Global rough solutions to the cubic nonlinear Boussinesq equation
Luiz Gustavo Farah, Felipe Linares
证明了在$2/3<s<1$的条件下，三次离焦非线性Boussinesq方程$u_{tt}-u_{xx}+u_{xxxx}-(|u|^2u)_{xx}=0$在实直线上的初值问题在$H^{s}(\R)$中是整体适定的.
We prove that the initial value problem (IVP) for the cubic defocusing nonlinear Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(|u|^2u)_{xx}=0$ on the real line is globally well-posed in $H^{s}(\R)$ provided $2/3<s<1$.
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In this paper, we consider the stochastic Boussinesq equations on $\mathbb T^3$ with transport noise and rough initial data. We first prove the existence and uniqueness of the local pathwise solution with initial data in $L^p(Ω;L^p)$ for $p>5$. By assuming additional smallness on the initial data and the noise, we establish the global existence of the pathwise solution.
On the inviscid Boussinesq system with rough initial data
本文研究了具有粗糙初始数据的二维无粘Boussinesq系统的局部适定性。由于涡度-密度公式中的某些项涉及里斯变换，因此这个问题在某种意义上是关键的。我们给出了一个积极的回答，一个特殊的子类Yudovich数据，包括光滑和奇异的涡补丁。对于后一种情况，我们假设，此外，初始密度是恒定的补丁边界的奇异部分周围。
This work concerns the global well-posedness problem for the 3D axisymmetric viscous Boussinesq system with critical rough initial data. More precisely, we aim to extending our recent result \cite{Hanachi-Houamed-Zerguine} to the case of initial data of measure type. To this end, we should first develop some notions of axisymmetric measures in a general context, then, in the spirit of \cite{Gallay-Sverak}, we prove the global wellposedness result provided that the atomic parts of the initial measures are small enough.
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Initial and initial-boundary value problems for a cubic sixth-order Boussinesq equation
本文针对三次六阶Boussaineseq方程，在Bourgain型空间中改进局部适定性，通过细化三线性估计实现，适用于实轴和半实轴。
Dispersive estimates and generalized Boussinesq equation on hyperbolic spaces with rough initial data
本文在实双曲空间的 Lorentz 粗糙框架下研究广义 Boussinesq 方程，通过内蕴 Fourier 分析与振荡积分估计建立相关群的色散估计，并推出适定性、散射及改进衰减。
On the global well-posedness of the Boussinesq system with zero viscosity
本文证明了具有粗糙初始数据的零粘性二维Boussinesq系统的全局适定性。
This article studies the global well-posedness for a class of defocusing, generalized sixth-order Boussinesq equations, extending a previous result obtained by Wang and Esfahani for the case when the nonlinear term is cubic.
Global well-posedness for a Boussinesq- Navier-Stokes System with critical dissipation
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We address the well-posedness for the two-dimensional Boussinesq equations with zero diffusivity in bounded domains. We prove global in time regularity for rough initial data: both the initial velocity and temperature have $ε$ fractional derivatives in $L^{q}$ for some $q>2$ and $ε>0$ arbitrarily small.
Global well-posedness for the 2D Boussinesq Equations with Zero Viscosity
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On the global well-posedness of a class of Boussinesq- Navier-Stokes systems
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Spatial decay and nonlinear smoothing of the sixth-order Boussinesq equation
本文针对任意空间维度下带二次与三次非线性项的六阶Boussinesq方程初值问题，通过Fourier限制范数方法与高低频分解建立其非线性光滑性，进而得到积分项的一致收敛性与空间衰减性。
The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in \cite{Gallay,Gallay-Sverak}. Roughly speaking, we show essentially that if the initial data $(v_0,ρ_0)$ is axisymmetric and $(ω_0,ρ_0)$ belongs to the critical space $L^1(Ω)\times L^1(\mathbb{R}^3)$, with $ω_0$ is the initial vorticity associated to $v_0$ …
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该研究针对Boussinesq方程的KdV极限问题，证明特定标度下其一大类粗糙解可由两个反向传播的KdV方程解之和逼近，拓展了已有结果的适用范围，证明采用非线性色散方程低正则性理论中的傅里叶分析方法。
Well-posedness of Boussinesq Equation with Linear Restoring Force
针对带线性恢复力的n维Boussinesq方程，在修正Sobolev和Bourgain空间中建立了首个局部适定性结果，通过双线性和三线性估计处理二次和三次非线性。
Improved local well-posedness for the periodic "good" Boussinesq equation
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An analytical proof for the stability of Heimburg-Jackson pulses
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In this paper, we study the cubic defocusing nonlinear wave equation on the three dimensional hyperbolic space. We use the Fourier truncation method to show that the equation is globally well-posed and scatters if the initial data lies in $H^s(\mathbb{H}^3)$, $s>\frac{182}{201}\approx 0.905$.
A refined long time asymptotic bound for 3D axially symmetric Boussinesq system with zero thermal diffusivity
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Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussinesq system on the Plane
本文在平面全无界区域上证明了具有大 Richardson 数的稳定分层 Couette 流在二维全耗散 Boussinesq 系统中的定量渐近稳定性，首次给出显式增强耗散、Taylor 色散和无粘阻尼估计。
A new stability result for the 2D Boussinesq equations with horizontal dissipation
针对带水平耗散的二维各向异性Boussinesq方程光滑解稳定性这一公开问题，研究在更粗糙的$H^{0,s}(\u211d^2)$空间中给出部分解答，并将正则性指数范围从$\frac12<s<1$推广到整数$s\ue0a11$，采用基础技术实现。
The Boussinesq system in 3-dimensional bounded rough domains: Well-posedness in critical spaces and long-time behavior
本文研究三维Boussinesq系统在有界Lipschitz和C^{1,α}领域中的临界空间适定性，证明了小初值下的全局存在性和任意初值下的局部唯一性，利用L^2和L^1最大正则性分析，揭示了流体速度指数衰减和温度收敛性。
Strong Uniqueness by Kraichnan Transport Noise for the 2D Boussinesq Equations with Zero Viscosity
本文研究了无粘二维Boussinesq方程在Kraichnan型传输噪声下的强唯一性，利用噪声引起的反常正则性补偿非线性项和耦合项的奇异性，证明了概率强解的存在性与唯一性。
Global well-posedness for the radial defocusing cubic wave equation on $\mathbb{R}^{3}$ and for rough data
利用$H^{s} \times H^{s-1}$，$1>s>{7/10}$中的数据证明了$3D$径向散焦三次波动方程的整体适定性.
Global well-posedness of cubic fractional Schrödinger equation with rough data
本文应用I-方法建立分数阶非线性薛定谔方程在能量阈值以下初值空间中的整体适定性，并通过修正Morawetz估计与I-方法结合改进径向初值结果，同时利用“倒置”I-方法推导高阶Sobolev范数的多项式时间增长界。
Cauchy theory for the gravity water waves system with non localized initial data
在这篇文章中，我们发展了重力水波系统的局部柯西理论，对于不衰减到无穷远的粗糙初始数据。我们工作的背景下，$L^2$为基础的一致本地索伯列夫空间介绍加藤。我们证明了一个经典的适定性结果（没有损失的衍生物）。我们的结果也意味着在Hölder空间（损失$d/2$衍生物）的局部适定性结果。作为一个例子，我们解决了Boussinesq提出的一个问题，在运河中的水波问题。我们利用一个基本的观察表明，布西内斯克建议的策略确实适用于这种设置。
Almost conservation laws and global rough solutions to a Nonlinear Schrödinger equation
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Global well-posedness of NLS with a rough potential below the energy norm
我们证明了在粗糙外势的存在下，当s>5/6时，三维立方离焦NLS在H^s中是整体适定的。证明是基于Colliander-Keel-Staffilani-Takaoka-Tao的方法，称为I-方法，但为了处理粗糙势，我们通过谱理论修改了调和分析工具。
In this article, we study long time dynamics for defocusing cubic NLS on three dimensional product space. First, we apply the decoupling method in Bourgain-Demeter \cite{BD} to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on three dimensional product space with rough initial data ($H^s$, $s>\frac{5}{6}$) based on I-method and the bilinear estimate. At last, we discuss the growth of higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon.
Instantaneous analytic smoothing of rough data for the modified and cubic gKdV equations
该研究针对修正型和三次gKdV方程，证明满足Nelson型条件的低正则奇异初始数据对应的解，在t≠0时瞬时实解析，扩展了KdV的光滑效应，采用Lorentz空间细化避免伪微分演算。
Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds
本文借助多重线性特征函数估计、I-method及Strichartz估计，证明三维Zoll流形与球面积上低正则三次NLS的整体适定性，并给出$H^s$范数多项式界。
Sharp global well-posedness for non-elliptic derivative Schrödinger equations with small rough data
我们证明了调制空间$M^s_{2,1}(\mathbb{R}^n)$ for $n\ge 3$（$n= 2$）中三次（四次）非椭圆型导数薛定谔方程Cauchy问题的尖锐的整体适定性.在二维立方情形下，利用Gabor框架，得到了各向异性Lebesgue空间中Schr dinger半群的时间整体色散估计，其中包括$L^2_{x_1}L^\infty_{x_2,t}$空间中的时间整体极大函数估计.利用光滑效应估计和各向异性Lebesgue空间中的色散估计，证明了当Feichtinger-Segal代数或加权索伯列夫空间中的初值足够小时，二维三次双曲导数NLS存在唯一的整体解.
The cubic nonlinear Schrödinger equation with rough potential
研究具有粗糙势能的立方非线性薛定谔方程，提出PDE适定性新最优结果及高效数值方法，通过收敛性分析与数值实验验证了理论正则性并证明新方案在粗糙势能下精度更高。
Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3
证明了在$H^s(\rr^3)$中的散焦三次非线性薛定谔方程对$s > {4/5}$的整体解的存在性和散射性.主要的新估计的论点是一个Morawetz型不等式的解决方案$ϕ$。这个估计限制了$\|ϕ(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$，而著名的Lin-Strauss的Morawetz型估计控制了$ \int_0^{\infty} \int_{\rr^3} \frac{(ϕ(x,t))^4}{|x|} dx dt
In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then pr…
We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schrödinger, Airy, Boussinesq, the fractional Schrödinger, and the gravity and gravity-capillary water wave equations. We also discuss applications to certain nonlinear dispersive equations. In particular, we obtain bounds for the dimension of the graph of the solution to cubic nonlinear Schrödinger and Korteweg-de Vries equations along oblique lines in space-time.
Existence and Uniqueness theory for the fractional Schrödinger equation on the torus
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Regularity properties of the cubic nonlinear Schrödinger equation on the half line
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Global well-posedness of one-dimensional cubic fractional nonlinear Schrödinger equations in negative Sobolev spaces
本文研究了实数轴和圆上三次分数阶非线性 Schrödinger 方程的 Cauchy 问题，通过规范变换与 Duhamel 公式部分迭代处理非线性项，证明了其在负 Sobolev 空间中的整体适定性，并将圆上结果扩展至端点正则性。
Almost sure scattering for the defocusing cubic nonlinear Schrödinger equation on $\mathbb{R}^3\times\mathbb{T}$
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Well-posedness in $L^2$ for nonlocal NLS in the full subcritical range of symbol growth
本文研究了一类非局部非线性Schrödinger方程在直线上的适定性，其非线性由具有亚临界高频增长的傅里叶乘子给出，证明了在L^2(R)中全尺度亚临界范围内的局部适定性。
On the 2d Zakharov system with L^2 Schrödinger data
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Remarks on nonlinear dispersive PDEs with rough dispersion management
本研究针对含粗糙时间系数色散管理的色散PDE，证明其在不同假设下的适定性可从无调制方程迁移，还发现粗糙调制的噪声正则化现象，完善拓展了现有结果。
We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schrödinger (HNLS) equations with cubic convolution (Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrödinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. This improves known results in Sobolev spaces in some sense. As …
Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus
针对四维环面上带加性时空白噪声的随机非线性梁方程（SNLB），引入时变重整化解决噪声粗糙性问题，结合I方法与Gronwall型论证，在能量空间以下建立了散焦三次SNLB的路径全局适定性，并证明了散焦随机阻尼非线性梁方程的几乎必然全局适定性和吉布斯测度不变性。
A continuum of invariant measures for the periodic KdV and mKdV equations
本文针对周期mKdV方程，利用Killip-Vişan-Zhang发现的守恒量构造了支撑在分数阶Sobolev空间上的不变测度，并通过Miura变换推广到KdV及散焦三次非线性Schrödinger方程，覆盖了流适定的最低正则性。
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