Yanping Cao, Edriss S. Titi
We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in $R$ and $R^2$ are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic problems in $\R^N$.
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In this paper, the invariant subspace method is applied to the time fractional modified Kuramoto-Sivashinsky partial differential equation. The obtained reduced system of nonlinear ordinary fractional equations is solved by the Laplace transform method and with using of some useful properties of Mittag-Leffler function. Then, some exact solutions of the time fractional nonlinear studied equation are found.
Review of some modified generalized Korteweg - de Vries - Kuramoto-Sivashinsky equations (mgKdV-KS)
本文综述了 KdV、KS、广义 KdV-KS 方程以及四分之一平面上 KdV-KS 非齐次边值问题解的存在性与唯一性结果。
We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove that ||u||_2 < C L^1.5. This result is slightly weaker than that recently announced by Giacomelli and Otto, but applies in the presence of an additional linear destabilizing term. We further show that for a large class of Lyapunov functions ϕthe exponent 1.5 is the best possible from this line of …
In this article we study the solution of the Kuramoto-Sivashinsky equation (for surface erosion or surface growth) on a bounded interval subject to a random forcing term. We show that a unique solution to the equation exists for all time and depends continuously on the initial data.
We introduce a new imaginary-Brownian-time-Brownian-angle process, which we also call the linear-Kuramoto-Sivashinsky process (LKSP). Building on our techniques in two recent articles involving the connection of Brownian-time processes to fourth order PDEs, we give an explicit solution to a linearized Kuramoto-Sivashinsky PDE in $\scriptstyle d$-dimensional space: $\scriptstyle{\eight{u_t=-\frac18Δ^2u-\frac12Δu-\frac12u}}$. The solution is given in terms of a functional of our LKSP.
Initial boundary value problems for the three dimensional Kuramoto-Sivashinsky equation posed on unbounded 3D grooves were considered. The existence and uniqueness of global strong solutions as well as their exponential decay have been established.
The two-dimensional anisotropic Kuramoto-Sivashinsky equation is a forth-order nonlinear evolution equation in two spatial dimensions that arises in sputter erosion and epitaxial growth on vicinal surfaces. A generalization of this equation is proposed and studied via group analysis methods. The complete group classification of this generalized Kuramoto-Sivashinsky equation is carried out, it is classified according to the property of the self-adjointness and the corresponding conservation laws are established.
Initial-boundary value problems for the $n$-dimensional ($n$ is a natural number from the interval [2,7]) Kuramoto-Sivashinsky equation posed on smooth bounded domains in $\mathbb{R}^n$ were considered. The existence and uniqueness of global regular solutions as well as their exponential decay have been considered.
The nonlocal Kuramoto-Sivashinsky equation arises in the modeling of the flow of a thin film of viscous liquid falling down an inclined plane, subject to an applied electric field. In this paper, the authors show that, as the coefficient of the nonlocal integral term goes to zero, the solution trajectories and the maximal attractor of the nonlocal Kuramoto-Sivashinsky equation converge to those of the usual Kuramoto-Sivashinsky equation.
An initial-boundary value problem for the n-dimensional ($n$ is a natural number from the interval [2,7]) Kuramoto-Sivashinsky-Zakharov-Kuznetsov equation posed on smooth bounded domains in $\mathbb{R}^n$ was considered. The existence and uniqueness of global regular solutions as well as their exponential decay have been established. A connection between an order of the stationary part of the equation and admissible dimensions of a domain has been revealed. Keywords: Kuramoto-Sivashinsky equation, Zakharov-Kuznetsov equation, Global solutions
We study the initial boundary value problem for one-dimensional Kuramoto-Sivashinsky equation with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results on the Cauchy problem, we obtain both the local well-posedness and the global well-posedness for the nonhomogeneous initial boundary value problem. It is shown that the Kuramoto-Sivashinsky equation is well-posed in Sobolev space $C([0,T]; H^s (R^+)) \bigcap L^2(0,T; H^{s+2}(R^+))$ for $s>-2$.
We introduce the planar helical flows on three dimensional torus and study the dissipation enhancement of such flows. We then use such flows as transport flows to solve the three dimensional advective Kuramoto-Sivashinsky and Keller-Segel equations. The global well-posedness of the Kuramoto-Sivashinsky equation is achieved when the linearized operator does not have growing mode in the direction orthogonal to the flow. The global classical solution of the three dimensional Keller-Segel is ensured for any size of the torus with arbitrarily large initial data.
We consider the Kuramoto-Sivashinsky equation (KSE) on the two-dimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in $L^2$, using a bootstrap argument. The initial data can be taken arbitrarily large.
In this paper we study the effects of a ``nonlocal'' term on the global dynamics of the Kuramoto-Sivashinsky equation. We show that the equation possesses a ``family of maximal attractors'' parameterized by the mean value of the initial data. The dimension of the attractor is estimated as a function of the coefficient of the nonlocal term and the width of the periodic domain.
In this note, we announce a general result resolving the long-standing question of nonlinear modulational stability, or stability with respect to localized perturbations, of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation, establishing that spectral modulational stability, defined in the standard way, implies nonlinear modulational stability with sharp rates of decay. The approach extends readily to other second- and higher-order parabolic equations, for example, the Cahn-Hilliard equation or more general thin film models.
We study a nonlocal equation, analogous to the Kuramoto-Sivashinsky equation, in which short waves are stabilized by a possibly fractional diffusion of order less than or equal to two, and long waves are destabilized by a backward fractional diffusion of lower order. We prove the global existence, uniqueness, and analyticity of solutions of the nonlocal equation and the existence of a compact attractor. Numerical results show that the equation has chaotic solutions whose spatial structure consists of interacting traveling waves resembling viscous shock profiles.
In this work, we study the initial-value problem associated with the Kuramoto-Sivashinsky equation. We show that the associated initial value problem is locally and globally well-posed in Sobolev spaces $H^s(\mathbb{R})$, where $s>1/2$. We also show that our result is sharp, in the sense that the flow-map data-solution is not $C^2$ at origin, for $s<1/2$. Furthermore, we study the behavior of the solutions when $μ\downarrow 0$.
This paper deals with the null controllability of a coupled parabolic system, which is Kuramoto-Sivashinsky-Korteweg-de Vries equation coupled with heat equation through first order derivative. More precisely, we prove the null controllability of the system with a single localized bilinear interior control acting on either of the components of the coupled system, and with a single periodic boundary control acting through zeroth order derivatives of either of the components. We employ the well-known moment method to study the controllability of the concerned system.
We propose and prove several regularity criteria for the 2D and 3D Kuramoto-Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution $ϕ$, the vector solution $u\triangleq\nablaϕ$, as well as the divergence $\text{div}(u)=Δϕ$, and each component of $u$ and $\nabla u$. We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates.
This paper presents an inverse problem for the nonlinear 1-d Kuramoto-Sivashinsky (K-S) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and at some positive time everywhere. Uniqueness and Lipschitz stability for this inverse problem are proven with the Bukhgeim-Klibanov method. The proof is based on a global Carleman estimate for the linearized K-S equation.
Global well-posedness of the initial-boundary value problem for the stochastic Kuramoto-Sivashinsky equation in a bounded domain $D$ with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data $u_0\in L^2(D\times Ω)$ this problem has a unique global solution $u$ in the space $L^2(Ω,C([0,T],L^2({D})))$ for any $T>0$, and the solution map $u_0\mapsto u$ is Lipschitz continuous.
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 …
In two spatial dimensions, there are very few global existence results for the Kuramoto-Sivashinsky equation. The majority of the few results in the literature are strongly anisotropic, i.e. are results of thin-domain type. In the spatially periodic case, the dynamics of the Kuramoto-Sivashinsky equation are in part governed by the size of the domain, as this determines how many linearly growing Fourier modes are present. The strongly anisotropic results allow linearly growing Fourier modes in only one of the spatial directions. We provide here the first proof of global solutions for the two-d…
We give a substantially simplified proof of near-optimal estimate on the Kuramoto-Sivashinsky equation from [F. Otto, "Optimal bounds on the Kuramoto-Sivashinsky equation", JFA 2009], at the same time slightly improving the result. The result in the above cited paper relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving…
Symbolic dynamics for Kuramoto-Sivashinsky PDE on the line --- a computer-assisted proof
本文研究了具有奇数和周期边界条件的Kuramoto-Sivashinsky PDE在直线上的符号动力学，通过计算机辅助证明了存在可数无穷多个具有任意大周期的周期轨道。
Symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line II
该研究通过计算机辅助的新严格积分算法，证明了特定参数与边界条件下直线上Kuramoto-Sivashinsky偏微分方程存在无穷多同宿和异宿轨道，为该类耗散PDE的符号动力学研究提供了支撑。
Parametric Reduced Order Models for the Generalized Kuramoto--Sivashinsky Equations
研究针对Kuramoto-Sivashinsky和广义Kuramoto-Sivashinsky方程的参数化降阶模型，通过多种POD和POD-DEIM投影方法及参数采样策略，构建能高效覆盖弱混沌、过渡和准周期动力学等不同参数区域的降阶模型，并强调准确表示短时瞬态行为的重要性。
This paper addresses sampled-data control of 2D Kuramoto-Sivashinsky equation over a rectangular domain. We suggest to divide the 2D rectangular into N sub-domains, where sensors provide spatially averaged or point state measurements to be transmitted through communication network to the controller. Note that differently from 2D heat equation, here we manage with sampled-data control under point measurements. We design a regionally stabilizing controller applied through distributed in space characteristic functions. Sufficient conditions ensuring regional stability of the closed-loop system ar…
The Kuramoto-Sivashinsky equation was introduced as a simple 1-dimensional model of instabilities in flames, but it turned out to mathematically fascinating in its own right. One reason is that this equation is a simple model of Galilean-invariant chaos with an arrow of time. Starting from random initial conditions, manifestly time-asymmetric stripe-like patterns emerge. As we move forward in time, it appears that these stripes are born and merge, but do not die or split. We pose a precise conjecture to this effect, which requires a precise definition of 'stripes'.
There is little analytical theory for the behavior of solutions of the Kuramoto-Sivashinsky equation in two spatial dimensions over long times. We study the case in which the spatial domain is a two-dimensional torus. In this case, the linearized behavior depends on the size of the torus -- in particular, for different sizes of the domain, there are different numbers of linearly growing modes. We prove that small solutions exist for all time if there are no linearly growing modes, proving also in this case that the radius of analyticity of solutions grows linearly in time. In the general case …
In this paper we treat controllability properties for the linear Kuramoto-Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the involved equation does not belong to a critical countable set of real numbers. We point out that the critical set for w…
In this paper we are interested in a rigorous derivation of the Kuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm, we consider a two-dimensional Stefan problem in a strip, a simplified version of a solid-liquid interface model. Near the instability threshold, we introduce a small parameter $\varepsilon$ and define rescaled variables accordingly. At fixed $\varepsilon$, our method is based on: definition of a suitable linear 1D operator, projection with respect to the longitudinal coordinate only, Lyapunov-Schmidt method. As a solvability condition, we derive a self…
In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a $ϕ(\triangle t)$-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter $\triangle t$ goes to 0. The proof relies on a known Carleman esti…
In this article the robust Stackelberg controllability (RSC) problem is studied for a nonlinear fourth-order parabolic equation, namely, the Kuramoto-Sivashinsky equation. When three external sources are acting into the system, the RSC problem consists essentially in combining two subproblems: the first one is a saddle point problem among two sources. Such an sources are called the "follower control" and its associated "disturbance signal". This procedure corresponds to a robust control problem. The second one is a hierarchic control problem (Stackelberg strategy), which involves the third for…
In this paper, we have studied the problem of determining the largest possible set of symmetries for an important example of nonlinear dynamical system: the Kuramoto-Sivashinsky (K-S) model in two spatial and one temporal dimensions. By applying the classical symmetry method for the K-S model, we have found the classical symmetry operators. Also, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras of the equation is constructed. The Lie invariants associated to the symmetry generators as well as the corresponding similarity reduced equations are al…
Symmetry reductions of a generalized Kuramoto-Sivashinsky equation via equivalence transformations
本文通过构造广义Kuramoto-Sivashinsky方程的等价群及最优系统，确定了所有非等价扩展方程，并利用附加对称性将偏微分方程约化为常微分方程，进而求得精确解。
We present numerical results and computer assisted proofs of the existence of periodic orbits for the Kuramoto-Sivashinky equation. These two results are based on writing down the existence of periodic orbits as zeros of functionals. This leads to the use of Newton's algorithm for the numerical computation of the solutions and, with some a posteriori analysis in combination with rigorous interval arithmetic, to the rigorous verification of the existence of solutions. This is a particular case of the methodology developed in [19] for several types of orbits. An independent implementation, cover…
Rigorous Numerics for Partial Differential Equations: the Kuramoto-Sivashinsky equation
本文提出基于自洽先验界的拓扑方法，严格证明Galerkin投影，将耗散PDE约化为低维ODE并用Conley指标分析，最终计算机辅助证明Kuramoto-Sivashinsky方程不动点的存在性。
A generalized time-fractional Kuramoto-Sivashinsky equation in the Schwartz space: global solvability and stability
本文研究了广义时间分数阶库马诺-西瓦斯基方程在Schwartz空间中的全局可解性，通过分离线性与非线性部分并应用分析方法，建立了拓扑函数空间中分数阶库马诺-西瓦斯基类型方程的严格分析框架。
Algebraic calming for the 2D Kuramoto-Sivashinsky equations
本文提出二维 KSE 的“镇定”近似模型，证明其全局适定且解在原方程适定区间内以代数速率收敛，并用模拟验证速率尖锐性。
Existence and analyticity of solutions of the Kuramoto-Sivashinsky equation with singular data
本文研究 Kuramoto-Sivashinsky 方程在低正则奇异数据下的适定性，分别在 Wiener 代数型空间和不同维数的伪测度空间中建立解的存在性、抛物正则化增益及正时刻解析性。
The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global existence and blow-up as well as uniform bounds are reviewed by using: (i) classic tools of interpolation theory and Galerkin methods, (ii) eigenfunction and nonlinear capacity methods, (iii) Henry's version of weighted Gronwall's inequalities, and (vi) two types of scaling (blow-up) arguments. For the IBVPs, existence of global solutions is proved for both Dirich…
Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations
本文研究了Kuramoto-Sivashinsky-KdV与椭圆方程耦合系统的零能控性，利用Carleman方法证明线性化系统的全局零能控性，并结合源项方法与Banach不动点定理得出非线性系统的小时间局部零能控性结论。
We study the Complex Ginzburg--Landau initial value problem $\partial_t u=(1+iα) \partial_x^2 u + u - (1+iβ) u |u|^2$, $u(x,0)=u_0(x)$ for a complex field $u\in{\bf C}$, with $α,β\in{\bf R}$. We consider the Benjamin--Feir linear instability region $1+αβ=-ε^2$ with $ε\ll1$ and $α^2<1/2$. We show that for all $ε\leq{\cal O}(\sqrt{1-2α^2} L_0^{-32/37})$, and for all initial data $u_0$ sufficiently close to 1 (up to a global phase factor $\ed^{i ϕ_0}, ϕ_0\in{\bf R}$) in the appropriate space, there exists a unique (spatially) periodic solution of space period $L_0$. These solutions are small {\em…
Exponential mixing for the stochastic Kuramoto-Sivashinsky equation on the 1D torus
本文研究加性高斯噪声扰动下一维环面随机 Kuramoto-Sivashinsky 方程的大时间行为，证明其唯一不变概率测度具指数吸引力，无需反扩散系数小性条件，成果推进了该方程的遍历性研究。
Data-based stochastic model reduction for the Kuramoto--Sivashinsky equation
本文提出基于数据的预测降阶模型，用于库莫-西瓦什insky方程，通过非线性自回归移动平均外生输入模型构建离散时间随机降阶系统。
In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin…
We study the stability and nonlinear local dynamics of spectrally stable periodic wave trains of the Korteweg-de Vries / Kuramoto-Sivashinsky equation when subjected to classes of periodic perturbations. It is known that for each $N\in\mathbb{N}$, such a $T$-periodic wave train is asymptotically stable to $NT$-periodic, i.e., subharmonic, perturbations, in the sense that initially nearby data will converge asymptotically to a small Galilean boost of the underlying wave, with exponential rates of decay. However, both the allowable size of initial perturbations and the exponential rates of decay…
Remarks on the stabilization of large-scale growth in the 2D Kuramoto-Sivashinsky equation
本文通过观察二维无旋 Burgers 方程中散度场的关键作用，提出基于散度的新正则性条件，并将其与 Ladyzhenskaya-Prodi-Serrin 型准则建立联系，进而用于证明修正 KSE 及二维 Burgers-Sivashinsky 方程的全局正则性。
We introduce $n$-parameter $\Rd$-valued Brownian-time Brownian sheet (BTBS): a Brownian sheet where each "time" parameter is replaced with the modulus of an independent Brownian motion. We then connect BTBS to a new system of $n$ linear, fourth order, and interacting PDEs and to a corresponding fourth order interacting nonlinear PDE. The coupling phenomenon is a result of the interaction between the Brownian sheet, through its variance, and the Brownian motions in the BTBS; and it leads to an intricate, intriguing, and random field generalization of our earlier Brownian-time-processes (BTPs) c…
The Kuramoto-Sivashinsky equations (KSE) arise in many diverse scientific areas, and are of much mathematical interest due in part to their chaotic behavior, and their similarity to the Navier-Stokes equations. However, very little is known about their global well-posedness in the 2D case. Moreover, regularizations of the system (e.g., adding large diffusion, etc.) do not seem to help, due to the lack of any control over the $L^2$ norm. In this work, we propose a new "reduced" 2D model that modifies only the linear part of (the vector form of) the 2D KSE in only one component. This new model s…
We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto-Sivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by space-time white noise in one-to-three dimensional space. Both classes were introduced---with Brownian-time-type kernel formulations---by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regul…
We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation $ \partial_t v+v\partial_x v+\partial_x^3 v+δ(\partial_x^2 v +\partial_x^4 v)=0$, $δ>0$, in the Korteweg-de Vries limit $δ\to 0$, a canonical limit describing small-amplitude weakly unstable thin film flow. More precisely, we carry out a rigorous singular perturbation analysis reducing the problem to the evaluation for each Bloch parameter $ξ\in [0,2π]$ of certain elliptic integrals derived formally (on an incomplete set of frequencies/Bloch parameters, hence as necessary …
Generalizing the L-Kuramoto-Sivashinsky (L-KS) kernel from our earlier work, we give a novel explicit-kernel formulation useful for a large class of fourth order deterministic, stochastic, linear, and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the Swift-Hohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal Hölder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on $\{\Rp\times\Rd\}_{d=1}^{3}$. The spatio-temporal Hölder exponents are exactl…
Insensitizing control problems for the stabilized Kuramoto-Sivashinsky system
本文研究稳定化 Kuramoto–Sivashinsky 耦合抛物系统的不敏感控制，通过将问题转化为加倍级联系统的零能控性，结合 Carleman 估计与局部反演定理证明其局部存在性。
Adaptive Boundary Control of the Kuramoto-Sivashinsky Equation Under Intermittent Sensing
本文针对间歇传感下的扰动Kuramoto-Sivashinsky方程，设计自适应边界控制器，在扰动上界已知时保证输入到状态稳定性，未知时保证全局一致最终有界性，全状态测量下可收敛至原点任意小邻域。
A bifurcation theory approach to the nonlocal Kuramoto-Sivashinsky equation
本文通过分支理论研究非局部库莫-西瓦斯基方程的稳态解存在性，证明了临界点处的分支及其全局延续性。
Nonlinear Instability in the 2D Kuramoto-Sivashinsky equation
研究二维Kuramoto-Sivashinsky方程在特定环面上非线性不稳定性，通过考虑最大增长模式\(\lambda_0\)，利用近似解构造、不动点论证及延拓论证，证明存在小尺寸初始数据流形，使解在特定时间尺度上有特定变化。
Many physical systems are modeled by nonlinear parabolic differential equations, such as the Kuramoto-Sivashinsky (KS) equation. In this paper, the existence of a concurrent optimal controller and actuator design is established for semilinear parabolic systems. Optimality equations are provided. The results are shown to apply to optimal controller/actuator design for the Kuramoto-Sivashinsky equation and also nonlinear diffusion.
Global Well-Posedness of the Kuramoto--Sivashinsky Equation in One and Two Dimensions under a Codimension-One Constraint
研究一维、二维和三维 Kuramoto--Sivashinsky 方程在余维一约束下的全局适定性；利用约束保持 L^2 范数，在一维和二维建立全局强解，并揭示三维的维度障碍。
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