arXiv subjects
Publications and source records attributed to Zihua Guo.
We prove that the derivative nonlinear Schrödinger equation is globally well-posed in $H^{\frac 12} (\mathbb{R})$ when the mass of initial data is strictly less than $4π$.
We prove scattering for the 2D cubic derivative Schrödinger equation with small data in the critical Besov space with one degree angular regularity. The main new ingredient is that we prove a spherically averaged maximal function estimate for the 2D Schrödinger equation. We also prove a global well-posedness result for the 2D Schrödinger map in the critical Besov space with one degree angular regularity. The key ingredients for the latter results are the spherically averaged maximal function estimate, null form structure observed in \cite{Bej}, as well as the generalised spherically averaged Strichartz estimates obtained in \cite{Guo2} in order to exploit the null form structure.
The Cauchy problem for the Zakharov system in four dimensions is considered. Some new well-posedness results are obtained. For small initial data, global well-posedness and scattering results are proved, including the case of initial data in the energy space. None of these results is restricted to radially symmetric data.
In this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption. We obtain the optimal $H^s$-$L^q$ estimate and its mixed norm generalization. As applications we prove some weighted Strichartz estimates for the propagator $e^{it(-Δ)^{α/2}}φ$, $α>0$.
We prove generalized Strichartz estimates with weaker angular integrability for the Schrödinger equation. Our estimates are sharp except some endpoints. Then we apply these new estimates to prove the scattering for the 3D Zakharov system with small data in the energy space with low angular regularity. Our results improve the results obtained recently in \cite{GLNW}.
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.
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.
We prove that the periodic modified Benjamin-Ono equation is locally well-posed in the energy space $H^{1/2}$. This ensures the global well-posedness in the defocusing case. The proof is based on an $X^{s,b}$ analysis of the system after gauge transform.
We obtain scattering for the 3D Zakharov system with non-radial small data in the energy space with angular regularity of degree one. The main ingredient is a generalized Strichartz estimate for the Schrödinger equation in the space of $L^2$ angular integrability.
We consider the global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy between scattering and finite time blow up.
We prove small energy scattering for the 3D Zakharov system with radial symmetry. The main ingredients are normal form reduction and the radial-improved Strichartz estimates.
We prove small energy scattering for the 3D Klein-Gordon-Zakharov system with radial symmetry. The idea of proof is the same as the Zakharov system studied in \cite{GN}, namely to combine the normal form reduction and the radial-improved Strichartz estimates
We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy u_t + u_{xxxxx} + c_1u_{x} u_{xx} + c_2u u_{x} = 0 x,t \in \R We prove a priori bound of solutions for H^s(\R) with s >= 5/4 and the local well-posedness for s >= 2. The method is a short time X^{s,b} space, which is first developed by Ionescu-Kenig-Tataru in the context of the KP-I equation. In addition, we use a weight on localized X^{s,b} structures to reduce the contribution of high-low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we have the fifth-order equation in the KdV hierarchy is globally well-posed in the energy space H^2.
We consider the global dynamics below the ground state energy for the Zakharov system in the 3D radial case. We obtain dichotomy between the scattering and the growup.
We implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation (NLS) in C_t L^2(T), without using any auxiliary function space. This allows us to construct weak solutions of NLS in C_t L^2(T)$ with initial data in L^2(T) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in H^s(T) for s \geq 1/6.
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.
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.
We prove that the Kawahara equation is locally well-posed in $H^{-7/4}$ by using the ideas of $\bar{F}^s$-type space \cite{GuoKdV}. Next we show it is globally well-posed in $H^s$ for $s\geq -7/4$ by using the ideas of "I-method" \cite{I-method}. Compared to the KdV equation, Kawahara equation has less symmetries, such as no invariant scaling transform and not completely integrable. The new ingredient is that we need to deal with some new difficulties that are caused by the lack symmetries of this equation.