Energy conservation for the Euler equations on T2 x R+ for weak solutions defined without reference to the pressure

We study weak solutions of the incompressible Euler equations on T2×R+; we use test functions that are divergence free and have zero normal component, thereby obtaining a definition that does not involve the pressure. We prove energy conservation under the assumptions that u∈L3(0,T;L3(T2×R+)), lim|y|→01|y|∫0T∫T2∫x3>|y|∞|u(x+y)−u(x)|3dxdt=0, and an additional continuity condition near the boundary: for some δ>0 we require u∈L3(0,T;C0(T2×[0,δ])). We note that all our conditions are satisfied whenever u(x,t)∈Cα, for some α>1/3, with Hölder constant C(x,t)∈L3(T2×R+×(0,T))

The Compressible to Incompressible Limit of 1D Euler Equations: the Non Smooth Case

We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal 1D Euler equations.Comment: 16 page

Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum

The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $\rm{GL}^+(3,\mathbb R)$. The evolution of the fluid domain is described by a family ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank $r=1$, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along $3-r$ of its principal axes. In the compressible case, the asymptotic limit has rank $r=3$, and asymptotic completeness holds, when the adiabatic index $\gamma$ satisfies $4/3<\gamma<2$. The number of possible degeneracies, $3-r$, increases with the value of the adiabatic index $\gamma$. In the incompressible case, affine motion reduces to geodesic flow in $\rm{SL}(3,\mathbb R)$ with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes