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Yuzhao Wang

Publications and source records attributed to Yuzhao Wang.

53 records · Page 3Linked to original sources
arXiv2018

On global well-posedness of the modified KdV equation in modulation spaces

We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $2\leq p < \infty$. For $s < \frac 14$, we show that the solution map for mKdV is not locally uniformly continuous in $M^{2,p}_{s}(\mathbb{R})$. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $2\leq p < \infty$.

math.AP↗
arXiv2018

Liouville-type theorems for the stationary MHD equations in 2D

This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some $L^p$ norm of the velocity-magnetic fields are finite. In particular, these results generalize the corresponding Liouville type properties for the 2D Navier-Stokes equations, such as Gilbarg-Weinberger \cite{GW1978} and Koch-Nadirashvili-Seregin-Sverak \cite{KNSS}, to the MHD setting.

math.AP↗
arXiv2018

Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces

We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces $H^s(\mathbb{T})$, $s > -\frac{1}{3}$, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in $H^s(\mathbb{T})$, $s > -\frac{9}{20}$, via the short-time Fourier restriction norm method. By following the argument in Guo-Oh for the cubic NLS, this also leads to non-existence of solutions for the (non-renormalized) 4NLS in negative Sobolev spaces. (ii) We then prove enhanced uniqueness in $H^s(\mathbb{T})$, $s > -\frac{1}{3}$, by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the $H^s$-energy functional, allowing us to introduce an infinite sequence of correction terms to the $H^s$-energy functional in the spirit of the $I$-method. In fact, the main novelty of this paper is this reduction of the $H^s$-energy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.

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arXiv2016

On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle

In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, adapting the argument by Christ-Colliander-Tao [14] to the periodic setting, we exhibit a norm inflation phenomenon for both the usual cubic NLS and the Wick ordered cubic NLS for $s \leq s_\text{crit} :=- \frac 12$. We also discuss norm inflation phenomena for general cubic fractional NLS on the circle.

math.AP↗
arXiv2015

Lower Bound Estimates for The First Eigenvalue of The Weighted $p$-Laplacian on Smooth Metric Measure Spaces

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the assumption of nonnegative $\infty$-Bakry--Émery Ricci curvature and the $m$-Bakry--Émery Ricci curvature bounded from below by a non-positive constant, the Li--Yau type lower bound estimates are given. The weighted $p$-Bochner formula and the weighted $p$-Reilly formula are derived as the key tools for the establishment of the above results.

math.DG↗
arXiv2015

Uniqueness of nonnegative weak solution to $u^p\le(-Δ)^\fracα{2}u$ on $\mathbb R^N$

This note shows that under $(p,α, N)\in (1,\infty)\times(0,2)\times\mathbb Z_+$ the fractional order differential inequality $$ (\dagger)\quad u^p \le (-Δ)^{\fracα{2}} u\quad\hbox{in}\quad\mathbb R^{N} $$ has the property that if $N\leα$ then a nonnegative solution to $(\dagger)$ is unique, and if $N>α$ then the uniqueness of a nonnegative weak solution to $(\dagger)$ occurs when and only when $p\le N/(N-α)$, thereby innovatively generalizing Gidas-Spruck's result for $u^p+Δu\le 0$ in $\R^N$ discovered in \cite{GS}.

math.AP↗
arXiv2014

Strichartz estimates for Schrödinger equations on irrational tori

In this paper, we prove new Strichartz estimates for linear Schrodinger equations posed on d-dimensional irrational tori. Then, we use these estimates to prove subcritical and critical local well-posedness results for nonlinear Schrodinger equations (NLS) on irrational tori.

math.AP↗
arXiv2013

On the energy-critical fractional Schödinger equation in the radial case

We consider the Cauchy problem for the energy-critical nonlinear Schrödinger equation with fractional Laplacian (fNLS) in the radial case. We obtain global well-posedness and scattering in the energy space in the defocusing case, and in the focusing case with energy below the ground state.

math.AP↗
arXiv2013

Periodic Cubic Hyperbolic Schrödinger equation on $\T^2$

We consider the cubic Hyperbolic Schrödinger equation \eqref{eq:nls} on torus $\T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that \eqref{eq:nls} is analytic locally well-posed in in $H^s(\T^2)$ with $s>1/2$, meanwhile, the ill-posedness in $H^s(\T^2)$ for $s<1/2$ is also obtained. The main difficulty comes from estimating the number of representations of an integer as a difference of squares.

math.AP↗
arXiv2012

Dispersive limit from the Kawahara to the KdV equation

We investigate the limit behavior of the solutions to the Kawahara equation $$ u_t +u_{3x} + \varepsilon u_{5x} + u u_x =0, $$ as $ 0<\varepsilon \to 0 $. In this equation, the terms $ u_{3x} $ and $ \varepsilon u_{5x} $ do compete together and do cancel each other at frequencies of order $ 1/\sqrt{\varepsilon} $. This prohibits the use of a standard dispersive approach for this problem. Nervertheless, by combining different dispersive approaches according to the range of spaces frequencies, we succeed in proving that the solutions to this equation converges in $ C([0,T];H^1(\R)) $ towards the solutions of the KdV equation for any fixed $ T>0$.

math.AP↗
arXiv2012

Periodic nonlinear Schrödinger equation in critical $H^s(\T^n)$ spaces

In this paper we prove some multi-linear Strichartz estimates for solutions to the linear Schrödinger equations on torus $\T^n$. Then we apply it to get some local well-posed results for nonlinear Schrödinger equation in critical $H^{s}(\T^n)$ spaces. As by-products, the energy critical global well-posed results and energy subcritical global well-posed results with small initial data are also obtained.

math.AP↗
arXiv2011

Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain up to some endpoints the full radial Strichartz estimates for the Schrödinger equation. The ideas of proof are based on Shao's ideas \cite{Shao} and some ideas in \cite{GPW} to treat the non-homogeneous case, while at the endpoint we need to use subtle tools to overcome some logarithmic divergence. We also apply the improved Strichartz estimates to the nonlinear problems. First, we prove the small data scattering and large data LWP for the nonlinear Schrödinger equation with radial critical $\dot{H}^s$ initial data below $L^2$; Second, for radial data we improve the results of the $\dot{H}^s\times \dot{H}^{s-1}$ well-posedness for the nonlinear wave equation in \cite{SmithSogge}; Finally, we obtain the well-posedness theory for the fractional order Schrödinger equation in the radial case.

math.AP↗
arXiv2010

Global well-posedness and scattering for Derivative Schrödinger equation

In this paper we study the Cauchy problem for the elliptic and non-elliptic derivative nonlinear Schrödinger equations in higher spatial dimensions ($n\geq 2$) and some global well-posedness results with small initial data in critical Besov spaces $B^s_{2,1}$ are obtained. As by-products, the scattering results with small initial data are also obtained.

math.AP↗
arXiv2010

Remark on Well-posedness of Quadratic Schrödinger equation with nonlinearity $u\bar u$ in $H^{-1/4}(\R)$

In this remark, we give another approach to the local well-posedness of quadratic Schrödinger equation with nonlinearity $u\bar u$ in $H^{-1/4}$, which was already proved by Kishimoto \cite{kis}. Our resolution space is $l^1$-analogue of $X^{s,b}$ space with low frequency part in a weaker space $L^{\infty}_{t}L^2_x$. Such type spaces was developed by Guo. \cite{G} to deal the KdV endpoint $H^{-3/4}$ regularity.

math.AP↗
arXiv2009

On the Well-posedness of the Schrödinger-Korteweg-de Vries system

We prove that the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(\R)\times H^{-{3/4}}(\R)$. The new ingredient is that we use the $\bar{F}^s$ type space, introduced by the first author in \cite{G}, to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares.

math.AP↗
arXiv2009

Local well-posedness for hyperbolic-elliptic Ishimori equation

In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem. We prove that such system is locally well-posed for small data in $H^{s}$ level space, for $s> 3/2$. The new ingredient is that we develop the methods of Ionescu and Kenig \cite{IK} and \cite{IK2} to approach the problem in a perturbative way.

math.AP↗