Jens Wirth
In this short note we discuss the influence of a time-periodic dissipation term on a-priori estimates for solutions to dissipative wave equations. The approach is based on a diagonalisation argument for high frequencies and results from spectral theory of periodic differential equations / Floquet theory for bounded frequencies.
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This paper establishes suficient conditions for the orbital stability of one-parameter spatially periodic traveling-wave solutions for one-dimensional dispersive equations. Our method of proof combines known techniques with some new ideas. As a consequence of our result, we give several applications for well known dispersive equations. Extension of the theory to regularized equations is also established.
In this paper, we establish a new criterion for the orbital stability of periodic waves related to a general class of regularized dispersive equations. More specifically, we present sufficient conditions for the stability without knowing the positiveness of the associated hessian matrix. As application of our method, we show the orbital stability for the fifth-order model. The orbital stability of periodic waves resulting from a minimization of a convenient functional is also proved.
We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.
We consider the stability and instability of periodic travling waves for Korteweg-de Vries type equations with fractional dispersion and other nonlinear dispersive equations. We establish that a constrained minimizer for the related variational problem is nonlinearly stable to period preserving perturbations. We also discuss when the associated linearized equation admits exponentially growing solutions. The proof utilizes the variational structure of the equation.
We consider weak solutions to dispersive partial differential equations with periodic boundary conditions and initial data with jump discontinuities. These are already known to be continuous at irrational times and piecewise constant at rational times; we show that as time approaches a rational value the solution exhibits a ringing effect, with the characteristic overshoot of fixed amplitude near the discontinuities. Furthermore this effect is the same whether the sequence of times follows rational or irrational values.
We study the question of existence of time-periodic, spatially periodic solutions for dispersive evolution equations, and in particular, we introduce a framework for demonstrating the nonexistence of such solutions. We formulate the problem so that doubly periodic solutions correspond to fixed points of a certain operator. We prove that this operator is locally contracting, for almost every temporal period, if the Duhamel integral associated to the evolution exhibits a weak smoothing property. This implies the nonexistence of nontrivial, small-amplitude time-periodic solutions for almost every…
In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic dnoidal waves associat…
We consider surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involving conservation of a certain family of moduli of continuity.
Dispersive estimates for discrete Klein-Gordon equations on one-dimensional lattice with quasi-periodic potentials
研究一维晶格上小实解析准周期势的离散 Klein-Gordon 方程的色散估计，证明 $\ell^{1}\!\to\!\ell^{\infty}$ 估计并获得 Strichartz 估计及非线性方程的小数据全局适定性。
We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive operator behaves for high frequencies as a Fourier multiplier by $ i |ξ|^αξ$, with $ 1\le α\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $ f $ is the sum of an entire series with infinite radius of convergence. Under these conditions, we prove the unconditional local well-posedness of the Cauchy problem in $H^{s}(\mathbb{T})$ for $ s\ge 1-\fracα{2(α+1)}$. This leads to some global existence …
In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in $H^{s}_{p}(\mathbb{T})$ with $s\in \mathbb{R}$. We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given deca…
Massera's theorems for a higher order dispersive system
本文针对有限区域上的高阶色散Kawahara系统，利用半群性质与解的衰减性建立了Massera型定理，证明了其解具有周期性、拟周期性和几乎周期性。
Low-regularity well-posedness for dispersive equations with derivative nonlinearity and quasi-periodic initial data
该研究证明了带导数非线性项与准周期初值的Korteweg-de Vries方程的低正则适定性，通过频率依赖时间局部化等方法推导双线性Strichartz估计，得到的解可保持初值的Sobolev正则性，且论证可推广至其他色散关系及高阶非线性项。
Well-posedness of the periodic dispersion-generalized Benjamin-Ono equation in the weakly dispersive regime
研究周期色散广义Benjamin-Ono方程，通过短时Fourier限制方法，在$\alpha\in(0,1)$时获得$H^s$（$s>3/2-\alpha$）中的局部适定性。
We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix $a^\varepsilon$ that is periodic with characteristic length scale $\varepsilon$; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions $u^\varepsilon$ well by a non-dispersive wave equation on fixed time intervals $(0,T)$. Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive eq…
Local Wellposedness of dispersive equations with quasi-periodic initial data
本文在准周期函数空间中证明了一类含导数非线性的色散方程的无条件局部适定性，覆盖广义KdV、广义Benjamin-Ono及导数非线性Schrödinger方程，并以更简短论证改进已有结果。
We study the phenomenon of revivals for the linear Schrödinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast with the case of the linear Schrödinger equation examined recently (which we develop further), we prove that, remarkably, the Airy equation does not generally exhibit revivals even for boundary conditions very close to periodic. We also describe a new, weaker form of revival phenomena, present in the case of certain Robin-type boundary conditions for the linear Schrödinger equation. In this weak revival, the dich…
In this paper we consider a wave model with non-effective mass and dissipation terms and provide asymptotic descriptions of its representation of solutions. In particular we conclude sharp estimates for a corresponding energy and estimates of dispersive type.
In this paper we obtain global well-posedness in low order Sobolev spaces of higher order KdV type equations with dissipation. The result is optimal in the sense that the flow-map is not twice continuously differentiable in rougher spaces. The solution is shown to be smooth for positive times.
Of concern is the a priori symmetry of traveling wave solutions for a general class of nonlocal dispersive equations \[ u_t +(u^2 +Lu)_x =0, \] where $L$ is a Fourier multiplier operator with symbol $m$. Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a class of symbols $m$ guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. Particularly, instead of considering waves with a unique crest and trough per period or a monotone structure near troughs as classically imposed in the water wave problem, we formu…
We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal $ α$-Hölder continuity in a class of fractional negative-order dispersive equations of the form \begin{equation*} u_t + (| \mathrm{D} |^{- α} u + n(u) )_x = 0 \end{equation*} for every $ α\in (0, 1) $ with homogeneous Fourier multiplier $ | \mathrm{D} |^{ - α} $. We tackle nonlinearities $ n(u) $ of the type $ | u |^p $ or $ u | u |^{p - 1} $ for all real $ p > 1 $, and show that when $ n $ is odd, the waves also feature antisymmetry and thus contain inverted cusps. Tools involve detailed point…
The present paper deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves. Our method can be seen as an extension to spatially periodic waves of the theory of solitary waves recently developed in \cite{st}. Firstly, our main result do not depend on the parametrization of the periodic wave itself. Secondly, motived by the well known orbital stability criterion for solitary waves, we show that the same criterion holds for periodic waves. In addition, we show that the positiveness of the principal ent…
We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schrödinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs; Second, for KDV type equations the loss of derivative in the nonlinear term is overcome in two complementary cases: (1) for smooth nonlinear…
Dispersive-Dissipative estimates for Boussinesq and other generalized wave equations
针对一类广义波动方程（特别是粘性Boussinesq方程），推导了与相位函数几何假设及低高频退化相关的色散-耗散估计，并分析了耗散对解衰减率的影响。
We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $\infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.
We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in $\R^n$, $n \in \{1, 2, 3\}$. Standard homogenization theory provides, for the limit of a small periodicity length $\eps>0$, an effective second order wave equation that describes solutions on time intervals $[0,T]$. In order to approximate solutions on large time intervals $[0,T\eps^{-2}]$, one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation, and an estimate for errors between the solution of the…
We consider the $2$D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the correponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.
Parameter-free higher-order Schrodinger systems with weak dissipation and forcing
本文针对含弱耗散和强迫的通用无限维色散PDE，提出一种无参数的高阶演化方程推导方法，利用Hilbert空间和Frechet导数避免复杂代数，适用于水波、非线性光学等场景。
New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations
本文研究双向色散方程在周期边界和间断初值下的复兴与分形化现象，证明其依赖于色散关系的大波数渐近，并通过非线性梁方程数值实验验证。
Large torus limit of global dynamics of the two-dimensional dispersive Anderson model
研究二维色散安德森模型在周期域和全空间的动力学收敛性，引入周期权重和加权函数空间控制噪声增长。
Considered herein is the reducibility of the quasi-periodically time dependent linear dynamical system with a diophantine frequency vector $ω\in \mathcal{O}_0 \subset \mathbb{R}^ν$. This system is derived from linearizing the dispersive Camassa-Holm equation with unbounded perturbations at a small amplitude quasi-periodic function. It is shown that there is a set $\mathcal{O}_{\infty} \subset \mathcal{O}_0$ of asymptotically full Lebesgue measure such that for any $ω\in \mathcal{O}_{\infty}$, the system can be reduced to the one with constant coefficients by a quasi-periodic linear transformat…
We consider transverse propagation of electromagnetic waves through a two-dimensional composite material containing a periodic rectangular array of circular cylinders. Propagation of waves is described by the Helmholtz equation with the continuity conditions the tangential components of the electric and magnetic fields on the boundaries of the cylinders. We assume that the dimensionless wave frequency $ν\ll 1$ that allows us to view the governing equation as a perturbation of the Laplace equation. We show that the eigenfunctions and the eigenvalues are even analytic functions of the magnitude …
Euler-flocking system with nonlocal dissipation in 1D: periodic entropy solutions
该研究针对一维周期域上带线性压力与全连接核的 Euler-flocking 系统，在有限总变差、密度正下界初值下建立周期熵弱解的全局存在性。
Optimal dispersion for discrete periodic Schrödinger operators
该研究证明了一维周期离散薛定谔算子的最优衰减速率色散估计，并借助标准方法推导了周期哈密顿量下小初值、合适非线性项的离散非线性薛定谔方程的色散估计。
A dispersive effective equation for transverse propagation of planar shallow water waves over periodic bathymetry
针对周期地形上的浅水波传播问题，采用多尺度摄动理论推导常系数有效方程组，结合Boussinesq型方程与地形积分函数描述波的特性，经数值验证其与变地形浅水方程吻合良好。
The quasigeostrophic equation is a prototypical geophysical fluid model. In this paper, we consider time-periodic motions of this model under dissipation and time-dependent wind forcing. We show that when the wind forcing is time-periodic and the spatial square-integral of the wind forcing is bounded in time, the full nonlinear quasigeostrophic model has time-periodic motions, under some conditions on $β$ parameter, Ekman number, viscousity and the domain size.
We consider a dissipative, dispersive system of Boussinesq type, describing wave phenomena in settings where dissipation has an effect. Examples include undular bores in rivers or oceans where dissipation due to turbulence is important for their description. We show that the model system admits traveling wave solutions known as diffusive-dispersive shock waves, and we categorize them into oscillatory and regularized shock waves depending on the relationship between dispersion and dissipation. Comparison of numerically computed solutions with laboratory data suggests that undular bores are accu…
We present local existence and uniqueness results for the following $2+1$ dispersive diffusive equation due to P. Hall arising in modeling of river braiding: $$u_{yyt} - γu_{xxx} -αu_{yyyy} - βu_{yy} + \left (u^2 \right)_{xyy} = 0$$ for $(x,y) \in [0, 2π] \times [0, π]$, $t> 0$, with boundary condition $u_{y}=0=u_{yyy}$ at $y=0$ and $y=π$ and $2π$ periodicity in $x$, using a contraction mapping argument in a Bourgain-type space $T_{s,b}$. We also show that the energy $\| u \|^2_{L^2} $ and cumulative dissipation $\int_0^t \| u_y \|_{L^2}^2 dt$ are globally controlled in time.
For both localized and periodic initial data, we prove local existence in classical energy space $H^s, s>\frac{3}{2}$, for a class of dispersive equations $u_{t}+(n(u))_{x}+Lu_{x}=0$ with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators $L$ whose symbol is of temperate growth, and $n(\cdot)$ in local Sobolev space $H^{s+2}_{\mathrm{loc}}(\mathbb{R})$. In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semi-group methods and a new composition result for Besov spaces. In…
Stability for the Boussinesq Equations with Horizontal Dissipation near the Hydrostatic Balance on $\mathbb{R}^2$
研究二维平面上近静水平衡处仅具水平耗散的Boussinesq方程，利用速度-温度耦合产生内重力波及色散衰减与水平耗散构成稳定机制，并对特定初始扰动建立经典解全局存在唯一性及给出速度和温度衰减率。
Global solutions of Euler-Maxwell equations with dissipation
该研究针对无离子背景的带阻尼Euler-Maxwell系统柯西问题，借鉴可压缩Euler系统相关研究方法，证明了满足色散条件的光滑初值下全局强解的存在唯一性及时间一致衰减性。
We consider the solution to the 2D Navier-Stokes equations around the Poiseuille flow $(y^2,0)$ on $\mathbb{T}\times\mathbb{R}$ with small viscosity $ν>0$. Via a hypocoercivity argument, we prove that the $x-$dependent modes of the solution to the linear problem undergo the enhanced dissipation effect with a rate proportional to $ν^{\frac{1}{2}}$. Moreover, we study the nonlinear enhanced dissipation effect and we establish a transition threshold of $ν^{\frac{2}{3}+}$. Namely, when the perturbation of the Poiseuille flow is size at most $ν^{\frac{2}{3}+}$, its size remains so for all times and…
We investigate the analyticity of the attractors of a class of Kuramoto-Sivashinsky type pseudo-differential equations in higher dimensions, which are periodic in all spatial variables and possess a universal attractor. This is done by fine-tuning the techniques used in a previous work of the second author, which are based on an analytic extensibility criterion involving the growth of $\nabla^n u$, as $n$ tends to infinity (here $u$ is the solution). These techniques can now be utilised in a variety of higher dimensional equations possessing universal attractors, including Topper--Kawahara equ…
We consider the initial value problem for the 2D quasi-geostrophic equation with weak dissipation term $κ(-Δ)^{α/2}θ (0<α\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a given initial data $θ_0$ which belongs to the scaling subcritical Sobolev space $H^s(\mathbb{R}^2)\ (s>2-α)$ if the size of dispersion parameter is sufficiently large. This phenomenon is so-called the global regularity. We also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. Moreover, we show the glob…
Bounds on unstable spectrum for dispersive Hamiltonian PDEs
针对 Hamiltonian PDE 周期行波的拟周期特征值问题，利用 Gershgorin 圆盘与谱对称性给出离轴不稳定特征值区域及数量的界。
We study an evolution equation that is the gradient flow in the $2$-Wasserstien metric of a non-convex functional for densities in $\mathbb{R}^n$ with $n \geq 3$. Like the Patlack-Keller-Segel system on $\mathbb{R}^2$, this evolution equation features a competition between the dispersive effects of diffusion, and the accretive effects of a concentrating drift. We determine a parameter range in which the diffusion dominates, and all mass leaves any fixed compact subset of $\mathbb{R}^n$ at an explicit polynomial rate.
Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg--de Vries equations
本文将KdV方程椭圆余弦波的线性化动力学研究推广到广义KdV方程中不受Benjamin–Feir不稳定性影响的小振幅周期行波，并证明在色散谱稳定性条件下，任意振幅波也具有Sobolev空间全局有界稳定性和色散型渐近稳定性。
In this paper, the local well-posedness of periodic fifth order dispersive equation with nonlinear term $P_1(u)\p_xu + P_2(u)\p_x u\p_xu $. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. We also get some new Strichartz estimates.
Asymptotic profiles of solutions for the generalized Fornberg-Whitham equation with dissipation
本文研究带耗散的广义 Fornberg-Whitham 方程 Cauchy 问题，证明其解收敛于修正热核，并构造第二渐近项以分析色散、耗散与非线性项的影响。
A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homogenisation limits to a class of time-dispersive media.
We show, by the means of several examples, how we can use Gibbs measures to construct global solutions to dispersive equations at low regularity. The construction relies on the Prokhorov compactness theorem combined with the Skorokhod convergence theorem. To begin with, we consider the non linear Schrödinger equation (NLS) on the tri-dimensional sphere. Then we focus on the Benjamin-Ono equation and on the derivative nonlinear Schrödinger equation on the circle. Next, we construct a Gibbs measure and global solutions to the so-called periodic half-wave equation. Finally, we consider the cubic …
Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$
本文研究二维环面上时间周期 Lipschitz 速度场驱动的对流-扩散方程，证明其在 $|\log ν|$ 时间尺度上具有最优增强耗散，并进一步得到无扩散情形的指数混合结果。
A dispersive extension result for a class of nonlocal nonlinearities
针对周期Sobolev空间上的一类非局部非线性项，通过将其重新表述为自由薛定谔算子生成的酉群下实解析函数的共振轨道平均，证明其可表示为受迫薛定谔方程系统温和解的迹，并将该结果应用于非局部振幅方程的等价局部重新表述。
Global results for weakly dispersive KP-II equations on the cylinder
本文研究圆柱上弱色散广义 KP-II 方程，利用 Fourier 解耦得到尖锐 Strichartz 估计，证明拟线性问题在 L² 中的全局适定性，并借助 Kato 光滑效应证明小质量解的长时间衰减。
Large-scale dispersive estimates for acoustic operators: homogenization meets localization
本工作结合均匀化算子的色散估计与波动方程的定量均匀化，推导出无序介质中波的大尺度色散估计，并应用于低能本征态扩展，证明了周期介质声学算子低谱的纯绝对连续性，为准周期或随机介质本征态局域化长度提供了新下界。
In this paper we characterize the Nazarov-Sklyanin hierarchy for the classical periodic Benjamin-Ono equation in two complementary degenerations: for the multi-phase initial data (the periodic multi-solitons) at fixed dispersion and for bounded initial data in the limit of small dispersion. First, we express this hierarchy in terms of a piecewise-linear function of an auxiliary real variable which we call a dispersive action profile and whose regions of slope $\pm 1$ we call gaps and bands, respectively. Our expression uses Kerov's theory of profiles and Kre\uın's spectral shift functions. Nex…
Jumps and cusps: a new revival effect in local dispersive PDEs
本文研究Airy方程与错位Laplacian Schrödinger方程的Dirichlet边值问题，证明初始跳跃间断会在适当时刻复兴为跳跃间断与对数尖点奇点，并阐明其成因。
Exponential mixing for random nonlinear wave equations: weak dissipation and localized control
本文提出随机动力系统指数混合的新判据，应用于具退化阻尼和局部化噪声的非线性波动方程，解决弱耗散与随机性交互的挑战。
We consider a one-dimensional Swift-Hohenberg equation coupled to a conservation law, where both equations contain additional dispersive terms breaking the reflection symmetry $x \mapsto -x$. This system exhibits a Turing instability and we study the dynamics close to the onset of this instability. First, we show that periodic traveling waves bifurcate from a homogeneous ground state. Second, fixing the bifurcation parameter close to the onset of instability, we construct modulating traveling fronts, which capture the process of pattern-formation by modeling the transition from the homogeneous…
Regularity results on the flow maps of periodic dispersive Burgers type equations and the Gravity-Capillary equations
本文证明周期色散Burgers方程和重力-毛细波方程的流映射在特定Sobolev空间中的Lipschitz正则性，并验证了先前结果的最优性。
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