Hartmut Pecher
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.
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We prove small energy scattering for the 3D Klein-Gordon-Zakharov system with radial symmetry. The idea of proof is the same as the Zakharov system studied in \cite{GN}, namely to combine the normal form reduction and the radial-improved Strichartz estimates
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.
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.
We are interested in the long time asymptotic behavoir of solutions to the scalar Zakharov system \[ i u_{t} + Δu = nu,\] \[n_{tt} - Δn= Δ|u|^2\] and the Klein-Gordon Zakharov system \[ u_{tt} - Δu + u = - nu,\] \[ n_{tt} - Δn= Δ|u|^2\] in one dimension of space. For these two systems, we give two results proving decay of solutions for initial data in the energy space. The first result deals with decay over compact intervals asuming smallness and parity conditions ($u$ odd). The second result proves decay in far field regions along curves for solutions whose growth can be dominated by an incre…
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…
The orbital instability of standing waves for the Klein-Gordon-Zakharov system has been established in two and three space dimensions under radially symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one space dimensional case, for the non-degenerate situation, we first check that the Klein-Gordon-Zakharov system satisfies Grillakis-Shatah-Strauss' assumptions on the stability and instability theorems for abstract Hamiltonian systems, see Grillakis-Shatah-Strauss (J. Funct. Anal. 1987). As to the degenerate case that the frequency $|ω|=1/\sqrt{2}$, we follow Wu (ArXiv: 1…
We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 5$ with initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0} \in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s_c=d/2-2$. By $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 5$.
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…
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 …
On the global well-posedness and scattering of the 3D Klein-Gordon-Zakharov system
本文研究了三维Klein-Gordon-Zakharov系统在小初始数据下的整体适定性，通过改进的Alinhac幽灵权重方法和正则形式型估计证明了全局解能量的一致有界性。
Multi-soliton solutions of Klein-Gordon-Zakharov system
本文研究Klein-Gordon-Zakharov系统的多孤子解，证明给定数量孤子的多孤子解在能量空间中渐近收敛到这些孤子之和，并通过局部化与调制技术消除方向性约束，改进了已有结果。
We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0}\in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s=s_c=d/2-2$. If the initial datum is radial, then we prove the small data global well-posedness and scattering at the critical space in $d \ge 4$ by applying the radial Strichartz estimates and $U^2, V^2$ type spaces. On the other hand, if the initial datum is non-radial,…
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.
Consider the Klein-Gordon-Zakharov equations in $\mathbb{R}^{1+2}$, and we are interested in establishing the small global solution to the equations and in investigating the pointwise asymptotic behavior of the solution. The Klein-Gordon-Zakharov equations can be regarded as a coupled semilinear wave and Klein-Gordon system with quadratic nonlinearities which do not satisfy the null conditions, and the fact that wave components and Klein-Gordon components do not decay sufficiently fast makes it harder to conduct the analysis. In order to conquer the difficulties, we will rely on the hyperboloi…
Existence of dipoles of Klein-Gordon-Zakharov system
研究Klein-Gordon-Zakharov系统的长时间行为，证明存在具有特殊性质的解，即偶极子解，其形式接近两个孤立波的叠加，且两者位移随时间对数增长。
In this note, we consider discrete nonlinear Klein-Gordon equations with potential. By the pioneering work of Sigal, it is known that for the "continuous" nonlinear Klein-Gordon equation, no small time periodic solution exists generically. However, for the discrete nonlinear Klein-Gordon equations, we show that there exist small time periodic solutions.
We prove global well-posedness for the 3D Klein-Gordon equation with a concentrated nonlinearity.
The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces - previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-Ono equations and also by Bejenaru, Herr, Holmer and Tataru for the 2D Zakharov system - and the null structure of the system detected by d'Ancona, Foschi and Selberg, and a refined bilinear St…
In this article we will prove the global existence of a type of wave-Klein-Gordon system in $2+1$ spacetime dimension. Some technical tools such as conformal energy estimate on hyperboloid, normal form transform on Klein-Gordon equations will be adapted to this $2+1$ dimensional case.
We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.
By exploiting estimates on Bloch functions obtained in a previous paper, we prove decay estimates for Klein Gordon equations with a time independent potential periodic in space in 1D and with generic mass
In this paper we consider the Klein-Gordon-Maxwell system in the electrostatic case, assuming the fall-off large-distance requirement on the gauge potential. We are interested in proving the existence of finite energy (and finite charge) standing waves, having the phase corresponding to the mass coefficient in the Klein-Gordon Lagrangian.
We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many standing waves.
We prove the existence of a mountain-pass solution and the a priori bound property for the electrostatic Klein-Gordon-Maxwell equations in high dimension.
We consider the decay rate of solutions to nonlinear Klein-Gordon systems with a critical type nonlinearity. We will specify the optimal decay rate for a specific class of Klein-Gordon systems containing the dissipative nonlinearites. It will turn out that the decay rate which is previously found in some models is optimal.
We consider a system of two coupled non-linear Klein-Gordon equations. We show the existence of standing waves solutions and the existence of a Lyapunov function for the ground state.
Static Klein-Gordon-Maxwell-Proca systems are massive versions of the electrostatic Klein-Gordon-Maxwell Systems. The vector field in these systems inherits a mass and is governed by the Proca action which generalizes that of Maxwell. Following previous work by Druet & al. we prove existence and uniform bounds for such systems in the double critical case of dimension 4.
On The Convergence of the Variational Iteration Method Applied to Variable Coefficient Klein-Gordon Problems
本文提出适用于变系数Klein-Gordon方程的变分迭代法表述，并证明该方法生成的近似解序列收敛到Airy函数。
We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay that the wave, unlike in the flat space.
We consider long time evolution of small solutions to general multispeed Klein-Gordon systems in 3+1 dimensions. We prove that such solutions are always global and scatter to a linear flow, thus extending previous partial results. The main new ingredients of our method is an improved linear dispersion estimate exploiting the asymptotic spherical symmetry of Klein-Gordon waves, and a corresponding bilinear oscillatory integral estimate.
We prove that in the nonrelativistic limit, solutions of the Klein-Gordon-Maxwell system in 1+3 dimensions converge in the energy space to solutions of a Schrodinger-Poisson system, under appropriate conditions on the initial data. This requires the splitting of the scalar Klein-Gordon field into a sum of two fields, corresponding, in the physical interpretation, to electrons and positrons.
In this paper, we prove the exponential decay of local energy for the Klein-Gordon equation with localized critical nonlinearity. The proof relies on generalized Strichartz estimates, and semi-group of Lax-Phillips.
We consider the Cauchy problem for the 2D and 3D Klein-Gordon-Schrödinger system. In 2D we show local well-posedness for Schrödinger data in H^s and wave data in H^σ x H^{σ-1} for s=-1/4 + and σ= -1/2, whereas ill-posedness holds for s<- 1/4 or σ<-1/2, and global well-posedness for s\ge 0 and s- 1/2 \le σ< s+ 3/2. In 3D we show global well-posedness for s \ge 0, s - 1/2 < σ\le s+1. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru, and Bejenaru and Herr for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schröding…
We prove small data global existence and scattering for quasilinear systems of Klein-Gordon equations with different speeds, in dimension three. As an application, we obtain a robust global stability result for the Euler-Maxwell equations for electrons.
We prove existence and uniform bounds for electrostatic Klein-Gordon-Maxwell systems in the inhomogeneous context of a compact Riemannian manifold when the mass potential, balanced by the phase, is small in a quantified sense.
We derive a uniform exponential decay of the total energy for the nonlinear Klein-Gordon equation with a damping around spatial infinity in the whole space or in the exterior of a star shaped obstacle.
In this work we consider the problem of global existence of small regular solutions to a type nonlinear wave-Klein-Gordon system with semi-linear interactions in two spatial dimension. We develop some new techniques on both wave equations and Klein-Gordon equations in order to get sufficient decay rates when energies are not uniformly bounded. These techniques are compatible with those introduced in previous work on two spatial-dimensional quasi-linear wave-Klein-Gordon systems, and can be applied in much general cases.
Finite time blow-up of semi-linear Klein-Gordon equations with positive initial energy in FLRW spacetimes
针对FLRW时空中半线性Klein-Gordon方程，论文将Minkowski时空的凹性方法推广至该背景，建立了解发生有限时间爆破的充分条件。
We study the scattering problems for the quadratic Klein-Gordon equations with radial initial data in the energy space. For 3D, we prove small data scattering, and for 4D, we prove large data scattering with mass below the ground state.
Asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions
本文研究高维Vlasov-Klein-Gordon系统中小解的渐进行为，采用向量场方法和双曲面叶结构分析，克服Klein-Gordon场质量带来的挑战。
In this paper we study the global existence and uniqueness of solution for a Klein-Gordon equations system with mixed boundary conditions. Also we analyze the asymptotic behavior of this solution.
We study linear Klein-Gordon equations with moving potentials motivated by the stability analysis of traveling waves and multi-solitons. In this paper, Strichartz estimates, local energy decay and the scattering theory for these models are established. The results and estimates obtained in this paper will be used to study the interaction of solitons and the stability of multisolitons of nonlinear Klein-Gordon equations.
Scattering for the Klein-Gordon-Zakharov system in two dimensions
针对二维Klein-Gordon-Zakharov系统，通过新型非线性变换与物理-频率空间结合方法，证明了小光滑局部初值解的全局存在性，揭示了修正散射与线性散射的二分现象。
We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many static solutions.
We look for three dimensional vortex-solutions, which have finite energy and are stationary solutions, of Klein-Gordon-Maxwell-Proca type systems of equations. We prove the existence of three dimensional cylindrically symmetric vortex-solutions having a least possible energy among all symmetric solutions. Moreover we show that, if the Proca mass disappears, then the solutions tends to a solution of the Klein-Gordon-Maxwell system.
We present a set of time quasi-periodic solutions to a nonlinear Klein-Gordon equation with a decaying nonlinear term on the torus in arbitrary dimensions. This paper generalizes the bifurcation method developed in [W2].
We prove that the Maxwell-Klein-Gordon equations on $\R^{1+4}$ relative to the Coulomb gauge are locally well-posed for initial data in $H^{1+ε}$ for all $ε> 0$. This builds on previous work by Klainerman and Machedon who proved the corresponding result for a model problem derived from the Maxwell-Klein-Gordon system by ignoring the elliptic features of the system, as well as cubic terms.
We show the ill-posedness of the Cauchy problem for the Dirac-Klein-Gordon system in one dimension in the critical Sobolev space. From this, we finish the classification of the regularities for which this problem is well-posed or ill-posed.
This article resolves some errors in the paper "Scattering threshold for the focusing nonlinear Klein-Gordon equation", Analysis & PDE 4 (2011) no. 3, 405-460. The errors are in the energy-critical cases in two and higher dimensions.
The Klein-Gordon - Schroedinger system with Yukawa coupling is shown to have a unique global solution for rough data, which not necessarily have finite energy. The proof uses a generalized bilinear estimate of Strichartz type and Bourgain's idea to split the data into low and high frequency parts.
In this paper, we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice, $h{\mathbb{Z}}^n.$ Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is well-posed in $\ell^2(h{\mathbb{Z}}^n).$ However, we also recover the well-posedness results in the certain Sobolev spaces in the limit of the semiclassical parameter $h\rightarrow 0.$
We consider the Cauchy problem for coupled systems of wave and Klein-Gordon equations with quadratic nonlinearity in three space dimensions. We show global existence of small amplitude solutions under certain condition including the null condition on self-interactions between wave equations. Our condition is much weaker than the strong null condition introduced by Georgiev for this kind of coupled system. Consequently our result is applicable to certain physical systems, such as the Dirac-Klein-Gordon equations, the Dirac-Proca equations, and the Klein-Gordon-Zakharov equations.
Strong instability of standing waves for a system of nonlinear Klein-Gordon equations with quadratic interaction
研究二维和三维空间中带二次相互作用的非线性Klein-Gordon方程组，在不假设质量共振条件时，证明驻波解的强不稳定性。
We give blow-up results for the Klein-Gordon equation and other perturbations of the semilinear wave equations with superlinear power nonlinearity, in one space dimension or in higher dimension under radial symmetry outside the origin.
This paper is devoted to study the dispersive properties of the linear Klein-Gordon and wave equations on a class of locally symmetric spaces. As a consequence, we obtain the Strichartz estimate and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on real hyperbolic spaces.
On dispersive estimates for one-dimensional Klein-Gordon equations
本文改进了1D Klein-Gordon方程色散衰减的结果，提出新方法以在更强的norm中建立衰减并减弱势函数的假设。
In this article we present some results on the existence of positive and ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. We introduce a general nonlinearity with subcritical and supercritical growth which does not require the usual Ambrosetti-Rabinowitz condition. The proof is based on variational methods and perturbation arguments.
Local well-posedness for the Dirac - Klein - Gordon equations is proven in one space dimension, where the Dirac part belongs to H^{-{1/4}+ε} and the Klein - Gordon part to H^{{1/4}-ε} for 0 < ε< 1/4, and global well-posedness, if the Dirac part belongs to the charge class L^2 and the Klein - Gordon part to H^k with 0 < k < 1/2 . The proof uses a null structure in both nonlinearities detected by d'Ancona, Foschi and Selberg and bilinear estimates in spaces of Bourgain-Klainerman-Machedon type.
We prove global well-posedness and scattering for the massive Dirac-Klein-Gordon system with small initial data of subcritical regularity in dimension three. To achieve this, we impose a non-resonance condition on the masses.
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