Well-posedness for Semi-relativistic Hartree Equations of Critical Type

By E. Lenzmann. 2005

196 citations recorded by the source corpus.

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 Hs(ℝ3), $s \geqslant 1/2$. 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 global-in-time existence for small initial data, where the smallness condition is expressed in terms of the L2-norm of solitary wave ground states. Our proof of well-posedness does not rely on Strichartz type estimates. As a major benefit from this, our method enables us to consider external potentials of a quite general class.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e3eeedc00a439286d05b417243cb4d023a963264

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In Mathematical Physics

Citations in the corpus

Sharp threshold of blow-up and scattering for the fractional Hartree equation (2017, background): “1) models the dynamics of (pseudo-relativistic) boson stars, where the potential 1 |x| is the Newtonian gravitational potential in the appropriate physical units (see[10, 30]).”
On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity (2012, methodology): “This result is an extension of the work of Lenzmann [22] and [8] to fractional NLS.”
Mean-Field Dynamics: Singular Potentials and Rate of Convergence (2009, unknown): “(In fact, as shown in [7], one has the bound φ(t) .”
Blowup for nonlinear wave equations describing boson stars (2005, methodology): “As an application, we exhibit instability of ground state solitary waves at rest if m = 0.”
Boson Stars as Solitary Waves (2005, background): “By conservation of N(ψ(t)) and of Ev(ψ(t)), whose proof can be done along the lines of [ 7 ] for the conservation of E(ψ(t)), we have that N(βn) = N(ψn(0)) and Ev(βn) = Ev(ψn(0)), which, by (3.2), implies”