Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type

Abstract

We prove local and global well-posedness for semi-relativistic, nonlinear Schrödinger equations $i{\partial }_{t}u=\sqrt{-\Delta +m^2}u+F(u)$ with initial data in $H^s({\mathbb{R}}^3)$, $s\ge 1/2$. Here $F(u)$ is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions. For focusing $F(u)$, which arise in the quantum theory of boson stars, we derive a sufficient condition for global-in-time existence in terms of a solitary wave ground state. Our proof of well-posedness does not rely on Strichartz type estimates, and it enables us to add external potentials of a general class.

18 pages; replaced with revised version; remark and reference on blow up added

References in corpus (4)

• Boson Stars as Solitary Waves
• Blow-Up for Nonlinear Wave Equations describing Boson Stars
• Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation
• Mean Field Dynamics of Boson Stars

Cited by in corpus (52)

• Boson Stars as Solitary Waves
• Mean-Field Dynamics: Singular Potentials and Rate of Convergence
• Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations
• Blow-Up for Nonlinear Wave Equations describing Boson Stars
• Numerical study of fractional Nonlinear Schrödinger equations
• On Singularity formation for the L^2-critical Boson star equation
• On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions
• Effective Dynamics for Boson Stars
• Sharp Threshold of Blow-up and Scattering for the fractional Hartree equation
• Minimizers for the Hartree-Fock-Bogoliubov Theory of Neutron Stars and White Dwarfs
• The Boson star equation with initial data of low regularity
• Dynamical Collapse of Boson Stars
• On Blowup for time-dependent generalized Hartree-Fock equations
• Modified scattering for the Boson Star Equation
• Derivation of the Hartree equation for compound Bose gases in the mean field limit
• Existence and mass concentration of pseudo-relativistic Hartree equation
• On ground states for the L^2-critical boson star equation
• Higher order Schrodinger and Hartree-Fock equations
• Classical limit for semi-relativistic Hartree systems
• Small data scattering for semi-relativistic equations with Hartree type nonlinearity
• A simple proof of convergence to the Hartree dynamics in Sobolev trace norms
• On Blow-up Profile of Ground States of Boson Stars with External Potential
• Rate of Convergence Towards Semi-Relativistic Hartree Dynamics
• On fractional Schrödinger equations with Hartree type nonlinearities
• Ground states for semi-relativistic Schrödinger-Poisson-Slater energies
• The Lieb-Yau Conjecture for Ground States of Pseudo-Relativistic Boson Stars
• A simple derivation of mean field limits for quantum systems
• Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
• Thermal effects in gravitational Hartree systems
• Nonrelativistic limit of standing waves for pseudo-relativistic nonlinear Schrödinger equations
• Optimal convergence rate of nonrelativistic limit for the nonlinear pseudo-relativistic equations
• Many-Body Blow-Up Profile of Boson Stars with External Potentials
• On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity
• Small data scattering of 2d Hartree type Dirac equations
• Global $H^s,s>0$ Large Data Solutions of 2D Dirac Equation with Hartree Type Interaction
• Uniqueness of ground states for the L^2-critical boson star equation
• Small data scattering for a cubic Dirac equation with Hartree type nonlinearity in ${\mathbb{R}}^{1+3}$
• Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates
• Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation
• On the generalized semi-relativistic Schrödinger-Poisson system in R^n
• Sharp Criteria of Scattering for the Fractional NLS
• Existence and nonlinear stability of stationary states for the semi-relativistic Schrödinger-Poisson system
• On the uniqueness of probability measure solutions to Liouville's equation of Hamiltonian PDEs
• A variational approach to the Brown-Ravenhall operator for the relativistic one-electron atoms
• Effective evolution equations from many body quantum dynamics
• Arrest of blowup for the 3-D semi-relativistic Schrödinger-Poisson system with pseudo-relativistic diffusion
• Local well-posedness of Dirac equations with nonlinearity derived from honeycomb structure in 2 dimensions
• Inverse scattering for the nonlinear Schrödinger equation with the Yukawa potential
• Ground states for pseudo-relativistic Hartree equations of critical type
• Small data scattering of semirelativistic Hartree equation
• Direct methods for pseudo-relativistic Schrödinger operators
• Boson Stars with Long-range Perturbations