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Changxing Miao

Publications and source records attributed to Changxing Miao.

At least 73 records · Page 4Linked to original sources
arXiv2022

The $W^{s,p}$-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential $$\mathcal L_a=-Δ+\tfrac{a}{|x|^2}, \quad a\geq -\tfrac{(d-2)^2}{4},$$ in dimension $d\geq 2$. We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are $W^{s,p}$-bounded for certain $p$ and $s$ which depend on $a$. As corollaries, we solve some open problems associated with the operator $\mathcal L_a$, which include the dispersive estimates and the local smoothing estimates in dimension $d\geq 2$. We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential.

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arXiv2021

Global Regularity of weak solutions to the generalized Leray equations and its applications

We investigate a regularity for weak solutions of the following generalized Leray equations \begin{equation*} (-Δ)^αV- \frac{2α-1}{2α}V+V\cdot\nabla V-\frac{1}{2α}x\cdot \nabla V+\nabla P=0, \end{equation*} which arises from the study of self-similar solutions to the generalized Naiver-Stokes equations in $\mathbb R^3$. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions $V$ in the weighted Hilbert space $H^α_ω(\mathbb R^3)$. Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from $H^α_ω(\mathbb R^3)$ to $H_ω^{1+α}(\mathbb R^3)$. This regularity result, together linear theory for the non-local Stokes system, lead to pointwise estimates of $V$ which allow us to obtain a natural pointwise property of the self-similar solution constructed in \cite{LXZ}. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Naiver-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai \cite[Comm. Math. Phys., 328 (2014), 29-44]{T}.

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arXiv2021

Threshold scattering for the focusing NLS with a repulsive potential

We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range potentials as previously considered in the work of Hong, as well as the inverse-square potential, previously considered in the work of the authors.

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arXiv2020

Restriction estimates for certain surfaces of finite type in $\mathbb{R}^3$

In this paper, we study the restriction estimate for a certain surface of finite type in $\mathbb{R}^3$, and partially improves the results of Buschenhenke-Müller-Vargas. The key ingredients of the proof include the so called generalized rescaling technique based on a decomposition adapted to finite type geometry, a decoupling inequality and reduction of dimension arguments.

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arXiv2020

On the $C_0$ semigroup generated by the Oseen operator around a steady flow exterior to a rotating obstacle

We consider the motion of an incompressible viscous fluid filling the whole space exterior to a moving with rotation and translation obstacle. We show that the Stokes operator around the steady flow in the exterior of this obstacle generates a $C_0$-semigroup in $L^p$ space and then develop a series of $L^p$-$L^q$ estimates of such semigroup. As an application, we give out the stability of such steady flow when the initial disturbance in $L^3$ and the steady flow are sufficiently small.

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arXiv2019

Scattering for the non-radial inhomogeneous NLS

We extend the result of Farah and Guzmán on scattering for the $3d$ cubic inhomogeneous NLS to the non-radial setting. The key new ingredient is a construction of scattering solutions corresponding to initial data living far from the origin.

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arXiv2019

On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion

In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the supercritical type cases can be evaluated explicitly.

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arXiv2019

Forward self-similar solutions of the fractional Navier-Stokes Equations

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion $(-Δ)^α.$ First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with $5/6<α\leq1$ for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in $\mathbb R^3\times (0,+\infty)$. In particular, when $α=1$, we prove that the solution constructed by Korobkov-Tsai [Anal. PDE 9 (2016), 1811-1827] satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in [Jia and Šverák, Invent. Math. 196 (2014), 233-265].

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arXiv2019

Linear adjoint restriction estimates for paraboloid

We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127-1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091-2102]. As an application, we show a local smoothing estimate for a solution of the linear Schrödinger equation under the assumption that the initial datum has additional angular regularity.

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arXiv2019

Improved variable coefficient square functions and local smoothing of Fourier integral operators

We establish certain square function estimates for a class of oscillatory integral operators with homogeneous phase functions. These results are employed to deduce a refinement of a previous result of Mockenhaupt Seeger and Sogge \cite{MSS-jams} on the local smoothing property for Fourier integral operators, which arise naturally in the study of wave equations on compact Riemannian manifolds. The proof is an adaptation of the bilinear approach of Tao and Vargas \cite{Tao-Vargas-II}, %as well as Garrigós and Seeger \cite{GaSe09}, and based on bilinear oscillatory integral estimates of Lee \cite{Lee-JFA}.

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arXiv2019

The two-dimensional Euler equations in Yudovich type space and $\mathrm{\textbf{bmo}}$-type space

We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a $\rm bmo$-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space involving an unbounded and non-decaying vorticity. Next, we establish an estimate with a logarithmic loss of regularity for the transport equation in a bmo-type space by developing classical analysis tool such as the John-Nirenberg inequality. We also optimize estimates of solutions to the vorticity-stream formulation of the two-dimensional Euler equations with a bi-Lipschitz vector field in bmo-type space by combining an observation introduced by Yodovich with the so-called "quasi-conformal property" of the incompressible.

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arXiv2018

Nonlinear Schrödinger equation with Coulomb potential

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential $i\partial_tu+Δu+\frac{K}{|x|}u=λ|u|^{p-1}u$ with $1 0$ and scattering theory when the Coulomb potential is repulsive i.e. $K\leq0$. The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms.

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arXiv2018

The energy-critical nonlinear wave equation with an inverse-square potential

We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocusing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.

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arXiv2018

Global well-posedness for the defocusing Hartree equation with radial data in $\mathbb R^4$

By $I$-method, the interaction Morawetz estimate, long time Strichartz estimate and local smoothing effect of Schrödinger operator, we show global well-posedness and scattering for the defocusing Hartree equation $$\left\{ \begin{array}{ll} iu_t + Δu &=F(u), \quad (t,x) \in \mathbb{R} \times \mathbb{R}^4 u(0) \\ &=u_0(x)\in H^s(\mathbb{R}^4), \end{array} \right. $$ where $F(u)= (V* |u|^2) u$, and $V(x)=|x|^{-γ}$, $3< γ<4$, with radial data in $H^{s}(\mathbb{R}^4)$ for $s>s_c:=γ/2-1$. It is a sharp global result except of the critical case $s=s_c$, which is a very difficult open problem.

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arXiv2018

Minimal blow-up initial data in critical Fourier-Herz spaces for potential Navier-Stokes singularities

In this paper, we mainly prove the existence of the minimal blow-up initial data in critical Fourier-Herz space $F\dot{B}^{2-{\frac3p}}_{p,q}(\RR^3)$ with $1<p\leq\infty$ and $1\leq q<\infty$ for the three dimensional incompressible potential Navier-Stokes equations by developing techniques of "localization in space" involving the partial regularity given by the De Giorgi iteration, weak-strong uniqueness, the short-time behaviour of the kinetic energy and stability of singularity of Calderón's solution.

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arXiv2018

Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case

In this paper, we develop the modulation analysis, the perturbation argument and the Virial identity similar as those in \cite{MartelM:Instab:gKdV} to show the orbital instability of the solitary waves $\Q\sts{x-ct}\e^{ıωt}$ of the generalized derivative nonlinear Schrödinger equation (gDNLS) in the degenerate case $c=2z_0\sqrtω$, where $z_0=z_0\stsσ $ is the unique zero point of $F\sts{z;~σ}$ in $\sts{-1, ~ 1}$. The new ingredients in the proof are the refined modulation decomposition of the solution near $\Q$ according to the spectrum property of the linearized operator $\Scal_{ω, c}"\sts{\Q}$ and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in \cite{Fukaya2017}.

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arXiv2018

The global well-posedness and scattering for the $5$D defocusing conformal invariant NLW with radial initial data in a critical Besov space

In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov space $\dot{B}^3_{1,1}\times\dot{B}^2_{1,1}(\mathbb{R}^5)$. This is the five dimensional analogue of \cite{dodson-2016}, which is the first result on the global well-posedness and scattering of the energy subcritical nonlinear wave equation without the uniform boundedness assumption on the critical Sobolev norms employed as a substitute of the missing conservation law with respect to the scaling invariance of the equation. The proof is based on exploiting the structure of the radial solution, developing the Strichartz-type estimates and incorporation of the strategy in \cite{dodson-2016}, where we also avoid a logarithm-type loss by employing the inhomogeneous Strichartz estimates.

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