5 problems

Let $I$ be an open interval, and let $(v,p)$ be a weak solution of the incompressible Euler equations on $I \times \mathbb{T}^3$. Assume that $v \in L_t^\infty C_x^{1/3}$ and that…

Let $(u,b)$ be a weak solution of the ideal MHD system, and define the total energy and cross helicity by … … Here $C_{x,t}^\alpha$ denotes joint Hölder regularity and…

Let $d\in\{2,3\}$. Let $u\in C^0([0,1];C^\alpha(\mathbb T^d))\cap C^\alpha([0,1];C^0(\mathbb T^d))$ be a weak solution of the incompressible Euler equations, for some Hölder expone…

Let $\theta$ be a weak solution of the active scalar equation, let $\delta$ be the parameter appearing in the equation, and set … The Hamiltonian is … with $T_0$ the Fourier multip…

Generic energy-profile regularity conjecture. A generic solution with regularity at most $1/3$ fails to conserve energy and has an energy profile of minimal regularity. More precis…