Blowup for nonlinear wave equations describing boson stars

By Juerg Froehlich, E. Lenzmann. 2005

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We consider the nonlinear wave equation $$i \partial_{t}u = \sqrt{-\Delta + m^{2}} \; u - (|{x}|^{-1} \ast |{u}|^{2})u \;\;\; {\rm on}\;\; {\tt R}^{3}$$ modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ C c∞ (ℝ3), with negative energy, we prove blowup of u(t, x) in the H1/2‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0. © 2007 Wiley Periodicals, Inc.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3150bcd7df6af22af061faf386a11e79138b00e6

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Citations in the corpus

Uniqueness of ground states for pseudorelativistic Hartree equations (2008, background): “…that the dispersive nonlinear PDE (1-7) exhibits a rich variety of phenomena, such as stable and unstable traveling solitary waves, as well as finite-time blowup solutions indicating the “gravitational collapse” of a boson star; see [Fröhlich et al. 2007a; 2007b; Fröhlich and Lenzmann 2007].”
On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity (2012, background): “For this purpose we adapt the Virial type argument of [15], in which the evolution of two quantities 〈u,Au〉 and 〈u,Mu〉 for A = − i 2 (∇ · x+ x · ∇), M = x ·D2−α m x.”
Boson Stars as Solitary Waves (2005, background): “as ǫ→ 0, (A.52) by part iii) of Lemma 2.4. Hence we conclude that |I 1 +··· +I 4| ≤ r 2(ǫ) → 0, as ǫ→ 0, (A.53) which proves Lemma A.2. A.3 Commutator Estimate An almost identical result is needed in [5], but we provide its proof again. Lemma A.3. Let m≥ 0, v∈ R3, and deﬁne Av := √ −∆+m2 + i(v· ∇). ”
Well-posedness for Semi-relativistic Hartree Equations of Critical Type (2005, background): “(4) It is shown in [ 7 ]t hat criterion (3) for having global-in-time solutions in the focusing case is optimal in the sense that there exist solutions, u(t), with � u0� 2 > � Q� 2 , which blow up within finite time.”
Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R (2010, background): “ and Benjamin-Bona-Mahony (BBM) water wave equations. 1. Introduction Fractional powers of the Laplacian arise in a numerous variety of equations in mathematical physics and related ﬁelds; see, e.g., [8, 1, 31, 26, 22, 13, 15, 18] and references therein. Here, a central role within these models is o”