Stability and large-time behavior for Euler-like equations

Abstract

This paper intends to understand the long-time existence and stability of solutions to an Euler-like equation. An Euler-like equation is the 2D incompressible Euler equation with an extra singular integral operator (SIO) type term. In contrast to the 2D Euler equation, the vorticity to the 2D Euler-like equation is not known to be bounded due to the unboundedness of the SIO on the space $L^{\infty }$. As a consequence, classical Yudovich theory fails on the Euler-like equation. The global existence, regularity and stability problems on the Euler-like equation are generally open. This paper makes progress on an Euler-like equation arising in the study of several fluids. We establish a long-time existence and stability result. When the Sobolev size of the initial data is of order ε, the solution is shown to live on a time interval of the size $1/{\epsilon }^2$. When the initial data is restricted to a class with special symmetry, we obtain the global existence and nonlinear stability.