Endpoint Strichartz Estimates Research Articles

Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

We obtain improved Strichartz estimates for solutions of the Schrödinger equation on negatively curved compact manifolds which improve the classical universal results of Burq, Gérard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss Lt,xqc-estimates on intervals of length log⁡λ⋅λ−1 for initial data whose frequencies are comparable to λ, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.

Strichartz and uniform Sobolev inequalities for the elastic wave equation

We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lamé operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.

Endpoint Strichartz estimates with angular integrability and some applications

The endpoint Strichartz estimate $$\Vert e^{it\Delta } f\Vert _{L_t^2 L_x^\infty } \lesssim \Vert f\Vert _{L^2}$$ is known to be false in two space dimensions. Taking averages spherically on the polar coordinates $$x=\rho \omega $$ , $$\rho >0$$ , $$\omega \in {{\mathbb {S}}}^1$$ , Tao showed a substitute of the form $$\Vert e^{it\Delta } f\Vert _{L_t^2L_\rho ^\infty L_\omega ^2} \lesssim \Vert f\Vert _{L^2}$$ . Here we address a weighted version of such spherically averaged estimates. As an application, the existence of solutions for the inhomogeneous nonlinear Schrödinger equation is shown for $$L^2$$ data.

Decay estimates for higher-order elliptic operators

This paper is mainly devoted to the study of time decay estimates of the higher-order Schrödinger-type operator H = ( − Δ ) m + V ( x ) H=(-\Delta )^{m}+V(x) in R n \mathbf {R}^{n} for n > 2 m n>2m and m ∈ N m\in \mathbf {N} . For certain decay potentials V ( x ) V(x) , we first derive the asymptotic expansions of resolvent R V ( z ) R_{V}(z) near zero threshold with the presence of zero resonance or zero eigenvalue, as well as identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen-type estimates and local decay estimates for higher-order Schrödinger propagator e − i t H e^{-itH} in the presence of zero resonance or zero eigenvalue. As a consequence, the endpoint Strichartz estimate and L p L^{p} -decay estimates can also be obtained. Finally, by a virial argument, a criterion on the absence of positive embedding eigenvalues is given for ( − Δ ) m + V ( x ) (-\Delta )^{m}+V(x) with a repulsive potential.

Endpoint Strichartz estimates for magnetic wave equations on two dimensional hyperbolic spaces

In this paper, we prove that the Kato smoothing effects for magnetic half wave operators can yield the endpoint Strichartz estimates for linear wave equations with magnetic potentials on the two dimensional hyperbolic spaces. As a corollary, we obtain the endpoint Strichartz estimates in the case of small potentials. This result serves as a cornerstone for the author's work [27] and collaborative work [29] in the study of asymptotic stability of harmonic maps for wave maps from $ \mathbb R\times \mathbb H^2$ to $ \mathbb H^2$.

Global Well-Posedness for the Massive Maxwell–Klein–Gordon Equation with Small Critical Sobolev Data

In this paper we prove global well-posedness and modified scattering for the massive Maxwell–Klein–Gordon equation in the Coulomb gauge on $$\mathbb {R}^{1+d}$$ $$(d \ge 4)$$ for data with small critical Sobolev norm. This extends to the general case $$ m^2 > 0 $$ the results of Krieger–Sterbenz–Tataru ( $$d=4,5 $$ ) and Rodnianski–Tao ( $$ d \ge 6 $$ ), who considered the case $$ m=0$$ . We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein–Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon–Sterbenz. To treat it one needs sharp $$L^{2}$$ null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru–Herr. To overcome logarithmic divergences we rely on an embedding property of $$ \Box ^{-1} $$ in conjunction with endpoint Strichartz estimates in Lorentz spaces.

Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions

In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [SIAM J. Math. Anal. 50 (2018), pp. 3251–3270], the first and second authors consider one- and two-dimensional cases and give a sufficient condition on the nonlinearity so that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converge to a given asymptotic profile in a faster rate than in the lower dimensional cases. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in the aforementioned article. Moreover, we present a candidate for the second asymptotic profile of the solution.

Scattering Theory for the Defocusing Fourth Order NLS with Potentials

Based on the endpoint Strichartz estimates for the fourth order Schrodinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schrodinger operator. J. Funct. Anal., 274, 605–658 (2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader, B.: The cubic fourth-order Schrodinger equation. J. Funct. Anal., 256, 2473–2517 (2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity $$1 + \frac{8}{n} < p < 1 + \frac{8}{{n - 4}}$$ in dimensions n ≥ 7.

Endpoint Strichartz estimates for charge transfer Hamiltonians

We prove the optimal endpoint Strichartz estimates for Schrodinger equation with charge transfer potentials and a general source term in $\mathbb{R}^n$ for $n\geq3$. The proof is based on using the projection on the scattering states defined implicitly in Rodnianski, Schlag and Soffer \cite{RSS1}, and asymptotic completeness for such systems. The method works as well in the matrix non-selfadjoint case.

Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions

We study the Cauchy problem for the Zakharov system in spatial dimension $d\ge 4$ with initial datum $ (u(0), n(0), \partial_t n(0) )\in H^k(\mathbb R^d)\times \dot{H}^l(\mathbb R^d)\times \dot{H}^{l-1}(\mathbb R^d)$. According to Ginibre, Tsutsumi and Velo ([9]), the critical exponent of $(k,l)$ is $ ((d-3)/2,(d-4)/2 ). $ We prove the small data global well-posedness and the scattering at the critical space. It seems difficult to get the crucial bilinear estimate only by applying the $U^2,\ V^2$ type spaces introduced by Koch and Tataru ([23], [24]). To avoid the difficulty, we use an intersection space of $V^2$ type space and the space-time Lebesgue space $E:=L^2_tL_x^{2d/(d-2)}$, which is related to the endpoint Strichartz estimate.

On the cubic Dirac equation with potential and the Lochak–Majorana condition

We study a cubic Dirac equation on R×R3i∂tu+Du+V(x)u=〈βu,u〉βu perturbed by a large potential with almost critical regularity. We prove global existence and scattering for small initial data in H1 with additional angular regularity. The main tool is an endpoint Strichartz estimate for the perturbed Dirac flow. In particular, the result covers the case of spherically symmetric data with small H1 norm. When the potential V has a suitable structure, we prove global existence and scattering for large initial data having a small chiral component, related to the Lochak–Majorana condition.

Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential

The purpose of this paper is to study the validity of global-in-time Strichartz estimates for the Schrödinger equation on Rn, n≥3, with the negative inverse-square potential −σ|x|−2 in the critical case σ=(n−2)2/4. It turns out that the situation is different from the subcritical case σ<(n−2)2/4 in which the full range of Strichartz estimates is known to be hold. More precisely, splitting the solution into the radial and non-radial parts, we show that (i) the radial part satisfies a weak-type endpoint estimate, which can be regarded as an extension to higher dimensions of the endpoint Strichartz estimate with radial data for the two-dimensional free Schrödinger equation; (ii) other endpoint estimates in Lorentz spaces for the radial part fail in general; (iii) the non-radial part satisfies the full range of Strichartz estimates.

On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schr\"{o}dinger equation. We assume the pair interaction is $N^{3\beta -1}V(N^{\beta}\bullet).$ For interaction parameter $\beta \in(0,\frac23)$, we prove that, as $N\rightarrow \infty ,$ the limit points of the solutions to the BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman-Machedon in 2008. This allows for the application of the Klainerman-Machedon uniqueness theorem, and hence implies that the limit is uniquely determined as a tensor product of solutions to the Gross-Pitaevski equation when the $N$-body initial data is factorized. The first result in this direction in 3D was obtained by T. Chen and N. Pavlovi\'{c} (2011) for $\beta \in (0,\frac14)$ and subsequently by X. Chen (2012) for $\beta\in (0,\frac27]$. We build upon the approach of X. Chen but apply frequency localized Klainerman-Machedon collapsing estimates and the endpoint Strichartz estimate in the estimate of the potential part to extend the range to $\beta\in (0,\frac23)$. Overall, this provides an alternative approach to the mean-field program by Erd\"os-Schlein-Yau (2007), whose uniqueness proof is based upon Feynman diagram combinatorics.

The Cubic Dirac Equation: Small Initial Data in $${{H^{\frac{1}{2}}} (\mathbb{R}^{2}}$$)

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space $H^{\frac12}(\mathbb{R}^2)$ is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein-Gordon equation in dimension $n=2$, which is captured by constructing an adapted systems of coordinate frames.

Small data scattering for semi-relativistic equations with Hartree type nonlinearity

We prove that the initial value problem for the equation−i∂tu+m2−Δu=(e−μ0|x||x|⁎|u|2)uin R1+3,m≥0,μ0>0 is globally well-posed and the solution scatters to free waves asymptotically as t→±∞ if we start with initial data which is small in Hs(R3) for s>12, and if m>0. Moreover, if the initial data is radially symmetric we can improve the above result to m≥0 and s>0, which is almost optimal, in the sense that L2(R3) is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, L2(R1+3) bilinear estimates for free waves and an application of the Up and Vp function spaces.

The Cubic Dirac Equation: Small Initial Data in $${H^1(\mathbb{R}^3)}$$ H 1 ( R 3 )

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space $H^1(\mathbb{R}^3)$. The main ingredient is obtaining a sharp end-point Strichartz estimate for the Klein-Gordon equation. In a classical sense this fails and it is related to the failure of the endpoint Strichartz estimate for the wave equation in space dimension three. In this paper, systems of coordinate frames are constructed in which endpoint Strichartz estimates are recovered and energy estimates are established.

On the Strichartz Estimates for the Kinetic Transport Equation

We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present an alternative approach to proving the non-endpoint cases using multilinear analysis.

The Schrödinger Equation with a Potential in Rough Motion

This paper proves endpoint Strichartz estimates for the linear Schrödinger equation in ℝ3, with a time-dependent potential that keeps a constant profile and is subject to a rough motion, which need not be differentiable and may be large in norm. The potential is also subjected to a time-dependent rescaling, with a non-differentiable dilation parameter. We use the Strichartz estimates to prove the non-dispersion of bound states, when the path is small in norm, as well as boundedness of energy. We also include a sample nonlinear application of the linear results.

A critical center‐stable manifold for Schrödinger's equation in three dimensions

AbstractConsider the focusing $\dot H^{1/2}$‐critical semilinear Schrödinger equation in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$ It admits an eight‐dimensional manifold of special solutions called ground state solitons.We exhibit a codimension‐1 critical real analytic manifold ${\cal N}$ of asymptotically stable solutions of (0.1) in a neighborhood of the soliton manifold. We then show that ${\cal N}$ is center‐stable, in the dynamical systems sense of Bates and Jones, and globally‐in‐time invariant.Solutions in ${\cal N}$ are asymptotically stable and separate into two asymptotically free parts that decouple in the limit—a soliton and radiation. Conversely, in a general setting, any solution that stays $\dot H^{1/2}$‐close to the soliton manifold for all time is in ${\cal N}$.The proof uses the method of modulation. New elements include a different linearization and an endpoint Strichartz estimate for the time‐dependent linearized equation.The proof also uses the fact that the linearized Hamiltonian has no nonzero real eigenvalues or resonances. This has recently been established in the case treated here—of the focusing cubic NLS in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$—by the work of Marzuola and Simpson and Costin, Huang, and Schlag. © 2012 Wiley Periodicals, Inc.

The cubic nonlinear Dirac equation

We present some results obtained in collaboration with prof. Piero D’Ancona concerning global existence for the 3D cubic non linear massless Dirac equation with a potential for small initial data in H 1 with slight additional assumptions. The new crucial tool is given by the proof of some refined endpoint Strichartz estimates.