Path-connectedness of incompressible Euler solutions

Abstract We study the incompressible Euler equation and prove that the set of weak solutions is path-connected. More precisely, we construct paths of Hölder regularity $$C^{1/2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> , valued in $$C^0_{t, loc} L^2_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>l</mml:mi> <mml:mi>o</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> endowed with the strong topology. The main result relies on a convex integration construction adapted from the seminal work of De Lellis and Székelyhidi (Ann of Math (2) 170:1417-1436, 2009), extending it to a more broader geometric framework, replacing balls with arbitrary convex compact sets.