Soliton resolution for energy-critical wave maps in the equivariant case
Jacek Jendrej, Andrew Lawrie
Abstract
We consider the equivariant wave maps equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript 1 plus 2 Baseline right-arrow double-struck upper S squared"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false"> → </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}^{1+2} \to \mathbb {S}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , in all equivariance classes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">k \in \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove that every finite energy solution resolves, continuously in time, into a superposition of asymptotically decoupling harmonic maps and free radiation.