Shuanglin Shao
In this paper, we establish the linear profile decomposition for the Airy equation with complex or real initial data in $L^2$, respectively. As an application, we obtain a dichotomy result on the existence of maximizers for the symmetric Airy-Strichartz inequality.
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In this paper, we first show that there exists a maximizer for the non-endpoint Strichartz inequalities for the Schrödinger equation in all dimensions based on the recent linear profile decomposition results. We then present a new proof of the linear profile decomposition for the Schröindger equation with initial data in the homogeneous Sobolev space; as a consequence, there exists a maximizer for the Sobolev-Strichartz inequality.
We prove the existence of maximizers of Sobolev-Strichartz estimates for a general class of propagators, involving relevant examples, as for instance the wave, Dirac and the hyperbolic Schrodinger flows.
We study the existence of maximizers for a one-parameter family of Strichartz inequalities on the torus. In general maximizing sequences can fail to be precompact in $L^2(\mathbb T)$, and maximizers can fail to exist. We provide a sufficient condition for precompactness of maximizing sequences (after translation in Fourier space), and verify the existence of maximizers for a range of values of the parameter. Maximizers for the Strichartz inequalities correspond to stable, periodic (in space and time) solutions of a model equation for optical pulses in a dispersion-managed fiber.
We compute explicitely the best constants and, by solving some functional equations, we find all maximizers for homogeneous Strichartz estimates for the Schrodinger equation and for the wave equation in the cases when the Lebesgue exponent is an even integer.
We prove the existence of maximizers for Strichartz inequalities for the wave equation in dimensions $d\geq 3$. Our approach follows the scheme given by Shao, which obtains the existence of maximizers in the context of the Schrödinger equation. The main tool that we use is the linear profile decomposition for the wave equation which we prove in $\mathbb{R}^d$, $d\geq 3$, extending the profile decomposition result of Bahouri and Gerard, previously obtained in $\mathbb{R}^3$.
This work is devoted to prove a linear profile decomposition for the Airy equation in $\dot{H}_x^{s_k}(\mathbb{R})$, where $s_k=(k-4)/2k$ and $k>4$. We also apply this decomposition to establish the existence of maximizers for a general class of Strichartz type inequalities associated to the Airy equation.
We prove that there exists an extremal function to the Airy Strichartz inequality, $e^{-t\partial_x^3}: L^2(\mathbb{R})\to L^8_{t,x}(\mathbb{R}^2)$ by using the linear profile decomposition. Furthermore we show that, if $f$ is an extremiser, then $f$ is extremely fast decaying in Fourier space and so $f$ can be extended to be an entire function on the whole complex domain. The rapid decay of the Fourier transform of extremisers is established with a bootstrap argument which relies on a refined bilinear Airy Strichartz estimate and a weighted Strichartz inequality.
In this paper, we give an improvement of the Strichartz estimate for Airy equation in the non-diagonal case. As an application, we prove the small data scattering and existence of a special non-scattering solutions, which are minimal in suitable sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation. Especially, we remove several technical restrictions on our previous work about the existence of a special non-scattering solution.
Consider the mass-critical nonlinear Schrödinger equations in both focusing and defocusing cases for initial data in $L^2$ in space dimension N. By Strichartz inequality, solutions to the corresponding linear problem belong to a global $L^p$ space in the time and space variables, where $p=2+4/N$. In 1D and 2D, the best constant for the Strichartz inequality was computed by D.~Foschi who has also shown that the maximizers are the solutions with Gaussian initial data. Solutions to the nonlinear problem with small initial data in $L^2$ are globally defined and belong to the same global $L^p$ spac…
We identify the compactness threshold for optimizing sequences of the Airy-- Strichartz inequality as an explicit multiple of the sharp constant in the Strichartz inequality. In particular, if the sharp constant in the Airy--Strichartz inequality is strictly smaller than this multiple of the sharp constant in the Strichartz inequality, then there is an optimizer for the former inequality. Our result is valid for the full range of Airy--Strichartz inequalities (except the endpoints) both in the diagonal and off-diagonal cases.
Strichartz type estimates of the Airy equation
本文给出Airy方程对应不等式$\\|u\\|_{L_t^qL_x^r}\lesssim \u\\\\|_{X^{s,b}}$成立的充要条件并完成指标关系分类，其结果可用于解决KdV方程及随机KdV方程的低正则性适定性问题。
Sharp Dispersive and Strichartz Estimates for the Neumann Cylindrical Wave Model
该研究针对三维圆柱凸域内带Neumann边界条件的线性波动方程，结合Airy函数相关分析建立精确色散与Strichartz估计，进而证明能量临界五次非线性波动方程的适定性。
In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on the $\mathbb{R} \times \mathbb{R}^2$; we provide a new proof to the characterization of the extremal functions. The only extremal functions are Gaussian functions up to the natural symmetry of the Strichartz inequality, which was investigated previously by Foschi \cite{Foschi:2007:maxi-strichartz-2d} and Hundertmark-Zharnitsky \cite{Hundertmark-Zharnitsky:2006:maximizers-Strichartz-low-dimensions}.
We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein-Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein-Tomas inequality. Our result is valid in any dimension.
Refinements of Strichartz estimates on tori and applications
本文在环面上证明依赖频率时间区间的三线性与新线性 Strichartz 估计，并将其用于 Airy 传播子，得到修正 KdV 方程解存在性的最优 Sobolev 正则性。
Let $u:\R \times \R^n \to \C$ be the solution of the linear Schrödinger equation $iu_t + Δu =0$ with initial data $u(0,x) = f(x)$. In the first part of this paper we obtain a sharp inequality for the Strichartz norm $\|u(t,x)\|_{L^{2k}_tL^{2k}_x(\R \times\R^n)}$, where $k\in \Z$, $k \geq 2$ and $(n,k) \neq (1,2)$, that admits only Gaussian maximizers. As corollaries we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi and Hundertmark - Zharnitsky) and also sharp forms of some Sobolev-Strichartz inequalities. In the second part of the paper we expres…
On profile decomposition for Airy type equation
本文研究了Airy类型方程的轮廓分解，探讨了奇曲线相关的Strichartz不等式及非齐次情况下的两个轮廓现象。
Gaussians Do Not Always Maximize Mixed-Norm Strichartz Inequalities for the Schrödinger Equation
本文研究薛定谔方程混合范数Strichartz不等式最大化问题，在低维1≤d≤5中确定了阈值ρ_d，证明高斯函数在r<ρ_d时为稳定局部最大化子，在r>ρ_d时不是，并给出特定情形下的全局稳定性不等式。
Conditional existence of maximizers for the Tomas-Stein inequality for the sphere
在球面与薛定谔方程Strichartz不等式最佳常数严格比较的条件下，利用精细Tomas-Stein不等式和剖面分解，证明了存在函数极大化该不等式，且任何极大化序列有子序列收敛到极大元。
We prove the pointwise decay of solutions to three linear equations: (i) the transport equation in phase space generalizing the classical Vlasov equation, (ii) the linear Schrodinger equation, (iii) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain $L^1$---$L^\infty$ decay through directly estimating the fundamental solution in physical space, or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines "vector field" commutators that capture the inherent symmetries of the relevant equati…
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