Strichartz Estimates Research Articles

Spectral multipliers and wave propagation for Hamiltonians with a scalar potential

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on R3 to the case of perturbed Hamiltonians of the form −Δ+V, where V is a scalar real-valued potential.In this paper, we prove resolvent estimates, a dispersive bound for the perturbed wave propagator, Mihlin multiplier and fractional integration bounds, and the full range of wave equation Strichartz estimates, under optimal or almost optimal scaling-invariant conditions on the potential and on the spectral multipliers themselves.

The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term.

Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations

We study the Cauchy problem for the inhomogeneous Hartree equation in this paper. Although its well-posedness theory has been extensively studied in recent years, much less is known compared to the classical Hartree model of homogeneous type. In particular, the problem of Sobolev initial data with the Sobolev critical index remains unsolved. The main contribution of this paper is to establish the local existence of solutions to the inhomogeneous equation in the critical cases. To do so, we obtain all possible Lp Strichartz estimates with singular weights.

A note on the inhomogeneous fractional nonlinear Schrödinger equation

This paper investigates some well-posedness issues of the fractional inhomogeneous Schrödinger equation iu˙-(-Δ)γu=±|x|ρ|u|p-1u,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} i\\dot{u}-(-\\Delta )^\\gamma u=\\pm |x|^\\rho |u|^{p-1}u, \\end{aligned}$$\\end{document}where 0<γ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\gamma <1$$\\end{document} and ρ<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho <0$$\\end{document}. Here, one considers the inter-critical regime 0<sc:=N2-2γ+ρp-1<γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<s_c:=\\frac{N}{2}-\\frac{2\\gamma +\\rho }{p-1}<\\gamma $$\\end{document}, where sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_c$$\\end{document} is the energy critical exponent, which is the only one real number satisfying ‖κ2γ+ρp-1u0(κ·)‖H˙sc=‖u0‖H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert \\kappa ^\\frac{2\\gamma +\\rho }{p-1}u_0(\\kappa \\cdot )\\Vert _{\\dot{H}^{s_c}}=\\Vert u_0\\Vert _{\\dot{H}^{s_c}}$$\\end{document}. In order to avoid a loss of regularity in Strichartz estimates, one assumes that the datum is spherically symmetric. First, using a sharp Gagliardo–Nirenberg-type estimate, one develops a local theory in the space H˙γ∩H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^\\gamma \\cap \\dot{H}^{s_c}$$\\end{document}. Then, one investigates the LN(p-1)ρ+2γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\frac{N(p-1)}{\\rho +2\\gamma }}$$\\end{document} concentration of finite-time blow-up solutions bounded in H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^{s_c}$$\\end{document}. Finally, one proves the existence of non-global solutions with negative energy. Since one considers the homogeneous Sobolev space H˙sc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{H}^{s_c}$$\\end{document}, the main difficulty here is to avoid the mass conservation law.

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension two. We construct a martingale solution using a modified Faedo–Galerkin’s method, following Brzeźniak et al (2019 Probab. Theory Relat. Fields 174 1273–338). Then by means of the Strichartz estimates deduced from Blair et al (2008 Proc. Am. Math. Soc. 136 247–56) but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov–Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler (1999 Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78). This is the first result of this type for stochastic nonlinear Schrödinger equation (NLS) on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well.

Strichartz estimates for Maxwell equations in media: The fully anisotropic case

We consider Maxwell equations in media in the uniformly fully anisotropic case in three dimensions and prove Strichartz estimates for Hölder-continuous material coefficients. To this end, we use the FBI transform to conjugate the problem to phase space. After reducing to a scalar estimate by means of a matrix symmetrizer, we show oscillatory integral estimates for a variable-coefficient Fourier extension operator. The characteristic surface has conical singularities for any non-vanishing time frequency. As a consequence of the Strichartz estimates, we improve the local well-posedness for certain fully anisotropic quasilinear Maxwell equations. For these we establish local well-posedness for initial data with Sobolev regularity, which could previously not be covered with energy methods.

Strichartz estimates for the 2D and 3D massless Dirac-Coulomb equations and applications

In this paper we continue the analysis of the dispersive properties of the 2D and 3D massless Dirac-Coulomb equations that has been started in [7] and [8]. We prove a priori estimates of the solution of the mentioned systems, in particular Strichartz estimates with an additional angular regularity, exploiting the tools developed in the previous works. As an application, we show local well-posedness results for a Dirac-Coulomb equation perturbed with Hartree-type nonlinearities.

Large time well-posedness for a Dirac–Klein–Gordon system

In this paper, we prove well-posedness for a Dirac equation perturbed with a moving potential W that satisfies a Klein–Gordon equation. This represents a “toy model” for atoms with relativistic corrections, as the wave function of the electrons interacts with an electric field generated by a nucleus with a given charge density. One of the main ingredients we need is a new family of Strichartz estimates for time-dependent perturbations of the Dirac equation: these represent a result of independent interest.

An alternative proof of Tataru’s dispersive estimates

Abstract The aim of this article is to give an alternative proof of Tataru’s dispersive estimates for wave equations posed on the hyperbolic space. Based on the formula for the wave kernel on ℍ n {\mathbb{H}^{n}} , we give the proof from the perspective of Bessel potentials, by exploiting various facts about Gamma functions, modified Bessel functions, and Bessel potentials. This leads to our proof being more self-contained than that in [D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc. 353 2001, 2, 795–807].

Dispersive estimates for nonhomogeneous radial phases: An application to weakly dispersive equations and water wave models

We study linear dispersive equations in dimension one and two for a class of radial nonhomogeneous phases. L1→L∞ type estimates, Strichartz estimates, local Kato smoothing and Morawetz type estimates are provided. We then apply our results to different water wave models.

Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields

Abstract In this paper, we combine the arguments of [L. Fanelli, J. Zhang and J. Zheng, Uniform resolvent estimates for Schrödinger operators in critical magnetic fields, Int. Math. Res. Not. IMRN 2023), 10.1093/imrn/rnac362] and [Y. Sire, C. D. Sogge, C. Wang and J. Zhang, Reversed Strichartz estimates for wave on non-trapping asymptotically hyperbolic manifolds and applications, Comm. Partial Differential Equations 47 2022, 6, 1124–1132] to prove the maximal estimates for fractional Schrödinger equations ( i ⁢ ∂ t + ℒ 𝐀 α 2 ) ⁢ u = 0 {(i\partial_{t}+\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})u=0} in the purely magnetic fields which includes the Aharonov–Bohm fields. The proof is based on the cluster spectral measure estimates. In particular, for α = 1 {\alpha=1} , the maximal estimate for wave equation is sharp up to the endpoint.

Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case

Abstract We consider the solutions of the Klein–Gordon equation in the 2 + 1 2{+}1 -dimensional space-time which gravity is analyzed, i.e., the manifold ℝ t × ℝ + × ℝ / 2 ⁢ π ⁢ α ⁢ ℤ {\mathbb{R}_{t}\times\mathbb{R}_{+}\times\mathbb{R}/2\pi\alpha\mathbb{Z}} created by a massive point particle. Using the Schwartz kernel of resolvent and spectral measure for Schrödinger operator on the spinless cone, we prove the dispersive estimates and Strichartz estimates for the Klein–Gordon equation. In a future paper, we will consider the problem on the spinning cone.

Dispersive estimates for the wave equation inside cylindrical convex domains

The dispersive and Strichartz estimates are essential for establishing well posedness results for nonlinear equations as well as long time behaviour of solutions to the equation. While in the boundary-less case these estimates are well understood, the case of boundary the situation can become much more difficult. In this work, we establish local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains \(\Omega\subset\mathbb{R}^3\) with smooth boundary \(\partial\Omega\neq \emptyset\). In this paper, we provide detailed proofs of the results established in [16, 17]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain \(\Omega\) have been proved by Blair, Smith, Sogge [4, 5]. Optimal estimates in strictly convex domains have been obtained in [12]. Our case of cylindrical domains is an extension of the result of [12] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions.

Strichartz estimates for the half wave/Klein–Gordon and Dirac equations on compact manifolds without boundary

In this paper we study Strichartz estimates for the half wave, the half Klein–Gordon and the Dirac Equations on compact manifolds without boundary, proving in particular for each of these flows local in time estimates both for the wave and Schrödinger admissible couples (in this latter case with an additional loss of regularity). The strategy for the proof is based on a refined version of the WKB approximation.

Local well‐posedness of the higher‐order nonlinear Schrödinger equation on the half‐line: Single‐boundary condition case

AbstractWe establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. We treat both high regularity () and low regularity () solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial‐boundary value problems, as it involves proving boundary‐type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial‐boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial‐boundary value problem for a partial differential equation associated with a multiterm linear differential operator.

Strichartz estimates for Maxwell equations in media: the structured case in two dimensions

We prove Strichartz estimates for the two dimensional Maxwell equations with diagonal Lipschitz permittivity of special structure. The estimates have no loss in regularity that occurs in general for C^1-coefficients. In the charge-free case, we recover Strichartz estimates local-in-time for Euclidean wave equations in two dimensions up to endpoints.

Global boundedness for the nonlinear Klein-Gordon-Schrödinger system with power nonlinearity

This paper is concerned with the Cauchy problem of the Klein-Gordon-Schrödinger (KGS) equations with a defocusing nonlinearity in three spatial dimensions. The global wellposedness at $H^2$-regularity level and the growth bounds for the corresponding Sobolev norm of the solutions are obtained by applying Koch-Tzvetkov type Strichartz estimates and modified energy, which removes the restriction of the smallness for the initial data in the previous literature and extends the exponential growth bounds to polynomial case.

Strichartz inequality for orthonormal functions associated with special Hermite operator

Abstract In this article, we derive the restriction theorem for the special Hermite transform and obtain the Strichartz estimate for the system of orthonormal functions associated with the special Hermite operator. Further, we discuss the optimal behavior of the constant as a limit of a large number of functions.

A sharpened energy‐Strichartz inequality for the wave equation

AbstractWe consider the sharp Strichartz estimate for the wave equation on in the energy space, due to Bez and Rogers. We show that it can be refined by adding a term proportional to the distance from the set of maximizers, in the spirit of the classical sharpened Sobolev estimate of Bianchi and Egnell.

Long‐time dynamics for the radial focusing fractional INLS

We consider the following fractional NLS with focusing inhomogeneous power‐type nonlinearity: where , , , and . We prove the ground state threshold of global existence and scattering versus finite time blowup of energy solutions in the inter‐critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson–Murphy. This method is based on Tao's scattering criteria and Morawetz estimates. We describe the threshold using some non‐conserved quantities in the spirit of the recent paper by Dinh. The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of a non‐local fractional Laplacian operator. The second one is the presence of a singular weight in the nonlinearity. The greater part of this paper is devoted to the scattering of global solutions in . The Lorentz spaces and the Strichartz estimates play crucial roles in our approach.