Luc Molinet, Jean-Claude Saut, Nikolay Tzvetkov
We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in all directions) traveling wave solution (e.g. the KdV line solitary wave or the Zaitsev solitary waves which are localized in $x$ and $y$ periodic or conversely).
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Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations : perturbations that are square integrable in $ \R\times \T $ and perturbations that are square integrable in $ \R^2 $. In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.
Localized energy estimates for the wave equation have been increasingly used to prove various other dispersive estimates. This article focuses on proving such localized energy estimates on $(1+4)$-dimensional Myers-Perry black hole backgrounds with small angular momenta. The Myers-Perry space-times are generalizations of higher dimensional Kerr backgrounds where additional planes of rotation are availabile while still maintaining axial symmetry. Once it is determined that all trapped geodesics have constant $r$, the method developed by Tataru and the fourth author, which perturbs off of the Sc…
Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the $3D$ finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in $3D$, which do exist for the same system in $2D$.
We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)$ being the characteristic function of an interval. we prove that there exists $p_0=\max\{1,\frac{m+1}2\}$ such that all global solution are bounded if $p>p_0$, while for $p\le p_0$ all the solution are global and unbounded. In the last case, we prove that if $p<m$ the grow-up rate is different to the one obtained when $a(x)\equiv1$, while if $p>m$ the grow-up rate coincides with that rate, but only inside the support of $a$; outside the interva…
Decomposition of global solutions of bi-laplacian Nonautonomous Schrödinger equations
该研究针对带一般时变相互作用项的双拉普拉斯薛定谔方程，证明其全局解在所有空间维度下可渐近分解为自由波与弱局部分量，且n≥9维时弱局部分量具空间局部性，采用适配的自由通道波算子构造完成证明。
Global well-posedness of the Navier-Stokes equations for small initial data in frequency localized Koch-Tataru's space
研究三维不可压Navier-Stokes方程小初值整体适定性，提出高频在BMO^{-1}、低频在Besov空间分别小的频率局部化条件，构造强L2取初值的整体光滑解，弱化Koch-Tataru假设并保证能量平衡。
We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove weak convergence to a nonlinear Schrödinger equation.
Global solutions of 2D non-resistive MHD system with constant background magnetic vorticity
本文通过分析线性化方程的耗散性质，证明了背景磁涡度对二维不可压缩无电阻MHD系统具有稳定化效应，并建立了光滑解的全局存在性。
Local and global regularity for the Stokes and Navier-Stokes equations with the localized boundary data in the half-space
研究半空间Stokes与Navier-Stokes方程在局部化边界数据下的正则性，证明时间光滑边界数据可导出边界附近时空光滑解，构造切向分量二阶法向导数奇异解，并给出特定边界数据下解至二阶导数的逐点界，表明其为全局强解但二阶法向导数在边界附近无界。
We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)=1$ for $x>0$, $a(x)=0$ for $x<0$. We first characterize the global existence exponent $p_0=1$ and the Fujita exponent $p_c=m+2$. Then we pass to study the grow-up rate in the case $p\le1$ and the blow-up rate for $p>1$. In particular we show that the grow-up rate is different as for global reaction if $p>m$ or $p=1\neq m$.
We derive exact solutions to the sine--Gordon equation describing localized structures on the background of librational and rotational travelling waves. In the case of librational waves, the exact solution represents a localized spike in space-time coordinates (a rogue wave) which decays to the periodic background algebraically fast. In the case of rotational waves, the exact solution represents a kink propagating on the periodic background and decaying algebraically in the transverse direction to its propagation. These solutions model the universal patterns in the dynamics of fluxon condensat…
We develop techniques to capture the effect of transport on the long-term dynamics of small, localized initial data in nonlinearly coupled reaction-diffusion-advection equations on the real line. It is well-known that quadratic or cubic nonlinearities in such systems can lead to growth of small, localized initial data and even finite time blow-up. We show that, if the components exhibit different velocities, then quadratic or cubic mix-terms, i.e. terms with nontrivial contributions from both components, are harmless. We establish global existence and diffusive Gaussian-like decay for exponent…
We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I equation which vanish for $x\to-\infty$ and study their large time asymptotic behavior. We prove that such solutions eject (for $t\to\infty$) a train of curved asymptotic solitons which move behind the basic wave packet.
It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of solutions with small, localized initial data and even finite time blow-up. It was recently proved, however, that, if the components of two nonlinearly coupled reaction-diffusion-advection equations propagate with different velocities, then quadratic or cubic mixed-terms, i.e.~nonlinear terms with nontrivial contributions from both components, do not affect global existence and Gaussian decay of small, localized initial data. The proof relied on pointwise estimates to capture th…
We introduce several mechanisms to dissipate the energy in the Benjamin-Bona-Mahony (BBM) equation. We consider either a distributed (localized) feedback law, or a boundary feedback law. In each case, we prove the global wellposedness of the system and the convergence towards a solution of the BBM equation which is null on a band. If the Unique Continuation Property holds for the BBM equation, this implies that the origin is asymp-totically stable for the damped BBM equation.
For the Stokes system in the half space, Kang [Math.~Ann.~2005] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives du…
Exponential mixing for the stochastic Allen--Cahn equation with localized white noise
本文研究了由局部白噪声驱动的1D随机Allen-Cahn方程，证明了相关马尔可夫过程具有唯一不变测度和指数混合性。主要挑战在于噪声的局部性质与系统非平凡全局动力学的相互作用，通过PDE控制理论中的稳定化和非线性方程的全局稳态可控性来克服。
In this paper, we initiate the study of the global stability of nonlinear wave equations with initial data that are not required to be localized around a single point. More precisely, we allow small initial data localized around any finite collection of points which can be arbitrarily far from one another. Existing techniques do not directly apply to this setting because they require norms with radial weights away from some center to be small. The smallness we require on the data is measured in a norm which does not depend on the scale of the configuration of the data. Our method of proof reli…
Global dynamics for the 1d quartic Klein-Gordon equation with an internal mode
本文研究一维四次克莱因-戈登方程，在费米黄金规则下通过辐射阻尼机制给出内模振幅的全局动力学描述，并证明辐射散射性，进而应用于φ^4模型扭结解的渐近稳定性。
Regimes with a singular peaking for a wide class of quasilinear second order parabolic equations are studied. On the basis of energy methods, precise estimates of a final profile of a weak solution in a neighborhood of the peaking time are established depending on the rate of increase of a global energy of this solution. Key words: quasilinear parabolic equations, peaking regimes, energy solutions.
Global solutions for 1D cubic defocusing dispersive equations: Part I
本文针对一维三次散焦NLS方程，提出一种新的非微扰方法，首次在小但非局部化的L^2初始数据下证明了全局适定性与散射，并得到全局L^6 Strichartz估计和双线性L^2界。
We prove global existence and modified scattering for the solutions of the generalized fifth-order KdV equation with critical nonlinearity for small and localized initial data. The proof is undergoing by using the space-time resonance method and the stationary phase argument.
We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the Strauss conjecture on the Schwarzschild and Kerr, with small angular momentum, black hole backgrounds. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and a localized energy estimate on the black hole background, which handles the behavior in the remaining compact set.
Global controllability of Boussinesq flows by using only a temperature control
针对二维不可压缩 Boussinesq 系统，本文证明仅依靠小测度区域内的局部温度控制即可实现全局近似能控，并给出具有显式结构的控制。
Localized and degenerate controls for the incompressible Navier-Stokes system
本文研究二维不可压缩Navier-Stokes系统在局部退化力驱动下的全局近似可控性，通过沿线性化涡量方程特征曲线追踪低模态控制，将问题归结为线性输运方程，从而尝试解决Agrachev提出的公开问题。
We consider the NLS on Schwarzschild manifold.For radial solutions with sufficiently localized initial data,global existence,L^p estimates and asymptotic completeness of the wave operators is proved
Local and global existence of localized solutions of a discrete nonlinear Schrodinger (DNLS) equation, with arbitrary on-site nonlinearity, is proved. In particular, it is shown that an initially localized excitation persists localized during infinite time. Moreover, if initial localization is stronger than |n|^{-d} with any power d, it maintains itself as such during infinite time. The results are generalized to various types of inter-side and saturable nonlinearities, to lattices with long range interactions, as well as DNLS with dissipation.
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