Simultaneous global inviscid Burgers flows with periodic Poisson forcing

Abstract

<jats:p> We study the inviscid Burgers equation on the circle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝕋</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mi>ℝ</mml:mi> <mml:mo>/</mml:mo> <mml:mi>ℤ</mml:mi> </mml:mrow> </mml:math> forced by the spatial derivative of a Poisson point process on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℝ</mml:mi> <mml:mo>×</mml:mo> <mml:mi>𝕋</mml:mi> </mml:mrow> </mml:math> . We construct global solutions with mean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> simultaneously for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℝ</mml:mi> </mml:mrow> </mml:math> , and in addition construct their associated global shocks (which are unique except on a countable set of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> ). We then show that as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> changes, the solution only changes through the movement of the global shock, and give precise formulas for this movement. This can be seen as an analogue of previous results by the author and Yu Gu in the viscous case with white-in-time forcing, which related the derivative of the solution in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> to the density of a particle diffusing in the Burgers flow. </jats:p>