On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation

In this article, we investigate the initial value problem for the hyperbolic geometric flow equation, which is derived from hyperbolic geometric flow for Riemannian metric. We find that some of the quadratic nonlinear terms satisfy the null condition by rearranging the equation. Based on the ghost weight and standard energy estimate, the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> </m:math> {L}¹-{L}∞ estimate of solutions, and the bootstrap argument, lifespan of smooth solutions is established under the assumption of small data.