J. Colliander, G. Staffilani
Spatial regularity properties of certain global-in-time solutions of the Zakharov system are established. In particular, the evolving solution $u(t)$ is shown to satisfy an estimate $\Hsup s {u(t)} \leq C {{|t|}^{(s-1)+}}$, where $H^s$ is the standard spatial Sobolev norm. The proof is an adaptation of earlier work on the nonlinear Schrödinger equation which reduces matters to bilinear estimates.
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On decay and regularity of solutions to the modified 2D Zakharov-Kuznetsov equation
本文研究了二维修正Zakharov-Kuznetsov方程的解在加权Sobolev空间中正则性与衰减之间的关系。
The two-dimensional Zakharov system is shown to have a unique global solution for data without finite energy if the L^2 - norm of the Schrödinger part is small enough. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao. A polynomial growth bound for the solution is also given.
In this work, we study some special properties of smoothness concerning to the initial value problem associated with the Zakharov-Kuznetsov-(ZK) equation in the $n-$ dimensional setting, $n\geq 2.$ It is known that the solutions of the ZK equation in the $2d$ and $3d$ cases verify special regularity properties. More precisely, the regularity of the initial data on a family of half-spaces propagates with infinite speed. Our objective in this work is to extend this analysis to the case in that the regularity of the initial data is measured on a fractional scale. To describe this phenomenon we pr…
In this paper we study the initial-value problem associated with the Benjamin-Ono-Zakharov-Kuznetsov equation. Such equation appears as a two-dimensional generalization of the Benjamin-Ono equation when transverse effects are included via weak dispersion of Zakharov-Kuznetsov type. We prove that the initial-value problem is locally well-posed in the usual $L^2(\R^2)$-based Sobolev spaces $H^{s}(\R^2)$, $s>11/8$, and in some weighted Sobolev spaces. To obtain our results, most of the arguments are accomplished taking into account the ones for the Benjamin-Ono equation.
We prove low-regularity global well-posedness for the 1d Zakharov system and 3d Klein-Gordon-Schrödinger system, which are systems in two variables $u:\mathbb{R}_x^d\times \mathbb{R}_t \to \mathbb{C}$ and $n:\mathbb{R}^d_x\times \mathbb{R}_t\to \mathbb{R}$. The Zakharov system is known to be locally well-posed in $(u,n)\in L^2\times H^{-1/2}$ and the Klein-Gordon-Schrödinger system is known to be locally well-posed in $(u,n)\in L^2\times L^2$. Here, we show that the Zakharov and Klein-Gordon-Schrödinger systems are globally well-posed in these spaces, respectively, by using an available conser…
This work aims to study some smoothness properties concerning the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation. More precisely, we prove that the solutions to this model satisfy the so-called propagation of regularity. Roughly speaking, this principle states that if the initial data enjoys some extra smoothness prescribed on a family of half-spaces, then the regularity is propagated with infinite speed. In this sense, we prove that regardless of the scale measuring the extra regularity in such hyperplane collection, then all this regul…
In this paper we study the local and global regularity properties of the Zakharov system on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. Smoothing means that the nonlinear part of the solution on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic Zakharov and the Zakharov on the real line. Uniqueness is proved in the class of smooth solutions. When the bounda…
We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by means of methods developed by Bejenaru-Herr for the Zakharov system and already applied by Kinoshita to the Klein-Gordon-Zakharov system in 2D.
The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces and it is proved that solutions scatter to free solutions as $t \to \pm \infty$. The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension $d=4$…
We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the $I$-method, we prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for any data in the defocusing case and under the assumption that the mass of the initial data is less than the mass of the ground state solution of $Δφ- φ+ φ^3 = 0$ in the focusing case. This improves the global well-posedness result of Linares and Pastor [20].
The Cauchy problem for the generalized Zakharov-Kuznetsov equation $$\partial_t u +\partial_xΔu=\partial_x u^{k+1}, \qquad \qquad u(0)=u_0$$ is considered in space dimensions $n=2$ and $n=3$ for integer exponents $k \ge 3$. For data $u_0 \in \dot{B}^{s_c}_{2,q}$, where $1\le q \le \infty$ and $s_c=\frac{n}{2}- \frac{2}{k}$ is the critical Sobolev regularity, it is shown, that this problem is locally well-posed and globally well-posed, if the data are sufficiently small. The proof follows ideas of Kenig, Ponce, and Vega and uses estimates for the corresponding linear equation, such as local smo…
Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space
该研究针对带耗散项的二维ZKB方程柯西问题，采用结合耗散效应的Fourier限制范数方法，将其适定性的Sobolev空间指标下限拓展至$s>-1/2$，改进了已有相关方程的适定性结果。
Low regularity analysis of the Zakharov--Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$
本文研究柱面上 Zakharov-Kuznetsov 方程的 Cauchy 问题，通过改进共振集的几何分析将局部适定性提升至 $s>1/2$ 且证明其最优性，并利用低正则性修正的 $X^{s,b}$ 空间在随机初始数据下获得 $s<0$ 的一般数据解。
Low-regularity global well-posedness theory for the generalized Zakharov-Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$ and polynomial growth of higher Sobolev norms
本文利用I方法和Strichartz型估计，研究了广义Zakharov-Kuznetsov方程在不同空间上的全局适定性，并证明了其高Sobolev范数的多项式增长特性。
We consider the focusing modified Zakharov-Kuznetsov (mZK) equation in two space dimensions. We prove that solutions which blow up in finite time in the $H^1(\R^{2})$ norm have the property that they concentrate a non-trivial portion of their mass (more precisely, at least the amount equal to the mass of the ground state) at blow-up time. For finite-time blow-up solutions in the $H^s(\R^2)$ norm for $\frac{17}{18} < s < 1$, we prove a slightly weaker result. Moreover, we prove that the stronger concentration result can be extended to the range $ \frac{17}{18} < s \le 1$ under an additional ass…
Local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in the fully periodic case with initial data in Sobolev spaces $H^s$, $s>1$, is proved. Frequency dependent time localization is utilized to control the derivative nonlinearity. The new ingredient to improve on previous results is a nonlinear Loomis-Whitney-type inequality.
The Zakharov System on the Upper Half-Plane
本文研究上半平面上具有非齐次边界条件的扎哈罗夫系统，利用受限范数和Fourier-Laplace方法获得低正则性局部适定性，并证明解的非线性部分更光滑，为二维初边值问题首次建立此类结果。
In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well posedness in low regularity spaces. Especially, the low regularity result for $k$ satisfies $-3/4<k\leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for "Sc…
This paper is concerned with the Cauchy problem of $2$D Klein-Gordon-Zakharov system with very low regularity initial data. We prove the bilinear estimates which are crucial to get the local in time well-posedness. The estimates are established by the Fourier restriction norm method. We utilize the bilinear Strichartz estimates and the nonlinear version of the classical Loomis-Whitney inequality which was applied to Zakharov system.
It is shown that the spatial Sobolev norms of regular global solutions of the (2+1),(3+1) and (4+1)-dimensional Klein-Gordon-Schroedinger system and the (2+1) and (3+1)-dimensional Zakharov system grow at most polynomially with the bound depending on the regularity class of the data. The proof uses the Fourier restriction norm method.
The subsonic limit of the 3D Zakharov system
研究三维Zakharov系统亚声速极限，通过精细范式分析与双线性Strichartz估计，在低正则性Sobolev空间中得到薛定谔和波动分量的最优收敛速率，改进了之前结果并解决了最优性问题。
The Zakharov system in dimension $d=2,3$ is shown to have a local unique solution for any initial values in the energy space $H^{s} \times H^{l} \times H^{l-1}$, where the range of regularity $(s, l)$ is extended, especially at $s=l-1$. The result is obtained mainly by the normal form reduction and the Strichartz estimates.
Multilinear Weighted Estimates and Quantum Zakharov System
本文研究了一维量子Zakharov系统的多线性加权估计，通过Bourgain范数方法证明了低正则性局部适定性，并在无小性假设下证明了全局适定性，同时探讨了量子极限在不同时间区间的行为。
This paper studies the regularity of solutions to the Zakharov and Klein-Gordon-Schrödinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial data. This relies on a new bilinear $X^{s,b}$ estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. Such smoothing estimates have a number of implications for the long-term dynamics of the system. In this work, we give a simplified proof of the existence of global attractors for the Klein-Gordon-Schrödinger flow in the …
We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a corollary, we obtain probabilistic global existence results in $L^2$-based Sobolev spaces. We also obtain global well-posedness in $H^{\frac12+} \times L^2$, which is sharp (up to endpoints) in the class of $L^2$-based Sobolev spaces.
The Zakharov system in dimension $d \geqslant 4$
本文研究维度$d\geqslant4$下经典Zakharov系统的柯西问题，确定了其适定性的Sobolev空间精确范围，还在特定条件下证明了全局适定性与散射性，覆盖能量临界及超临界维度。
We study in this article the stochastic Zakharov-Kuznetsov equation driven by a multiplicative noise. We establish, in space dimensions two and three the global existence of martingale solutions, and in space dimension two the global pathwise uniqueness and the existence of pathwise solutions. New methods are employed in the passage to the limit on a special type of boundary conditions and in the verification of the pathwise uniqueness of martingale solutions with a lack of regularity, where both difficulties arise due to the partly hyperbolic feature of the model.
We show an improved global well-posedness result for the Zakharov system in two space dimensions with minimal regularity assumptions for the data. Especially we are able to allow Schroedinger and wave data, which do not belong to H^1 and L^2, respectively, thus with infinite energy. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao and bilinear estimates by Bejenaru, Herr, Holmer and Tataru. A polynomial growth bound for the solution is also given.
We consider the final-state problem for the Zakharov system in the energy space in three space dimensions. For $(u_+, v_+) \in H^1 \times L^2$ without any size restriction, symmetry assumption or additional angular regularity, we perform a physical-space randomization on $u_+$ and an angular randomization on $v_+$ yielding random final states $(u_+^ω, v_+^ω)$. We obtain that for almost every $ω$, there is a unique solution of the Zakharov system scattering to the final state $(u_+^ω, v_+^ω)$. The key ingredient in the proof is the use of time-weighted norms and generalized Strichartz estimates…
In this paper we consider the Zakharov system with periodic boundary conditions in dimension one. In the first part of the paper, it is shown that for fixed initial data in a Sobolev space, the difference of the nonlinear and the linear evolution is in a smoother space for all times the solution exists. The smoothing index depends on a parameter distinguishing the resonant and nonresonant cases. As a corollary, we obtain polynomial-in-time bounds for the Sobolev norms with regularity above the energy level. In the second part of the paper, we consider the forced and damped Zakharov system and …
We obtain scattering for the 3D Zakharov system with non-radial small data in the energy space with angular regularity of degree one. The main ingredient is a generalized Strichartz estimate for the Schrödinger equation in the space of $L^2$ angular integrability.
The stochastic Zakharov system in dimension $d \geq 4$: Local well-posedness and regularization by noise for scattering
本文研究了高维随机扎哈罗夫系统在超越能量空间的局部适定性及噪声正则化散射效应，揭示了大初始数据下的全局存在性和长期动力学。
The 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schrödinger data u_0 \in \hat{H^{k,p}} and wave data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain assumptions on the parameters k,l and 1<p\le 2, where \|u_0\|_{\hat{H^{k,p}}} := \| < ξ>^k \hat{u_0}\|_{L^{p'}}, generalizing the results for p=2 by Ginibre, Tsutsumi, and Velo. Especially we are able to improve the results from the scaling point of view, and also allow suitable k<0, l<-1/2, i.e. data u_0 \not\in L^2 and (n_0,n_1)\not\in H^{-1/2}\times H^{-3/2}, which was excluded…
We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cu…
Low-regularity well-posedness for the ZK equation on a half-strip
研究半条带上带非齐次边界条件的Zakharov-Kuznetsov方程，利用适应ZK色散结构的Bourgain型空间、各向异性光滑和边界迹估计，建立了$L^2$中的局部适定性。
We consider the vectorial Zakharov system describing Langmuir waves in a weakly magnetized plasma. In its original derivation the evolution for the electric field envelope is governed by a Schrödinger type equation with a singular parameter which is usually large in physical applications. Motivated by this, we study the rigorous limit as this parameter goes to infinity. By using some Strichartz type estimates to control separately the fast and slow dynamics in the problem, we show that the evolution of the electric field envelope is asymptotically constrained onto the space of irrotational vec…
Remark on the Adiabatic Limit of Quantum Zakharov System
本文研究量子 Zakharov 系统绝热极限所得修正 NLSE 的低正则适定性，并严格证明量子参数趋于零时其收敛于标准 NLSE。
In this work we study some regularity properties associated to the initial value problem (IVP) \begin{equation}\label{main1} \left\{ \begin{array}{ll} \partial_{t}u-\partial_{x_{1}}(-Δ)^{α/2} u+u\partial_{x_{1}}u=0, \quad 0< α\leq 2,& \\ u(x,0)=u_{0}(x),\quad x=(x_{1},x_{2},\dots,x_{n})\in \mathbb{R}^{n},\, n\geq 2,\quad t\in\mathbb{R},& \\ \end{array} \right. \end{equation} where $(-Δ)^{α/2}$ denotes the $n-$dimensional fractional Laplacian. We show that solutions to the IVP (0.1) with initial data in a suitable Sobolev space exhibit a local smoothing effect in the spatial variable of $\fracα…
Construction of blow-up solution with minimal mass for 2D cubic Zakharov--Kuznetsov equation
针对二维三次Zakharov--Kuznetsov方程，构造其最小质量爆破解，研究低正则性解在质量阈值处的动力学并证明有限时间爆破可能发生，补充了能量空间中最小质量爆破元不存在的结果。
Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system
本文通过改进抛物面在壳型限制下的解耦不等式，建立新的三线性离散Fourier限制估计，从而在低正则性条件下证明了环面上Zakharov系统的局部适定性，并改进了Kishimoto的结果。
Optimal Decay Estimate and Asymptotic Profile for Solutions to the Generalized Zakharov-Kuznetsov-Burgers Equation in 2D
本文研究二维广义Zakharov-Kuznetsov-Burgers方程柯西问题，证明解的L^∞最优衰减速率为t^(-3/4)，并结合抛物与薛定谔方程技巧推导出显式渐近轮廓。
Recently, we have given the $l^2$ bijectivity for defocusing Ablowitz-Ladik systems in the discrete Sobolev space $l^{2,1}$ by inverse spectral method. Based on these results, the goal of this article is to investigate the long-time asymptotic property for the initial-valued problem of the defocusing Ablowitz-Ladik system with initial potential in lower regularity. The main idea is to perform proper deformations and analysis to the corespondent Riemann-Hilbert problem with the unit circle as the jump contour $Σ$. As a result, we show that when $|\frac{n}{2t}|\le 1<1$, the solution admits Zakha…
On the algebraic lower bound for the radius of spatial analyticity for the Zakharov-Kuznetsov and modified Zakharov-Kuznetsov equations
本文研究了二维广义Zakharov-Kuznetsov方程的初值问题，证明了在时间区间内解的空间解析性半径保持不变，并给出了在不同情况下解析性半径衰减的代数下界。
We consider an initial boundary value problem for a quantum version of the Zakharov system arising in plasma physics. We prove the global well-posedness of this problem in some Sobolev type classes and study properties of solutions. This result confirms the conclusion recently made in physical literature concerning the absence of collapse in the quantum Langmuir waves. In the dissipative case the existence of a finite dimensional global attractor is established and regularity properties of this attractor are studied. For this we use the recently developed method of quasi-stability estimates. I…
On the modified Zakharov--Kuznetsov equation on the cylinder: A trilinear Bourgain-space estimate and its application to local well-posedness
本文针对柱面上的修正Zakharov--Kuznetsov方程，建立三线性Bourgain空间估计，并用于证明$s>1$时的局部适定性，给出了自包含的二进证明。
The Klein-Gordon-Schrödinger system in 3D is shown to be locally well-posed for Schrödinger data in H^s and wave data in H^σ \times H^{σ-1}, if s > - 1/4, σ> - 1/2, σ-2s > 3/2 and σ-2 < s < σ+1 . This result is optimal up to the endpoints in the sense that the local flow map is not C^2 otherwise. It is also shown that (unconditional) uniqueness holds for s=σ=0 in the natural solution space C^0([0,T],L^2) \times C^0([0,T],L^2) \times C^0([0,T],H^{-1/2}) . This solution exists even globally by Colliander, Holmer and Tzirakis. The proofs are based on new well-posedness results for the Zakharov sy…
On the local well-posedness of the Benjamin-Ono-Zakharov-Kuznetsov equation
该研究在$\boldsymbol{\text{R}}^2$上针对BO-ZK方程的柯西问题，通过改进的估计与方法将其局部适定性阈值从$s>5/4$降至$s>19/16$，并建立了解的存在性、唯一性与连续依赖性。
We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases…
From Stochastic Zakharov System to Multiplicative Stochastic Nonlinear Schr{ö}dinger Equation
研究时间白噪声驱动、空间有色噪声的 Zakharov 系统在离子声速趋于无穷时向乘性随机非线性 Schr{ö}dinger 方程的收敛性，通过扩散近似框架与扰动测试函数方法克服噪声导致的奇异性，并证明一维情形下解的存在唯一性与依概率收敛。
Dynamics of the collision of two nearly equal solitary waves for the Zakharov-Kuznetsov equation
研究2维和3维Zakharov-Kuznetsov方程中两个近似等速孤立波碰撞的动力学，通过构造近似解、修正能量泛函和精细调制估计证明了解的稳定性。
Modified Strichartz Estimate of the Periodic Fourth Order NLS
本文在一维环面上建立适配四阶色散关系的修正Strichartz估计，证明四阶非线性Schrödinger方程整体适定性，并揭示解对量子参数的连续依赖仅局部一致而非整体一致。
On decay properties for solutions of the Zakharov-Kuznetsov equation
该研究将Zakharov-Kuznetsov方程解的空间衰减结果推广至任意维数与非整数衰减幂次，证明了指数型初值衰减对应的解的高阶正则性，还得到各向异性加权Sobolev空间适定性，且结论可推广至Korteweg-de Vries方程。
We establish sharp geometric $C^{1+α}$ regularity estimates for bounded weak solutions of evolution equations of $p$-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.
Our goal is to establish a hidden regularity result for solutions of time fractional Petrovsky system where the Caputo fractional derivative is of order $α\in(1,2)$. We achieve such result for a suitable class of weak solutions.
A regularity theory for evolution equations with time-measurable pseudo-differential operators in weighted mixed-norm Sobolev-Lipschitz spaces
该研究针对带时间可测伪微分算子的发展方程，在加权混合范数Sobolev-Lipschitz空间框架下分析解的存在性、唯一性、正则性，并探究迹嵌入与解的连续性，丰富了相关方程的正则性理论。
Regularity Analysis for Two Coupled Second Order Evolution Equations
本文研究具有间接阻尼的两个耦合二阶发展方程组的半群正则性，通过特征值渐近分析划分参数空间，并结合虚轴预解式估计，给出了解析性与Gevrey类正则性的完整精确刻画。
We consider parabolic nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains and study existence, uniqueness and regularity in (anisotropic) weighted Sobolev spaces of the solution.
In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We will extend well-known results for standard evolution equations such as the blow-up alternative, to the time-fractional evolution equations. We also prove the differentiability with respect to time of the solution when the perturbation is sufficiently smooth. The differentiability enables us to use the maximum principle. The theory on general Banach spaces en…
We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, hav…
We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived from three-dimensional Prandtl-Reuss plasticity. We extend the previous existence result by introducing non-zero external forces in the model, and we discuss the regularity of the solutions thus obtained. In particular, we show that the first derivatives with respect to space of the stress tensor are locally square integrable.
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