Spacetime integral bounds for the energy-critical nonlinear wave equation

Abstract

In this paper we prove a global spacetime bound for the quintic, nonlinear wave equation in three dimensions. This bound depends on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript t Superscript normal infinity Baseline upper L Subscript x Superscript 2"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">L_{t}^{\infty } L_{x}^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript t Superscript normal infinity Baseline ModifyingAbove upper H With dot squared"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L_{t}^{\infty } \dot {H}^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norms of the solution to the quintic problem. The main motivation for this paper is the use of an interaction Morawetz estimate for the nonlinear wave equation.

Locations

• Proceedings of the American Mathematical Society - View
• arXiv (Cornell University) - View - PDF