On smoothness of extremizers of the Tomas-Stein inequality for $S^1$
We prove that the extremizers to the Tomas-Stein inequality for the one dimension sphere are smooth. This is achieved by studying the associated Euler-Lagrange equation.
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arXiv subjects
Shuanglin Shao
Publications and source records attributed to Shuanglin Shao.
Conditional existence of maximizers for the Tomas-Stein inequality for the sphere
The Tomas-Stein inequality for a compact subset $Γ$ of the sphere $S^d$ states that the mapping $f\mapsto \widehat{fσ}$ is bounded from $L^2(Γ,σ)$ to $L^{2+4/d}(\R^{d+1})$. Then conditional on a strict comparison between the best constants for the sphere and for the Strichartz inequality for the Schrödinger equations, we prove that there exist functions which extremize this inequality, and any extremising sequence has a subsequence which converges to an extremizer. The method is based on the refined Tomas-Stein inequality for the sphere and the profile decompositions. The key ingredient to establish orthogonality in profile decompositions is that we use Tao's sharp bilinear restriction theorem for the paraboloids beyond the Tomas-Stein range. Similar results have been previously established by Frank, Lieb and Sabin \cite{Frank-Lieb-Sabin:2007:maxi-sphere-2d}, where they used the method of the missing mass.
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A remark on the Strichartz inequality in one dimension
In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on $\mathbb{R}^2$. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi and Hundertmark-Zharnitsky.
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A remark on the two dimensional water wave problem with surface tension
We consider the motion of a two-dimensional interface between air (above) and an irrotational, incompressible, inviscid, infinitely deep water (below), with surface tension present. We propose a new way to reduce the original problem into an equivalent quasilinear system which are related to the interface's tangent angle and a quantity related to the difference of tangential velocities of the interface in the Lagrangian and the arc-length coordinates. The new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts. Then if assuming that waves are periodic, we establish a priori energy inequality.
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Linear profile decompositions for a family of fourth order Schrödinger equations
We establish linear profile decompositions for the fourth order Schrödinger equation and for certain fourth order perturbations of the Schrödinger equation, in dimensions greater than or equal to two. We apply these results to prove dichotomy results on the existence of extremizers for the associated Stein--Tomas/Strichartz inequalities; along the way, we also obtain lower bounds for the norms of these operators.
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On one dimensional Quantum Zakharov system
In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well posedness in low regularity spaces. Especially, the low regularity result for $k$ satisfies $-3/4<k\leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for "Schrödinger" waves.
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On characterization of the sharp Strichartz inequality for the Schrödinger Equation
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}.
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On existence of extremizers for the Tomas-Stein inequality for $S^1$
The Tomas-Stein inequality or the adjoint Fourier restriction inequality for the sphere $S^1$ states that the mapping $f\mapsto \hat{fσ}$ is bounded from $L^2(S^1)$ to $L^6(\mathbb{R}^2)$. We prove that there exists an extremizer for this inequality. We also prove that any extremizer satisfies $|f(-x)|=|f(x)|$ for almost every $x\in S^1$.
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On extremisers to a bilinear Strichartz inequality
In this note, we show that a pair of Gaussian functions are extremisers to a bilinear Strichartz inequality, and unique up to the symmetry group of the inequality.
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Pointwise convergence of the Schrödinger maximal operator
This paper has been withdrawn by the author due to an error estimate in Lemma 3.1.
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Nonexistence of soliton-like solutions for defocusing generalized KdV equations
In this short note, we consider the global dynamics of the defocusing generalized KdV equations: u_t + u_{xxx} = (|u|^{p-1}u)_x. We use Tao's theorem that the energy moves faster than mass to prove a moment type dispersion estimate. As an application of the dispersion estimate, we show that there is no soliton-like solutions with decaying assumption.
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Analyticity of extremisers to the Airy Strichartz inequality
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.
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On the extremizers of an adjoint Fourier restriction inequality
The adjoint Fourier restriction inequality for the sphere $S^2$ states that if $f\in\lt(S^2,σ)$ then $\widehat{fσ}\in L^4(\reals^3)$. We prove that all critical points $f$ of the functional $\norm{\widehat{fσ}}_{L^4}/\norm{f}_{\lt}$ are smooth; that any complex-valued extremizer for the inequality is a nonnegative extremizer multiplied by the character $e^{ix\cdotξ}$ for some $ξ$; and that complex-valued extremizing sequences for the inequality are precompact modulo multiplication by characters.
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Existence of Extremals for a Fourier Restriction Inequality
The adjoint Fourier restriction inequality of Tomas and Stein states that the mapping $f\mapsto \widehat{fσ}$ is bounded from $\lt(S^2)$ to $L^4(\reals^3)$. We prove that there exist functions which extremize this inequality, and that any extremizing sequence of nonnegative functions has a subsequence which converges to an extremizer.
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On localization of the Schrödinger maximal operator
In \cite{Lee:2006:schrod-converg}, when the spatial variable $x$ is localized, Lee observed that the Schrödinger maximal operator $e^{itΔ}f(x)$ enjoys certain localization property in $t$ for frequency localized functions. In this note, we give an alternative proof of this observation by using the method of stationary phase, and then include two applications: the first is on is on the equivalence of the local and the global Schrödinger maximal inequalities; secondly the local Schrödinger maximal inequality holds for $f\in H^{3/8+}$, which implies that $e^{itΔ}f$ converges to $f$ almost everywhere if $f\in H^{3/8+}$. These results are not new. In this note we would like to explore them from a slightly different perspective, where the analysis of the stationary phase plays an important role.
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The low regularity global solutions for the critical generalized KdV equation
We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces $H^s(\R)$ for $s>6/13$. Of course, we require that the mass is strictly less than that of the ground state in the focusing case. The main approach is the "I-method" together with the multilinear correction analysis. Moreover, we use some "partially refined" argument to lower the upper control of the multiplier in the resonant interactions. The result improves the previous works of Fonseca, Linares, Ponce (2003) and Farah (2009).
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The linear profile decomposition for the fourth order Schrödinger equation
In this paper, we establish the linear profile decomposition for the one dimensional fourth order Schrödinger equation $$ iu_t-μΔu+Δ^2u=0, t\in\mathbb{R}, x\in\mathbb{R}, u(0,x)=f(x)\in L^2, $$ where $μ\ge 0$. As an application, we establish a dichotomy result on the existence of extremals to the symmetric Schrödinger Strichartz inequality.
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On the mass-critical generalized KdV equation
We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear Schrödinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution.
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