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Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure

Changxing Miao;  Youbin Zhu — 2006

Applicationes Mathematicae

We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧ u t t - Δ u + m ² u = - F ₁ ( | u | ² , | v | ² ) u , ⎨ ⎩ v t t - Δ v + m ² v = - F ₂ ( | u | ² , | v | ² ) v for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local L²-norm, and local energy also decays to zero if the initial energy satisfies E ( u , v , ℝ ⁿ , 0 ) = 1 / 2 ∫ ℝ ⁿ ( | ∇ u ( 0 ) | ² + | u t ( 0 ) | ² + m ² | u ( 0 ) | ² + | ∇ v ( 0 ) | ² + | v t ( 0 ) | ² + m ² | v ( 0 ) | ² + F ( | u ( 0 ) | ² , | v ( 0 ) | ² ) ) d x < ∞ , and F₁(|u|²,|v|²)|u|² + F₂(|u|²,|v|²)|v|² - F(|u|²,|v|²) ≥ aF(|u|²,|v|²) ≥ 0, a > 0.

Scattering theory for a nonlinear system of wave equations with critical growth

Changxing Miao;  Youbin Zhu — 2006

Colloquium Mathematicae

We consider scattering properties of the critical nonlinear system of wave equations with Hamilton structure ⎧uₜₜ - Δu = -F₁(|u|²,|v|²)u, ⎨ ⎩vₜₜ - Δv = -F₂(|u|²,|v|²)v, for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). By using the energy-conservation law over the exterior of a truncated forward light cone and a dilation identity, we get a decay estimate for the potential...

The Cauchy problem for the coupled Klein-Gordon-Schrödinger system

Changxing Miao;  Youbin Zhu — 2006

Annales Polonici Mathematici

We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural...

On the blow-up phenomenon for the mass-critical focusing Hartree equation in ℝ⁴

Changxing Miao;  Guixiang Xu;  Lifeng Zhao — 2010

Colloquium Mathematicae

We characterize the dynamics of the finite time blow-up solutions with minimal mass for the focusing mass-critical Hartree equation with H¹(ℝ⁴) data and L²(ℝ⁴) data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we analyze the mass concentration phenomenon of such blow-up solutions.

Factorization theorem for product Hardy spaces

Wengu Chen;  Yongsheng Han;  Changxing Miao — 2006

Studia Mathematica

We extend the well known factorization theorems on the unit disk to product Hardy spaces, which generalizes the previous results obtained by Coifman, Rochberg and Weiss. The basic tools are the boundedness of a certain bilinear form on ℝ²₊ × ℝ²₊ and the characterization of BMO(ℝ²₊ × ℝ²₊) recently obtained by Ferguson, Lacey and Sadosky.

An improved maximal inequality for 2D fractional order Schrödinger operators

Changxing Miao;  Jianwei Yang;  Jiqiang Zheng — 2015

Studia Mathematica

The local maximal operator for the Schrödinger operators of order α > 1 is shown to be bounded from H s ( ℝ ² ) to L² for any s > 3/8. This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain’s argument in the case of α = 2. The extension from α = 2 to general α > 1 faces three essential obstacles: the lack of Lee’s reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable...

On the real analyticity of the scattering operator for the Hartree equation

Changxing Miao;  Haigen Wu;  Junyong Zhang — 2009

Annales Polonici Mathematici

We study the real analyticity of the scattering operator for the Hartree equation i ∂ t u = - Δ u + u ( V * | u | ² ) . To this end, we exploit interior and exterior cut-off in time and space, together with a compactness argument to overcome difficulties which arise from absence of good properties as for the Klein-Gordon equation, such as the finite speed of propagation and ideal time decay estimate. Additionally, the method in this paper allows us to simplify the proof of analyticity of the scattering operator for the nonlinear...

The Cauchy problem for the magneto-hydrodynamic system

Marco Cannone;  Changxing Miao;  Nicolas Prioux;  Baoquan Yuan — 2006

Banach Center Publications

We study the uniqueness and regularity of Leray-Hopf's weak solutions for the MHD equations with dissipation and resistance in different frameworks. Using different kinds of space-time estimates in conjunction with the Littlewood-Paley-Bony decomposition, we present some general criteria of uniqueness and regularity of weak solutions to the MHD system, and prove the uniqueness and regularity criterion in the framework of mixed space-time Besov spaces by applying Tao's trichotomy method.

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