Global solutions to the 3D half-wave maps equation with angular regularity

Abstract The half-wave maps equation is a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. Global wellposedness for small data in the critical space $${\dot{B}}^{n/2}_{2,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> is known since [11, 15] for dimensions $$n\geqslant 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . There is a major obstruction to extending these results to $$n=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> due to the loss of the $$L^2_tL^\infty _x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> Strichartz estimate. In this work, we prove that the equation admits global solutions for small smooth data in $${\dot{B}}^{3/2}_{2,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> which also possesses some angular regularity and weighted decay of derivatives. We use the improved Strichartz estimates of [24] to develop trilinear estimates in Strichartz spaces weighted by commuting vector fields which avoid the use of the $$L^2_tL^\infty _x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> endpoint.