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1
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Scattering theory for a nonlinear system of wave equations with critical growth

100%
Miao C., Zhu Y.
Colloquium Mathematicum
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2006
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tom 106
|
nr 1
69-81
EN
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 energy. The resulting global-in-time estimates imply immediately the existence of the wave operators and the scattering operator.
2
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Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure

100%
Miao C., Zhu Y.
Applicationes Mathematicae
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2006
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tom 33
|
nr 3-4
323-344
EN
We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧$u_{tt} - Δu + m²u = -F₁(|u|²,|v|²)u$, ⎨ ⎩$v_{tt} - Δv + m²v = -F₂(|u|²,|v|²)v$ 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). 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 $E(u,v,ℝⁿ,0) = 1/2 ∫_{ℝⁿ}(|∇u(0)|² + |u_t(0)|² + m²|u(0)|² + |∇v(0)|² + |v_t(0)|² + m²|v(0)|² + F(|u(0)|²,|v(0)|²))dx < ∞$, 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.
3
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The Cauchy problem for the coupled Klein-Gordon-Schrödinger system

100%
Miao C., Zhu Y.
Annales Polonici Mathematici
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2006
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tom 89
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nr 2
163-195
EN
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 restriction on the power of interactions. In the last part of this paper, we use special admissible pairs and Strichartz estimates to remove the restriction, thereby generalizing previous results and obtaining the well-posedness of the system.
4
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An improved maximal inequality for 2D fractional order Schrödinger operators

81%
Miao C., Yang J., Zheng J.
Studia Mathematica
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2015
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tom 230
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nr 2
121-165
EN
The local maximal operator for the Schrödinger operators of order α > 1 is shown to be bounded from $H^{s}(ℝ²)$ 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 Galilean transformation in the deduction of the main theorem. We get around these difficulties by establishing a new reduction lemma and analyzing all the possibilities in using the separation of the segments to obtain the analogous bilinear L²-estimates. To compensate for the absence of Galilean invariance, we resort to Taylor's expansion for the phase function. The Bourgain-Guth inequality (2011) is also generalized to dominate the solution of fractional order Schrödinger equations.
5
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The Cauchy problem for the magneto-hydrodynamic system

81%
Cannone M., Miao C., Prioux N., Yuan B.
Banach Center Publications
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2006
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tom 74
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nr 1
59-93
EN
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.
6
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Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case

81%
Miao C., Xu G., Zhao L.
Colloquium Mathematicum
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2009
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tom 114
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nr 2
213-236
EN
We establish global existence and scattering for radial solutions to the energy-critical focusing Hartree equation with energy and Ḣ¹ norm less than those of the ground state in $ℝ × ℝ^{d}$, d ≥ 5.
7
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On the real analyticity of the scattering operator for the Hartree equation

81%
Miao C., Wu H., Zhang J.
Annales Polonici Mathematici
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2009
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tom 95
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nr 3
227-242
EN
We study the real analyticity of the scattering operator for the Hartree equation $i∂_tu = -Δu + u(V*|u|²)$. 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 Klein-Gordon equation with cubic nonlinearity.
8
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On the blow-up phenomenon for the mass-critical focusing Hartree equation in ℝ⁴

81%
Miao C., Xu G., Zhao L.
Colloquium Mathematicum
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2010
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tom 119
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nr 1
23-50
EN
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.
9
Content available remote

Factorization theorem for product Hardy spaces

81%
Chen W., Han Y., Miao C.
Studia Mathematica
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2006
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tom 177
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nr 3
235-249
EN
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.
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