Energy conservation for the Euler equations on T2×R+\mathbb{T}^2\times \mathbb{R}_+ for weak solutions defined without reference to the pressure

We study weak solutions of the incompressible Euler equations on $\mathbb{T}^2\times \mathbb{R}_+$; 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 $u\in L^3(0,T;L^3(\mathbb{T}^2\times \mathbb{R}_+))$, $$\lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\mathbb{T}^2}\int^\infty_{x_3>|y|} |u(x+y)-u(x)|^3\mathrm{d} x\, \mathrm{d} t=0,$$ and an additional continuity condition near the boundary: for some $\delta>0$ we require $u\in L^3(0,T;C^0(\mathbb{T}^2\times [0,\delta])))$. We note that all our conditions are satisfied whenever $u(x,t)\in C^\alpha$, for some $\alpha>1/3$, with H\"older constant $C(x,t)\in L^3(\mathbb{T}^2\times\mathbb{R}^+\times(0,T))$.Comment: 21 page

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))

On Double H\"older Regularity of the Hydrodynamic Pressure in Bounded Domains

We prove that the hydrodynamic pressure $p$ associated to the velocity $u\in C^\theta(\Omega)$, $\theta\in(0,1)$, of an inviscid incompressible fluid in a bounded and simply connected domain $\Omega\subset \mathbb R^d$ with $C^{2+}$ boundary satisfies $p\in C^{\theta}(\Omega)$ for $\theta \leq \frac12$ and $p\in C^{1,2\theta-1}(\Omega)$ for $\theta>\frac12$. Moreover, when $\partial \Omega\in C^{3+}$, we prove that an almost double H\"older regularity $p\in C^{2\theta-}(\Omega)$ holds even for $\theta<\frac12$. This extends and improves the recent result of Bardos and Titi obtained in the planar case to every dimension $d\ge2$ and it also doubles the pressure regularity. Differently from Bardos and Titi, we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the $d$-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.Comment: Improved version after referee comments. Version accepted in Calc Var & PDE

Full double H\"older regularity of the pressure in bounded domains

We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class $C^{2,1}$ then the corresponding pressure satisfies $p\in C^{2\gamma}_*(\Omega)$ in the case $\gamma\in (0,\frac{1}{2}]$, where $C^{2\gamma}_*$ is the H\"older-Zygmund space, which coincides with the usual H\"older space for $\gamma<\frac12$. This result, together with our previous one in [11] covering the case $\gamma\in(\frac12,1)$, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding $C^{2\gamma}_*$ estimate for the pressure on the whole space $\mathbb{R}^d$, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case $\gamma=\frac{1}{2}$. The boundary regularity features the use of local normal geodesic coordinates, pseudodifferential calculus and a fine Littlewood-Paley analysis of the modified equation in the new coordinate system. We also discuss the relation between different notions of weak solutions, a step which plays a major role in our approach.Comment: Extended version published on IMRN. Section 2.1 and Section 6.1 have been added, some minor mistakes have been corrected after the referee report