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Changxing Miao

Publications and source records attributed to Changxing Miao.

At least 91 records · Page 5Linked to original sources
arXiv2017

On the 4D Nonlinear Schrödinger equation with combined terms under the energy threshold

In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schrödinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension. This extends the results in Miao et al. (Commun Math Phys 318(3):767-808, 2013, The dynamics of the NLS with the combined terms in five and higher dimensions. Some topics in harmonic analysis and applications, advanced lectures in mathematics, ALM34, Higher Education Press, Beijing, pp 265-298, 2015) to four dimension without radial assumption and the proof of scattering is based on the interaction Morawetz estimates developed in Dodson (Global well-posedness and scattering for the focusing, energy-critical nonlinear Schröinger problem in dimension $d=4$ for initial data below a ground state threshold, arXiv:1409.1950), the main ingredients of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

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arXiv2017

Dynamics of nonlinear wave equations

In this lecture, we will survey the study of dynamics of the nonlinear wave equation in recent years. We refer to some lecture notes including such as C. Kenig \cite{Kenig01,Kenig02}, C. Kenig and F. Merle \cite{KM1} J. Shatah and M. Struwe \cite{SS98}, and C. Sogge \cite{sogge:wave} etc. This lecture was written for LIASFMA School and Workshop on Harmonic Analysis and Wave Equations in Fudan universty.

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arXiv2017

The interctitical defocusing nonlinear Schrödinger equations with radial initial data in dimensions four and higher

In this paper, we consider the defocusing nonlinear Schrödinger equation in space dimensions $d\geq 4$. We prove that if $u$ is a radial solution which is \emph{priori} bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot{H}^{s_c}_x$, then $u$ is global and scatters. In practise, we use weighted Strichartz space adapted for our setting which ultimately helps us solve the problems in cases $d\geq 4$ and $0<s_c<\f{1}{2}$. The results in this paper extend the work of \cite[Comm. in PDEs, 40(2015), 265-308]{M3} to higher dimensions.

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arXiv2017

Solitary waves for nonlinear Schrödinger equation with derivative

In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesn't enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters $4ω>c^2$ and the critical parameters $4ω=c^2, c>0$, we show the existence and uniqueness of the solitary waves for (DNLS), up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters $4ω=c^2, c\leq 0$ and the supercritical parameters $4ω 0$ or $4ω>c^2$. On one hand, different with the scattering result for the $L^2$-critical NLS in \cite{Dod:NLS_sct}, the scattering result of (DNLS) doesn't hold for initial data in $\mathcal{K}^+_{ω,c}$ because of the existence of infinity many small solitary/traveling waves in $\mathcal{K}^+_{ω,c},$ with $4ω=c^2, c>0$ or $4ω>c^2$. On the other hand, our global result improves the global result in \cite{Wu-DNLS, Wu-DNLS2} (see Corollary \ref{cor:gwp}).

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arXiv2017

Stability of the traveling waves for the derivative Schrödinger equation in the energy space

In this paper, we continue the study of the dynamics of the traveling waves for nonlinear Schrödinger equation with derivative (DNLS) in the energy space. Under some technical assumptions on the speed of each traveling wave, the stability of the sum of two traveling waves for DNLS is obtained in the energy space by Martel-Merle-Tsai's analytic approach in \cite{MartelMT:Stab:gKdV, MartelMT:Stab:NLS}. As a by-product, we also give an alternative proof of the stability of the single traveling wave in the energy space in \cite{ColinOhta-DNLS}, where Colin and Ohta made use of the concentration-compactness argument.

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arXiv2017

Scattering in $H^1$ for the intercritical NLS with an inverse-square potential

We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.

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arXiv2016

Global well-posedness for the two-dimensional Maxwell-Navier-Stokes equations

In this paper, we investigate Cauchy problem of the two-dimensional full Maxwell-Navier-Stokes system, and prove the global-in-time existence and uniqueness of solution in the borderline space which is very close to $L^2$-energy space by developing the new estimate of $\sup_{j\in\mathbb Z} 2^{2j} \int_0^t \sum_{k\in\mathbb{Z}^2} \big\| \sqrt{ϕ_{i,k}} u(τ) \big\|^2_{L^2(\mathbb{R}^2)} \text{d}τ< \infty$. This solves the open problem in the framework of borderline space purposed by Masmoudi in \cite{Masmoudi-10}.

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arXiv2016

The dynamics of the NLS with the combined terms in five and higher dimensions

In this paper, we continue the study in \cite{MiaoWZ:NLS:3d Combined} to show the scattering and blow-up result of the solution for the nonlinear Schrödinger equation with the energy below the threshold $m$ in the energy space $H^1(\R^d)$, iu_t + Δu = -|u|^{4/(d-2)}u + |u|^{4/(d-1)}u, \; d\geq 5. \tag{CNLS} The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + Δu = -|u|^{4/(d-2)}u$. Compared with the argument in \cite{MiaoWZ:NLS:3d Combined}, the new ingredient is that we use the double duhamel formula in \cite{Kiv:Clay Lecture, TaoVZ:NLS:mass compact} to lower the regularity of the critical element in $L^{\infty}_tH^1_x$ to $L^{\infty}\dot H^{-ε}_x$ for some $ε>0$ in five and higher dimensions and obtain the compactness of the critical element in $L^2_x$, which is used to control the spatial center function $x(t)$ of the critical element and furthermore used to defeat the critical element in the reductive argument.

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arXiv2015

Scattering theory below energy space for two dimensional nonlinear Schrödinger equation

The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schr\'[o]dinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see [J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Commun. Pure Appl. Math. 62(2009)920-968; F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Ecole Norm. Sup. 42(2009)261-290]), establish global well-posedness and scattering for low-regularity solutions of the equation $iu_t + Δu = λ_1|u|^{p_1} u + λ_2|u|^{p_2} u$ under certain assumptions on parameters. This is the first result of this type for an equation which is not scale-invariant. In the first step, we establish global well-posedness and scattering for low regularity solutions of the equation $iu_t + Δu = |u|^p u$, for a suitable range of the exponent $p$ extending the result of Colliander, Grillakis and Tzirakis [Commun. Pure Appl. Math. 62(2009)920-968].

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arXiv2015

An improved maximal inequality for 2D fractional order Schrödinger operators

The local maximal inequality for the Schrödinger operators of order $\a>1$ is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. 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 case of $\a=2$. The extension from $\a=2$ to general $\a>1$ confronts 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 at our disposal and analyzing all the possibilities in using the separateness of the segments to obtain the analogous bilinear $L^2-$estimates. To compensate the absence of Galilean invariance, we resort to Taylor's expansion for the phase function. The Bourgain-Guth inequality in \cite{ref Bourgain Guth} is also rebuilt to dominate the solution of fractional order Schrödinger equations.

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arXiv2015

Scattering theory for the radial $\dot H^{1/2}$-critical wave Equation with a cubic convolution

In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution $\partial_{t}^2u-Δu=\pm(|x|^{-3}\ast|u|^2)u$ in dimensions $d\geq4$. We prove that if the radial solution $u$ with life-span $I$ obeys $(u,u_t)\in L_t^\infty(I;\dot H^{1/2}_x(\mathbb R^d)\times\dot H^{-1/2}_x(\mathbb R^d))$, then $u$ is global and scatters. By the strategy derived from concentration compactness, we show that the proof of the global well-posedness and scattering is reduced to disprove the existence of two scenarios: soliton-like solution and high to low frequency cascade. Making use of the No-waste Duhamel formula and double Duhamel trick, we deduce that these two scenarios enjoy the additional regularity by the bootstrap argument of [Dodson,Lawrie, Anal.PDE, 8(2015), 467-497]. This together with virial analysis implies the energy of such two scenarios is zero and so we get a contradiction.

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arXiv2015

The defocusing quintic NLS in four space dimensions

We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

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arXiv2015

On Wolff's $L^{\frac{5}{2}}-$Kakeya maximal inequality in $\R^3$

We reprove Wolff's $L^{\frac{5}2}-$ bound for the $\R^3-$Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right.

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arXiv2014

The Defocusing Energy-Critical Wave Equation with a Cubic Convolution

In this paper, we study the theory of the global well-posedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity $u_{tt}-Δu+(|x|^{-4}\ast|u|^2)u=0$ in spatial dimension $d \geq 5$. The main difficulties are the absence of the classical finite speed of propagation (i.e. the monotonic local energy estimate on the light cone), which is a fundamental property to show the global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity $u_{tt}-Δu+|u|^\frac4{d-2}u=0$. To compensate it, we resort to the extended causality and utilize the strategy derived from concentration compactness ideas. Then, the proof of the global well-posedness and scattering is reduced to show the nonexistence of the three enemies: finite time blowup; soliton-like solutions and low-to-high cascade. We will utilize the Morawetz estimate, the extended causality and the potential energy concentration to preclude the above three enemies.

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arXiv2014

Maximal estimates for Schroedinger equation with inverse-square potential

In this paper, we consider the maximal estimates for the solution to an initial value problem of the linear Schroedinger equation with a singular potential. We show a result about the pointwise convergence of solutions to this special variable coefficient Schroedinger equation with initial data $u_0\in H^{s}(\R^n)$ for $s>1/2$ or radial initial data $u_0\in H^{s}(\R^n)$ for $s\geq1/4$ and the solution does not converge when $s<1/4$.

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arXiv2014

Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case

We consider the Cauchy problem for the nonlinear Schrödinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in $H^1(\Bbb R^d)$ and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in $H^1(\Bbb R^d)$ below the threshold for radial data when $d\leq4$.

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