spirit of Bogoliubov and Mitropolsky for ODE's and Henry for parabolic equations are used together with topological methods such as fixed point index and topological degree.
We present and summarize the results of recent studies on non-Euclidean plates with imposed constant negative Gaussian curvature in both the Föppl - von Kármán and Kirchhoff approximations. Motivated by experimental results we focus on annuli with a periodic profile. We show that in the Föppl - von Kármán approximation there are only two types of global minimizers -- flat and saddle shaped deformations with localized regions of stretching near the boundary of the annulus. We also show that there exists local minimizers with $n$-waves that have regions of stretching near their lines of inflecti…
We construct solutions to the randomly-forced Navier--Stokes--Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns of the problem. Additionally, these solutions dissipate energy, satisfies a relative energy inequality in the sense of [4] and satisfy a renormalized form of the continuity equation in the sense…
