A deformation-curl decomposition for three dimensional steady Euler equations

In this paper, we provide a deformation-curl decomposition of three dimensional steady Euler equation. The key issue in this new decomposition is based on a simple observation that the density can be represented as a function of the Bernoullis function, the entropy and the speed, and hence the density equation can be rewritten as a Frobenius inner product of a symmetric matrix with the deformation matrix: $A(\boldsymbol{u}):~\mathcal{D}(\boldsymbol{u})=0$, where $\mathcal{D}(\boldsymbol{u})=(\frac12(\partial_j~u_k+~\partial_k~u_j))_{j,k=1}^3$ is the deformation matrix. Furthermore, employing the momentum equations, we find the vorticity can be resolved by a transport equation with another two algebraic equations involving the Bernoullis function and entropy.We also obtain the velocity by solving the deformation and curl system. The most important advantage of this decomposition is the velocity, the Bernoullis function and entropy we obtained share the same regularity as the pressure or density. Based on this decomposition, we prove the existence and uniqueness of subsonic flows in a finitely long rectangular nozzle by prescribing suitable physical boundary conditions.