IUMJ

Title: A new formulation for the 3-D Euler equations with an application to subsonic flows in a cylinder

Authors: Shangkun Weng

Issue: Volume 64 (2015), Issue 6, 1609-1642

Abstract:

In this paper, a new formulation for the three-dimensional Euler equations is derived. Since the Euler system is hyperbolic-elliptic coupled in a subsonic region, an effective decoupling of the hyperbolic and elliptic modes is essential for any development of the theory. The key idea in our formulation is to use Bernoulli's law to reduce the dimension of the velocity field by defining new variables $(1,\beta_2=u_2/u_1,\beta_3=u_3/u_1)$ and replacing $u_1$ by Bernoulli's function $B$ through \[ u_1^2=\frac{2(B-h(\rho))}{1+\beta_2^2+\beta_3^2}. \] We find a conserved quantity for flows with a constant Bernoulli function, which behaves like the scaled vorticity in the two-dimensional case. More surprisingly, a system of new conservation laws can be derived, which is new even in the two-dimensional case. We use this new formulation to construct a smooth subsonic Euler flow in a rectangular cylinder, which is also required to be adjacent to some special subsonic states. The same idea can be applied to obtain similar information for the three-dimensional incompressible Euler equations, the self-similar Euler equations, the steady Euler equations with damping, the steady Euler-Poisson equations, and the steady Euler-Maxwell equations.