The 1-dimensional Zakharov system is shown to have a unique global solution for data without finite energy. The proof uses the " I-method " introduced by Colliander, Keel, Staffilani, Takaoka, and Tao in connection with a refined bilinear Strichatz estimate.
The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in $H^s({\mathbb{R}^2})$ for all $\frac{5}{7}<s<1$ via using $I$-method in the context of atomic spaces. By means of the increment of modified energy, the exsitence of global attractor for weakly damped, forced Zakharov-Kuznetsov equation is also established in $H^s({\mathbb{R}^2})$ for $\frac{10}{11}<s<1$.
We consider the global dynamics below the ground state energy for the Zakharov system in the 3D radial case. We obtain dichotomy between the scattering and the growup.
An initial-boundary value problem for the 3D Zakharov-Kuznetsov equation posed on bounded domains is considered. Existence and uniqueness of a global regular solution as well as exponential decay of the $H^2$-norm for small initial data are proven.
We consider the global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy between scattering and finite time blow up.
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 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.
We consider a class of wave-Schroedinger systems with a Zakharov-Schulman type coupling. This class of systems is indexed by a parameter gamma which measures the strength of the null form in the nonlinearity of the wave equation. The case gamma = 1 corresponds to the well-known Zakharov system, while the case gamma = -1 corresponds to the Yukawa system. Here we show that sufficiently smooth and localized Cauchy data lead to pointwise decaying global solutions which scatter, for any gamma in (0,1].
The Zakharov system was originally proposed to study the propagation of Langmuir waves in an ionized plasma. In this paper, motivated by earlier work of the first and third authors, we numerically and analytically investigate the dynamics of the dissipative Zakharov system on the torus in 1 dimension. We find an interesting family of stable periodic orbits and fixed points, and explore bifurcations of those points as we take weaker and weaker dissipation.
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].
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.
This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a corollary, we obtain global well-posedness in $L^2({\mathbb{R}}^2)$.
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 consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\partial_x Δu+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces $H^s(\mathbb{R}^2)$, where $s$ is greater than the critical scaling index $s_k=1-2/k$. For $k\geq 3$ we also establish a sharp criteria to obtain global $H^1(\R^2)$ solutions. A nonlinear scattering result in $H^1(\R^2)$ is also established assuming the initial data is small and belongs to a suitable Lebesgue space.
The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shown to be global well-posed for the initial date in $H^{s}$ provided $s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below $L^2$, we construct an almost conserved quantity using multilinear correction term following the $I$-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to KdV equation, the main difficulty is to handle the resonant interactions which are significant due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear…
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
Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf（杜塞尔多夫海因里希·海涅大学数学研究所）
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
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 show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation$$ u\_t-D\_x^αu\_{x} + u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le α\le 2,$$is locally well-posed in the spaces $E^s$, $s\textgreater{}\frac 2α-\frac 34$, endowed with the norm$\|f\|\_{E^s} = \|\langle |ξ|^α+μ^2\rangle^s\hat{f}\|\_{L^2(\R^2)}.$As a consequence, we get the global well-posedness in the energy space $E^{1/2}$ as soon as $α\textgreater{}\frac 85$. The proof is based on the approach of the short time Bourgain spaces developed by Ionescu, Kenig and Tataru \cit…
Initial-boundary value problems in a strip with different types of boundary conditions for two-dimensional generalized Zakharov--Kuznetsov equation are considered. Results on global existence and uniqueness of weak solutions in certain weighted spaces are established.
Initial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov-Kuznetsov equation are considered. Results on global existence, uniqueness and long-time decay of weak and regular solutions are establishrd.
An initial-boundary value in a strip with homogeneous Dirichlet boundary conditions for two-dimensional Zakharov-Kuznetsov-Burgers equation is considered. Results on global well-posedness and long-time decay of solutions in Sobolev spaces are established.
An initial-boundary value problem with homogeneous Dirichlet boundary conditions for three-dimensional Zakharov-Kuznetsov equation is considered. Results on global existence, uniqueness and large-time decay of weak solutions in certain weighted spaces are established.
An initial-boundary value problem in a strip with homogeneous Diriclet boundary conditions for two-dimensional generalized Zakharov-Kuznetsov equation is considered. In particular, dissipative and absorbing degenerate terms can be supplemented to the original Zakharov-Kuznetsov equation. Results on global existence, uniqueness and long-time decay of weak silutions are established.
We prove the invariant of the symplectic capacity for the Zakharov system on a torus. If the Zakharov solution map is well-defined, then it can be regarded as a symplectomorphism. Thus, we first show the global well-posedness via the local well-posedness and the conservation law. The invariant of the symplectic capacity can be obtained using an approximation method. Many authors use an approximation method to obtain the nonsqueezing theorem, instead of an invariant of the symplectic capacity. However, the conditions of the Hamiltonian system introduced by Kuksin can be relaxed by a new modifie…
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 the global well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^{s}({\mathbb{R}^2})$ when $\frac{11}{13}<s<1$ via the I-method. Additionally, local well-posedness for the symmetrized ZK equation in $ B^\frac{1}{2}_{2,1}(\mathbb{R}^2)$ is established by using atomic spaces.
We are interested in the Klein-Gordon-Zakharov system in $\mathbb{R}^{1+2}$, which is an important model in plasma physics with extensive mathematical studies. The system can be regarded as semilinear coupled wave and Klein-Gordon equations with nonlinearities violating the null conditions. Without the compactness assumptions on the initial data, we aim to establish the existence of small global solutions, and in addition, we want to illustrate the optimal pointwise decay of the solutions. Furthermore, we show that the Klein-Gordon part of the system enjoys linear scattering while the wave par…
We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t^{-1}, whereas the Schrödinger component decays almost at a rate of t^{-7/6}.
Initial-boundary value problems in a bounded rectangle with different types of boundary conditions for two-dimensional Zakharov-Kuznetsov equation are considered. Results on global well-posedness in the classes of weak and regular solution are established. As applications of the developed technique results on boundary controllability and long-time decay of weak solutions are also obtained.
This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely, \{equation*} \quad \left\{\{array}{lll} {\displaystyle u_t+\partial_x Δu+u^ku_x = 0,}\qquad (x,y) \in \mathbb{R}^2, \,\,\,\, t>0, {\displaystyle u(x,y,0)=u_0(x,y)}. \{array} \right. \{equation*} For $2\leq k \leq 7$, the IVP above is shown to be locally well-posed for data in $H^s(\mathbb{R}^2)$, $s>3/4$. For $k\geq8$, local well-posedness is shown to hold for data in $H^s(\mathbb{R}^2)$, $s>s_k$, where $s_k=1-3/(2k-4)$. Furthermore, for $k\geq3$, …
We are interested in the Klein-Gordon-Zakharov equations in $\mathbb{R}^{1+3}$, and we aim to show that the energy for the global solution to the equations is uniformly bounded, and we do not require the compactness assumption on the initial data. To achieve these goals, the key is to apply Alinhac's ghost weight energy estimates adapted to the Klein-Gordon equations.
The initial value problem of the Zakharov system on two-dimensional torus with general period is considered in this paper. We apply the $I$-method with some 'resonant decomposition' to show global well-posedness results for small-in-$L^2$ initial data belonging to some spaces weaker than the energy class. We also consider an application of our ideas to the initial value problem on $\mathbb{R}^2$ and give an improvement of the best known result by Pecher (2012).
Initial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov-Kuznetsov equation are considered. Results on global well-posedness in classes of regular solutions in the cases of periodic and Neumann boundary conditions, as well as on internal regularity of solutions for all types of boundary conditions are established. Also in the case of Dirichlet boundary conditions one result on long-time decay of regular solutions is obtained.
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…
We consider the Zakharov system in the energy critical dimension $d=4$ with energy below the ground state. It is known that below the ground state solutions exist globally in time, and scatter in the radial case. Scattering below the ground state in the non-radial case is an open question. We show that if scattering fails, then there exists a minimal energy non-scattering solution below the ground state. Moreover the orbit of this solution is precompact modulo translations. The proof follows by a concentration compactness argument, together with a refined small data theory for energy dispersed…
We consider the Zakharov-Kutznesov (ZK) equation posed in $\mathbb R^d$, with $d=2$ and $3$. Both equations are globally well-posed in $L^2(\mathbb R^d)$. In this paper, we prove local energy decay of global solutions: if $u(t)$ is a solution to ZK with data in $L^2(\mathbb R^d)$, then \[ \liminf_{t\rightarrow \infty}\int_{Ω_d(t)}u^{2}({\bf x},t)\mathrm{d}{\bf x}=0, \] for suitable regions of space $Ω_d(t)\subseteq \mathbb R^d$ around the origin, growing unbounded in time, not containing the soliton region. We also prove local decay for $H^1(\mathbb R^d)$ solutions. As a byproduct, our results…
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 prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^s(\mathbb{R}^2)$, for $s\in [1,2]$, on the background of an $L^\infty(\mathbb{R}^3)$-function $Ψ(t,x,y)$, with $Ψ(t,x,y)$ satisfying some natural extra conditions. This result not only gives us a framework to solve the ZK equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of periodic solutions. Additionally, we show the global well-posedness in the energy space $H^1(\mathbb{R}^2)$.
We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2. Solutions u with initial data u\_0 $\in$ H s are known to be global if s $\ge$ 1. We prove that for any integer s $\ge$ 2, u(t) H s grows at most polynomially in t for large times t. This result is related to wave turbulence and how a solution of (ZK) can move energy to high frequencies. It is inspired by analoguous results by Staffilani on the non linear Schr{ö}dinger Korteweg-de-Vries equation. The main ingredients are adequate bilinear estimates in the context of Bourgain's spaces.
