Bilinear virial identities and applications

By F. Planchon, Luis Vega. 2007

173 citations recorded by the source corpus.

We prove bilinear virial identities for the nonlinear Schrodinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries.

Subfields: Algebra and Number Theory

View Paper: https://www.semanticscholar.org/paper/9e0494271de656ce68894a82b74d51b80998fee7

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In Algebra and Number Theory

1. Rowan Killip
2. B. Dodson

Citations in the corpus

• Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, methodology): “To do this we modify the proof in [72] of ∥ ∥∂x|u(t,x)|(2) ∥ ∥2 L(2)t,x([0,T ]×R) ‖u(t)‖L∞t Ḣ1([0,T ]×R)‖u(t)‖ 3 Lt L 2 x([0,T ]×R), (6.”
• Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state (2011, unknown): “The chief remaining difficulty is that the interaction Morawetz estimates of [11], [45], [9], and [37] are heavily reliant on μ = +1, and fail to be positive definite when μ = −1.”
• Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$ (2010, methodology): “Remark: The interaction Morawetz estimates of [6], [25] are not positive definite when μ = −1.”
• Nonlinear Schrodinger Equations at Critical Regularity (2013, unknown): “See for instance [67].”
• Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d =3$ (2009, background): “Remark: The interaction Morawetz estimates of [32], [24], [?], and [8] rely heavily on μ = +1.”
• Quintic NLS in the exterior of a strictly convex obstacle (2012, background): “Using this argument, Strichartz estimates have been proved for merely non-trapping obstacles; for further discussion see [6, 11, 38, 54, 64].”