Mathematical Research Letters

Volume 20 (2013)

Number 1

Singularities of solutions to compressible Euler equations with vacuum

Pages: 41 – 50



Yi Du (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Zhen Lei (Contemporary Applied Mathematics, Fudan University, Shanghai, China)

Qingtian Zhang (Department of Mathematics, Pennsylvania State University, Penn., U.S.A.)


Presented are two results on the formation of finite-time singularities of solutions to the compressible Euler equations in two and three space dimensions for isentropic, polytropic, ideal fluid flows. The initial velocity is assumed to be symmetric and the initial sound speed is required to vanish at the origin. They are smooth in Sobolev space $H^3$, but not required to have a compact support. It is shown that the $H^3$ norm of the velocity field and the sound speed will blow up in a finite time.


finite-time singularities, compressible Euler equations, vacuum

Full Text (PDF format)