Federico Cacciafesta, Anne-Sophie de Suzzoni
We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.
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The endpoint Strichartz estimates for the Schrödinger equation are known to be false in two dimensions. However, if one averages the solution in $L^2$ in the angular variable, we show that the homogeneous endpoint and the retarded half-endpoint estimates hold, but the full retarded endpoint fails. In particular, the original versions of these estimates hold for radial data.
In this paper we prove Strichartz estimates for the Dirac equation on asymptotically flat manifolds. The proof combines the weak dispersive estimates proved by the first two authors with the Strichartz and smoothing estimates for the wave and Klein-Gordon flows, exploiting some recent results in the same geometrical setting.
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 {\bf draft version} we prove inhomogeneous Strichartz estimates with spherical symmetry in the abstract setting via duality arguments. Then we derive some new explicit estimates in the context of the wave equation. This allows us to prove global existence of spherically symmetric solutions to the Dirac-Klein-Gordon (DKG) system in two space dimensions.
In this paper we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq3$. In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric, and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds.
We prove generalized Strichartz estimates for wave and massless Dirac equations in Aharonov-Bohm magnetic fields. Following a well established strategy to deal with scaling critical perturbations of dispersive PDEs, we make use of Hankel transform and rely on some precise estimates on Bessel functions. As a complementary result, we prove a local smoothing estimate for the Klein-Gordon equation in the same magnetic field.
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 Schroedinger 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.
The endpoint Strichartz estimate $\|e^{itΔ} f\|_{L_t^2 L_x^\infty} \lesssim \|f\|_{L^2}$ is known to be false in two space dimensions. Taking averages spherically on the polar coordinates $x=ρω$, $ρ>0$, $ω\in\mathbb{S}^1$, Tao showed a substitute of the form $\|e^{itΔ} f\|_{L_t^2L_ρ^\infty L_ω^2} \lesssim \|f\|_{L^2}$. 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.
In this paper we obtain some new Strichartz estimates for the Dirac flow in the context of Wiener amalgam spaces which control the local regularity of a function and its decay at infinity separately unlike $L^p$ spaces. While it is well understood recently for some flows such as the Schrödinger and wave flows that work in the non-relativistic regime, nothing is known about the Dirac flow which governs a physical system in the case of relativistic fields.
Strichartz estimates for the half Klein-Gordon equation on asymptotically flat backgrounds and applications to cubic Dirac equations
研究渐近平坦时空上半克莱因 - 戈登方程的\(L^2_t\)端点斯特里赫茨估计，通过拟基本解构造等方法，证明了全次临界范围内大质量三次狄拉克方程的小数据全局适定性和散射性。
Strichartz estimates for the 2D and 3D massless Dirac-Coulomb equations and applications
本文利用既有工具证明了二维和三维无质量 Dirac-Coulomb 方程带有附加角正则性的 Strichartz 估计，并以此展示了带 Hartree 型非线性项的 Dirac-Coulomb 方程的局部适定性结果。
Generalized Strichartz estimates for the massive Dirac equation with critical potentials
本文针对二维阿哈罗诺夫-玻姆磁势和三维库仑势下的大质量狄拉克方程，利用相对论性Hankel变换证明广义Strichartz估计，首次获得该情形下的色散估计。
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 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.
We consider an $n$-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schrödiner equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension $n$ and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary $β>0$ loss. This is in contras…
Dispersive and Strichartz estimates for Dirac equation in a cosmic string spacetime
该研究针对宇宙弦时空背景下的狄拉克方程，完成狄拉克算子自伴延拓分类与传播子显式核推导，证明带权及不带权色散估计，并得到异于经典欧氏情形的精确受限指标Strichartz估计。
In this paper we consider Dirac operators in $\mathbb R^n$, $n\geq2$, with a potential $V$. Under mild decay and continuity assumptions on $V$ and some spectral assumptions on the operator, we prove a limiting absorption principle for the resolvent, which implies a family of Strichartz estimates for the linear Dirac equation. For large potentials the dynamical estimates are not an immediate corollary of the free case since the resolvent of the free Dirac operator does not decay in operator norm on weighted $L^2$ spaces as the frequency goes to infinity.
We extend to general dimension $n\ge1$ the virial identity proved by Boussaid-D'Ancona-Fanelli for the 3D magnetic Dirac equation. As an application we deduce smoothing and Strichartz estimates for an $n$-dimensional Dirac equation perturbed with a magnetic potential.
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^\infty$ estimates fail at the critical regularity in high dimensions by using stable Lévy process in $\R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^\infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.
We analyze the dispersive properties of a Dirac system perturbed with a magnetic field. We prove a general virial identity; as applications, we obtain smoothing and endpoint Strichartz estimates which are optimal from the decay point of view. We also prove a Hardy-type inequality for the perturbed Dirac operator.
We show that global Strichartz estimates for magnetic Dirac operators generally fails, if the potentials do not decay fast enough at infinity. In order to prove this, we construct some explicit examples of homogeneous magnetic potentials with less than Coulomb decay, i. e. with homogeneity-degree more than -1, such that the magnetic field points to a fixed direction, which does not depend on $x\in \mathbb{R}^3$.
We describe the self-adjoint realizations of the operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$, for $m\in\mathbb R $, and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/{|x|}\,β)$, for $ν,μ,λ\in \mathbb R$. We characterize the self-adjointness in terms of the behaviour of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension.
We prove global smoothing and Strichartz estimates for the Schroedinger, wave, Klein-Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.
In this paper we prove well posedness for a system coupling a nonlinear Dirac with a Klein-Gordon equation that represents a toy model for the Helium atom with relativistic corrections: 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.
The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in H^s for s > 1/2. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.
We prove that the Chern-Simons-Dirac equations in the Coulomb gauge are locally well-posed from initial data in H^s with s > 1/4 . To study nonlinear Wave or Dirac equations at this regularity generally requires the presence of null structure. The novel point here is that we make no use of the null structure of the system. Instead we exploit the additional elliptic structure in the Coulomb gauge together with the bilinear Strichartz estimates of Klainerman-Tataru.
We consider the Dirac equation with cubic Hartree-type nonlinearity derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data scattering result in full subcritical range. Main ingredients of the proof are the localized Strichartz estimates, improved bilinear estimates thanks to null-structure hidden in Dirac operator and $Up,Vp$ function spaces. We apply the projection operator and get a system which of linear part is the Klein-Gordon type. It enables us to exploit the null-structures in equation. This result is shown to be almost optimal by showing that iteration method based …
Orthonormal Strichartz estimate for dispersive equations with potentials
本文利用 Kato 光滑扰动理论和微局部分析，建立了带位势的高阶及分数阶 Schrödinger、波、Klein-Gordon 和 Dirac 方程的正交 Strichartz 估计，并应用于 Hartree 方程全局存在性和精细 Strichartz 估计。
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 S…
We prove local smoothing and weighted Strichartz estimates for the Dirac equation with a Aharonov-Bohm potential. The proof relies on an explicit representation of the solution built in terms of spectral projections.
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.
We construct a conformal scattering theory for Dirac fields in the interior of a Reissner-Nordström-like black hole, between the black hole event horizon and the Cauchy horizon. The main result is a resolution of the characteristic Cauchy problem for the Dirac equation on the horizons by solving a system of wave equations and re-interpreting its solution as the components of the required Dirac solution.
Gauge Transforms and Strichartz Estimates for Benjamin--Ono-Type Dirac Equations
本文研究Benjamin-Ono型Dirac方程，通过构造单位模规范映射推导耦合Schrödinger系统，结合Strichartz估计得到$n\geq3$时$\psi$的先验Strichartz型估计。
The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces - previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-Ono equations and also by Bejenaru, Herr, Holmer and Tataru for the 2D Zakharov system - and the null structure of the system detected by d'Ancona, Foschi and Selberg, and a refined bilinear St…
Massless Dirac equation on spinor bundles over real hyperbolic spaces
该研究证明实双曲空间旋量丛狄拉克方程的时间精确色散估计，揭示其短/长时间差异及负曲率影响，指出欧几里得等价性失效，建立无角导数损失且可允许集更大的全局Strichartz估计。
We prove that the initial value problem for the Dirac equation $ \left ( -iγ^μ\partial_μ+ m \right) ψ = \left(\frac{e^{- |x|}}{|x|} \ast ( \overline ψψ)\right) ψ\quad \text{in } \ \R^{1+3} $ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$, if we start with initial data that is small in $H^s$ for $s>0$. This is an almost critical well-posedness result in the sense that $L^2$ is the critical space for the equation. The main ingredients in the proof are Strichartz estimates, space-time bilinear null-form estimates for free waves in $L^2$…
In this paper we prove some uniform asymptotic estimates for confluent hypergeometric functions making use of the steepest-descent method. As an application, we obtain Strichartz estimates with loss of angular derivatives for the massless Dirac-Coulomb equation in $3D$.
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.
We study a static, spherically symmetric system of (2j+1) massive Dirac particles, each having angular momentum j, j=1,2,..., in a classical gravitational and SU(2) Yang-Mills field. We show that for any black hole solution of the associated Einstein-Dirac-Yang/Mills equations, the spinors must vanish identically outside of the event horizon.
Cauchy problem for Dirac equations with Chern-Simons-Proca gauge field
本文通过时空共振与正规形方法，克服Strichartz估计失效和自旋零结构缺失的困难，证明了带Chern-Simons-Proca规范场的Dirac方程小初值解的整体存在性与散射结果。
Decay estimates for massive Dirac equation in a constant magnetic field
研究恒定磁场下大质量 Dirac 方程的衰减与Strichartz估计，证明微局域化衰减估计及局部时间Strichartz估计。
The aim of this work is to prove by a perturbation method the existence of solutions of the coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. We relate the solutions of our equations to those of the nonlinear Choquard equation and we show that the nondegenerate solution of Choquard's equation generates solutions for Einstein-Dirac equations.
We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a $t^{-1}$ decay rate as an operator from the Hardy space $H^1$ to $BMO$, the space of functions of bounded mean oscillation. This estimate, along with the $L^2$ conservation law allows one to deduce a family of Strichartz estimates. We classify the structure of threshold obstructions as being composed of s-wave resonances, p-wave resonances and eigenfunctions. We show that, as in the case of the Schrödinger evolution, the presence of a threshol…
The aim of this Note is to prove by a perturbation method the existence of solutions of the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state and with the electromagnetic coupling constant $(\frac{e}{m})^2<1$. We show that the nondegenerate solution of Choquard's equation generates a branch of solutions of the Einstein-Dirac-Maxwell equations.
We determine explicitly a boundary triple for the Dirac operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/|x|\,β)$, with $ν,μ,λ\in \mathbb R$. Consequently we determine all the self-adjoint realizations of $H$ in terms of the behaviour of the functions of their domain in the origin. When $\sup_{x} |x||\mathbb V(x)| \leq 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form.
Dispersive estimates for Dirac equations in Aharonov-Bohm magnetic fields: massless case
本文研究了受 Aharonov-Bohm 磁场扰动的二维无质量 Dirac 方程的色散性质，利用相对论 Hankel 变换证明了一族逐点衰减估计和全范围 Strichartz 估计。
We study a cubic Dirac equation on $\mathbb{R}\times\mathbb{R}^{3}$ \begin{equation*} i \partial _t u + \mathcal{D} u + V(x) u = \langle βu,u \rangle βu \end{equation*} perturbed by a large potential with almost critical regularity. We prove global existence and scattering for small initial data in $H^{1}$ 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 $H^{1}$ norm. When the potential $V$ has a suitable structure, we prove global existence and s…
On the one dimensional Dirac equation with potential
该研究针对带势一维Dirac方程的$L^1\to L^\infty$色散估计展开，分析了阈值结构与共振影响，借助改进的高能论证得到近最优高能界，并推导出一致极限吸收原理等相关结果。
Construction of a Gibbs measure for the zonal Dirac equation
本文提出了一种构造球面上狄拉克方程吉布斯测度的框架，并通过紧性论证证明了该测度的存在性。
We study bilinear $L^2$ Fourier restriction estimates which are related to the 2d wave equation in the sense that we restrict to subsets of thickened null cones. In an earlier paper we studied the corresponding 3d problem, obtaining several refinements of the Klainerman-Machedon type estimates. The latter are bilinear generalizations of the $L^4$ estimate of Strichartz for the 3d wave equation. In 2d there is no $L^4$ estimate for solutions of the wave equation, but as we show here, one can nevertheless obtain $L^2$ bilinear estimates for thickened null cones, which can be viewed as analogues …
The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions
该研究针对带势三维无质量Dirac方程的色散估计展开，得到不依赖零能本征函数的L¹到L^∞衰减率、带空间权重的改进衰减率，归类阈值障碍结构并证明低正则条件下演化的全局有界性。
On the dispersive estimates for the discrete Schrödinger equation on a honeycomb lattice
本文研究蜂窝晶格上离散薛定谔方程，通过刻画Dirac点处的锥形奇点，证明了$O(|t|^{-2/3})$衰减的色散估计和Strichartz估计，并应用于非线性模型的小数据散射。
We consider a spherically symmetric, static system of a Dirac particle interacting with classical gravity and an SU(2) Yang-Mills field. The corresponding Einstein-Dirac-Yang/Mills equations are derived. Using numerical methods, we find different types of soliton-like solutions of these equations and discuss their properties. Some of these solutions are stable even for arbitrarily weak gravitational coupling.
We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU(2) Yang-Mills field. We prove that the only black-hole solutions of the corresponding Einstein-Dirac-Yang/Mills equations are the Bartnik-McKinnon black-hole solutions of the SU(2) Einstein-Yang/Mills equations; thus the spinors must vanish identically. This indicates that the Dirac particles must either disappear into the black-hole or escape to infinity.
We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-Δ)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2<s<1$, $0<b<2s$ and $1+\frac{2(2s-b)}{N}<p<1+\frac{2(2s-b)}{N-2s}$. We prove the ground state threshold of global existence and scattering versus finite time blow-up of energy solutions in the inter-critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson-Murphy ({Proc. Am. Math. Soc.} {145}: {4859--4867}, 2017). This method is base…
On the Cauchy problem for the Hartree approximation in quantum dynamics
本文证明了量子动力学中含时 Hartree 近似的存在性与唯一性，通过 Strichartz 估计处理 Coulomb 势，并采用受对称拟线性双曲方程启发的递归构造解决了非线性耦合不可视为微扰的一般情形。
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