On the energy of inviscid singular flows

It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space $B^{1/3}_{3,\infty}$. When the singular set of the solution is (or belongs to) a smooth manifold, we derive various $L^p$-space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of $u$ is conserved provided the one sided non-tangential limits to the surface exist and the non-tangential maximal function is $L^3$ integrable, while the maximal function of the pressure is $L^{3/2}$ integrable. The results directly apply to prove energy conservation of the classical vortex sheets in both 2D and 3D at least in those cases where the energy is finite.Comment: 19 page

Continuous spectrum of the 3D Euler equation is a solid annulus

In this note we give a description of the continuous spectrum of the linearized Euler equations in three dimensions. Namely, for all but countably many times $t\in \R$, the continuous spectrum of the evolution operator $G_t$ is given by a solid annulus with radii $e^{t\mu}$ and $e^{t M}$, where $\mu$ and $M$ are the smallest and largest, respectively, Lyapunov exponents of the corresponding bicharacteristic-amplitude system of ODEs

The essential spectrum of advective equations

A description of the essential spectrum is given for a general class of linear advective PDE with pseudodifferential bounded perturbation. We prove that every point in the Sacker-Sell spectrum of the corresponding bicharacteristic-amplitude system exponentiates into the spectrum of PDE. Exact spectral pictures are found in various cases. Applications to instability are presented.Comment: This replaces the earlier version of the paper. The content of the original submission appeared in two publications -- this present one and the other one entitled "Cocycles and Ma\~{n}e sequences with an application to ideal fluids