Davit Martirosyan
The paper is devoted to studying the stochastic nonlinear wave (NLW) equation in a bounded domain D $\subset$ R3. We show that the Markov process associated with the flow of solution has a unique stationary measure $μ$, and the law of any solution converges to $μ$ with exponential rate in the dual-Lipschitz norm
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We prove the existence of the global attractor in $ \dot H^s$, $s > 11/12$ for the weakly damped and forced mKdV on the one dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space is established. We directly apply the I-method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresp…
The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from $L^2$ and mean-zero initial data we prove that the solution decomposes into two parts; a linear one which decays to zero as time goes to infinity and a nonlinear one which always belongs to a smoother space. As a corollary we prove that all solutions are attracted by a ball in $H^s$, $s\in(0,1)$, whose radius depends only on $s$, the $L^2$ norm of the forcing term and the damping parameter. This gives a new proof for the existence of a smooth global attractor and provide…
Global attractor for the weakly damped forced Kawahara equation on the torus
研究环面上弱阻尼受迫Kawahara方程解的长时间行为，证明其在$L^2$中存在全局吸引子且位于$H^2$中并有界，还给出仅依赖阻尼参数和强迫项范数的$H^2$吸引子大小上界。
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 $H^{s+3}(\T)$.
We study two damped and forced discrete nonlinear Schrödinger equations on the one-dimensional infinite lattice. Without damping and forcing they are represented by the integrable Ablowitz-Ladik equation (AL) featuring non-local cubic nonlinear terms, and its standard (nonintegrable) counterpart with local cubic nonlinear terms (DNLS). The global existence of a unique solution to the initial value problem for both, the damped and forced AL and DNLS, is proven. It is further shown that for sufficiently close initial data, their corresponding solutions stay close for all times. Concerning the as…
Forced oscillation of a damped BBM equation posed on whole line in low regularity spaces
本文在低正则性空间中研究了受迫阻尼BBM方程的时间周期解的存在性和稳定性，采用I能量方法推导估计并处理对流项作为扰动。
Donsker-Varadhan large deviation principle for locally damped and randomly forced NLS equations
本文提出了一种新的抽象准则，用于研究局部阻尼和随机强迫非线性薛定谔方程的大偏差原理，并应用于波动方程和纳维-斯托克斯系统。
Polynomial mixing for infinite-energy solutions of the white-forced Navier--Stokes equation in an infinite pipe
本文研究无限管道中受白噪声驱动的纳维-斯托克斯方程无限能量解的长时间行为，在噪声满足适当假设下证明其不变测度唯一性与多项式混合性，在零截面通量情形下肯定回答了Kuksin和Shirikyan的开放问题1。
We consider the three-dimensional magnetohydrodynamics system forced by noise that is white in both time and space. Its complexity due to four non-linear terms makes its analysis very intricate. Nevertheless, taking advantage of its structure and adapting the theory of paracontrolled distributions from \cite{GIP15}, we prove its local well-posedness. A first challenge is to find an appropriate paracontrolled ansatz which must consist of both the velocity and the magnetic fields. Second challenge is that for some non-linear terms, renormalizations cannot be achieved individually; we overcome th…
The magnetohydrodynamics system consists of the Navier-Stokes and Maxwell's equations, coupled through multiples of nonlinear terms. Such a system forced by space-time white noise has been studied by physicists for decades, and the rigorous proof of its solution theory has been recently established in Yamazaki (2019, arXiv:1910.04820 [math.AP]) using the theory of paracontrolled distributions and a technique of coupled renormalizations. When an equation is well-posed, and it is approximated by replacing the differentiation operator by reasonable discretization schemes with a parameter, it is w…
Global unique solution to the perturbation of the Burgers' equation forced by derivatives of space-time white noise
本文研究受时空白噪声分数阶导数驱动的一维Burgers方程，依托Anderson Hamiltonian与二维Navier-Stokes方程的相关方法，证明了其温和解与弱解的全局时间存在性与唯一性。
Remarks on the three-dimensional Navier-Stokes equations with Lions' exponent forced by space-time white noise
本文研究受时空白噪声驱动且处于能量临界情形的三维Navier-Stokes方程，沿用Hairer和Rosati的方法证明了其全局解理论。
Polynomial mixing for the white-forced hyperviscous Burgers equation on the whole line
研究全直线上白噪声驱动的超粘性伯格斯方程的遍历性，在相空间多个方向受随机强迫假设下，用耦合方法及准则证明动力学以任意幂次多项式速率吸引到唯一不变概率测度。
Polynomial mixing for the white-forced Navier-Stokes system in the whole space
本文研究二维全空间白噪声驱动 Navier-Stokes 系统，利用耦合、Foiaş-Prodi 估计及加权轨道与 Leray 投影估计，证明平稳测度唯一且在对偶-Lipschitz 度量下多项式混合。
Level-3 large deviations for the white-forced 2D Navier-Stokes system in a bounded domain
该研究针对有界区域白噪声驱动的Navier-Stokes系统，在噪声非退化假设下，借助改进的Kifer准则等多种方法建立了Donsker-Varadhan型的二级与三级大偏差原理，给出了对应速率函数。
Periodic solutions for weakly damped systems
本文研究了在周期性强迫下，弱阻尼演化偏微分方程周期解的存在性和性质，特别关注对应自由方程能量衰减非均匀的情况，探讨了衰减速度与强迫方程周期解存在的关系，并刻画了解算子增长与相关正则性损失的关系，通过线性和非线性阻尼波方程以及耦合双曲-抛物方程组的例子加以说明，最后给出了一个反例，展示了共振现象中即使有阻尼，正则但无界解也会出现。
Remarks on the two-dimensional magnetohydrodynamics system forced by space-time white noise
本文针对时空白噪声驱动的二维磁流体力学系统，借助 Maxwell 方程结构与仿受控方法证明了解的全局时间存在唯一性。
Exponential mixing for the white-forced complex Ginzburg--Landau equation in the whole space
本文研究全空间实直线上白噪声驱动的阻尼复Ginzburg--Landau方程，在噪声充分非退化条件下证明了平稳测度唯一性与对偶Lipschitz度量下的指数混合，主要方法为耦合技术、Foiaş--Prodi估计推广及时空加权估计。
Polynomial mixing for the white-forced wave equation on the whole line
本文研究全直线上白噪声驱动波动方程的遍历性，证明其平稳测度唯一且在双-Lipschitz度量下多项式混合，克服了无光滑耗散及域无界的困难，依赖四项关键技术完成证明。
We consider the damped nonlinear wave (NLW) equation driven by a spatially regular white noise. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation and seems to be new in the context of randomly forced PDE's. The proof is based on an extension of methods developed in \cite{JNPS-2012} and \cite{JNPS-2014} in the case of kick forced dissipative PDE's with parabolic regularisati…
We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein-Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.
The paper is devoted to studying the asymptotics of the family $(μ^\varepsilon)$ of stationary measures of the Markov process generated by the flow of stochastic 2D Navier-Stokes equation with smooth white noise. By using the large deviations techniques, we prove that this family is exponentially tight in $H^{1-γ}(D)$ for any $γ>0$ and vanishes exponentially outside any neighborhood of the set $\cal O$ of $ω$-limit points of the deterministic equation. In particular, any of its weak limits is concentrated on the closure $\bar{\cal O}$. A key ingredient of the proof is a new formula that allows…
Remarks on the two-dimensional Cahn-Hilliard equation forced by divergence of space-time white noise
该研究针对由时空白噪声散度驱动的二维Cahn-Hilliard方程，通过结合最少重整化程序与确定性分析工具，建立了其全局时间唯一解理论，解决了非线性项乘积无定义的问题。
Inspired by applications, we consider reaction-diffusion equations on $\mathbb{R}$ that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in [Hamster & Hupkes 2017] and [Hamster & Hupkes 2018] for noise processes…
We consider the non-homogeneous generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2} = η,\ t \geq 0,\ x \in S^1. Here f is strongly convex and satisfies a growth condition, νis small and positive, while ηis a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to t>=T_1, where T_1 depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of $u$ averaged in time and in ensembl…
Spectral submanifold reduction for PDEs describing nonlinear continuum vibrations
研究描述非线性连续体振动的偏微分方程，证明谱子流形存在，可据此从方程中严格提取骨干曲线和强迫响应曲线，无需先验离散化，通过弹性梁和薄板例子展示可直接从方程手动计算相关曲线。
Stochastically forced Navier-Stokes equations interacting with an elastic structure
针对二维不可压缩Navier-Stokes流体与一维阻尼Kirchhoff板耦合的随机流固耦合问题，证明了全局强适定性，通过线性随机部分与非线性确定性部分的分解，结合算子\(\mathcal{H}^\infty\)-演算和先验估计。
Local large deviations for randomly forced nonlinear wave equations with localized damping
针对有界噪声扰动的局部阻尼非线性波动方程，本文利用渐近紧性与渐近指数胎紧性建立经验分布的局部型LDP，并提出随机动力系统的新LDP方法。
Long-time evolution of forced waves in the low viscosity regime
该研究针对带二阶黏度项阻尼的零阶伪微分哈密顿量内波模型，在经典动力学满足Morse-Smale等弱全局条件下，证明小黏度区域内耗散效应最早出现于$t\thicksim ν^{-1/3-}$时间尺度，刻画了受迫波的长时间演化行为。
We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity o…
We study a damped stochastic non-linear Schrödinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markovian transition semi-group toward a unique invariant probability measure. This kind of method was originally developped to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schrödinger equation in the one dimensional cubic case. We prove that the mixing property holds and that the rat…
Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation
该研究拓展非线性色散PDE概率适定性理论至方差爆破外，以BBM方程为对象，通过两种重整化方法证明解的依分布收敛性，且结果适用于任意低正则高斯初值，还推广到分数阶噪声驱动情形。
We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum $\sim 1/k$, where $k$ is the wave number. From very-high-resolution numerical simulations, in the limit of vanishing viscosity, we find evidence for multiscaling of velocity structure functions which cannot be falsified by standard tests. We find a new artifact in which logarithmic corrections can appear disguised as anomalous scaling and conclude that bifractal scaling is likely.
We consider the two-dimensional stochastic damped nonlinear wave equation (SdNLW) with the cubic nonlinearity, forced by a space-time white noise. In particular, we investigate the limiting behavior of solutions to SdNLW with regularized noises and establish triviality results in the spirit of the work by Hairer, Ryser, and Weber (2012). More precisely, without renormalization of the nonlinearity, we establish the following two limiting behaviors; (i) in the strong noise regime, we show that solutions to SdNLW with regularized noises tend to 0 as the regularization is removed and (ii) in the w…
A critical stochastic heat equation with long-range noise
研究空间维数至少为3且受时间白噪声、空间协方差按$|x|^{-2}$衰减的临界半线性随机热方程，证明在带对数衰减的扩散标度下解的逐点统计可由FBSDE刻画。
Exponential mixing for Korteweg-de Vries equation with localized noise
本文研究受随机强迫且弱阻尼的KdV方程，利用概率框架、非线性光滑性等方法建立其指数混合性，延续了色散偏微分方程遍历性与混合性的相关研究。
We consider a damped/driven nonlinear Schrödinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions \[ u_t-νΔu+i|u|^2u=\sqrtνη(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad ν>0, \] where $η(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ \| u(t)\|_m^2 \le Cν^{-m}, $ uniformly in $t\ge0$ and $ν>0$. In this work we prove that for small $ν>0$ and any initial data, with large probability the Sobolev norms $\|u(t,\cdot)\|_m$ of the solutions with $m>2$ beco…
Polynomial mixing for the weakly damped stochastic nonlinear Schrödinger equation on the whole space
本文研究实直线上弱阻尼随机非线性薛定谔方程，在噪声非退化条件下证明唯一平稳测度存在及多项式混合，方法结合耦合论证、Foiaş-Prodi 估计与截断 Poincaré 不等式。
Mixing at the Batchelor Scale for White-In-Time Flows
研究二维环面上白噪声时间速度场对流扩散方程的混合性质，证明在扩散系数趋零时几乎必然指数耗散率有正下界，验证了Batchelor尺度猜想，并刻画了无扩散时的指数混合速率，且推广至三维。
Well-posedness of a parametrically forced nonlinear Schrödinger equation driven by translation-invariant noise
该研究证明了带乘性平移不变Stratonovich噪声的一维参量强迫非线性薛定谔方程在H^σ(ℝ)空间的适定性，通过不动点构造局部解、建立爆破准则并推导L²范数路径估计，结果也适用于标准三次NLS方程。
Stability of Cnoidal Waves for the Damped Nonlinear Schrödinger Equation
研究一维环面上带线性阻尼的立方非线性薛定谔方程，证明椭圆余弦波族在耗散动力学下轨道稳定，通过一阶近似和指数递减Lyapunov泛函控制扰动。
For the damped-driven KdV equation $$ \dot u-ν{u_{xx}}+u_{xxx}-6uu_x=\sqrtνη(t,x), x\in S^1, \int u dx\equiv \intηdx\equiv0, $$ with $0<ν\le1$ and smooth in $x$ white in $t$ random force $η$, we study the limiting long-time behaviour of the KdV integrals of motions $(I_1,I_2,...)$, evaluated along a solution $u^ν(t,x)$, as $ν\to0$. We prove that %if $u=u^ν(t,x)$ is a solution of the equation above, for $0\leτ:= νt \lesssim1$ the vector $ I^ν(τ)=(I_1(u^ν(τ,\cdot)),I_2(u^ν(τ,\cdot)),...), $ converges in distribution to a limiting process $I^0(τ)=(I^0_1,I^0_2,...)$. The $j$-th component $I_j^0$ e…
Exponential mixing and limit theorems of quasi-periodically forced 2D stochastic Navier-Stokes Equations in the hypoelliptic setting
该研究针对拟周期与退化噪声驱动的二维随机Navier-Stokes方程，证明了存在指数吸引所有解分布的拟周期不变测度，并建立了带显式收敛速率的强大数定律与中心极限定理，揭示了收敛速率与拟周期频率丢番图性质的关联。
We consider three-dimensional stochastically forced Navier-Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper and lower bounds of the mean value of the time-averaged energy dissipation rate, $\mathbb{E} [\langle\varepsilon \rangle] $, are derived directly from the equations. First, we show that for a weak (martingale) solution to the stochastically forced Navier-Stokes equations, \[ \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L},\] where $G^2$ is the total energy rate supplied by the random force, $U$ is…
On the long-time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation
本文研究弱阻尼随机驱动KdV方程的长时间统计行为，利用运动积分建立Lyapunov结构与弱Foias-Prodi估计，证明不变测度的存在性、光滑支集及不同阻尼下的遍历性质。
We consider the one-dimensional Burgers equation perturbed by a stochastic forcing, which is assumed to be white in time and localised and low-dimensional in space. We establish a mixing property for the Markov process associated with the problem in question. The proof is based on a general criterion for mixing and a recent result on global approximate controllability to trajectories for damped conservation laws.
Stochastic complex Ginzburg-Landau equation on compact surfaces
本文研究紧致曲面上受时空白噪声驱动的随机复Ginzburg-Landau方程，通过重整化证明局部适定性，并证明弱色散区域中散焦方程的全局适定性。
We establish the convergence of statistically invariant states for the stochastic Boussinesq Equations in the infinite Prandtl number limit and in particular demonstrate the convergence of the Nusselt number (a measure of heat transport in the fluid). This is a singular parameter limit significant in mantle convection and for gasses under high pressure. The equations are subject to a both temperature gradient on the boundary and internal heating in the bulk driven by a stochastic, white in time, gaussian forcing. Here, the stochastic source terms have a strong physical motivation for example a…
We study stochastically forced semilinear parabolic PDE's of the Ginzburg-Landau type. The class of forcings considered are white noises in time and colored smooth noises in space. Existence of the dynamics in $L^\infty$, as well as existence of an invariant measure are proven. We also show that the solutions are with high probability analytic in a strip around the real axis and give estimates on the width of that strip.
We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $λβe^{βu }$, forced by an additive space-time white noise. We prove local and global well-posedness of these equations, depending on the sign of $λ$ and the size of $β^2 > 0$, and invariance of the associated Gibbs measures. See the abstract of the paper for a more precise abstract. (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here.)
On generalised d'Alembert-type integral representations for damped wave equations on the quarter-plane
该研究将Fokas统一变换方法扩展到半无限区间的双曲-抛物问题，构造了四分之一平面上阻尼波动方程的闭式解，揭示了保证光滑解存在的相容性条件，其积分表示可用于数值基准及多领域建模研究。
Uniform attraction and exit problems for stochastic damped wave equations
本文研究受乘性时空白噪声扰动的一维随机阻尼波动方程，利用大偏差和可控性方法刻画局部解从一致吸引区域逃逸的时间和位置的拟势渐近，并引入正则边界点概念处理无穷维边界光滑性问题。
The objective of this note is to present the results from the two recent papers. We study the Navier--Stokes equation on the two--dimensional torus when forced by a finite dimensional white Gaussian noise. We give conditions under which both the law of the solution at any time t>0, projected on a finite dimensional subspace, has a smooth density with respect to Lebesgue measure and the solution itself is ergodic. In particular, our results hold for specific choices of four dimensional white Gaussian noise. Under additional assumptions, we show that the preceding density is everywhere strictly …
We prove that the stochastic Burgers equation on $\mathbf{R}^{d}$, $d<4$, forced by gradient noise that is white in time and smooth in space, admits spacetime-stationary solutions. These solutions are thus the gradients of solutions to the KPZ equation on $\mathbf{R}^{d}$ with stationary gradients. The proof works by proving tightness of the time-averaged laws of the solutions in an appropriate weighted space.
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