7 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 $(M,\mathsf{g})$ be a closed, compact Riemannian manifold without boundary, and let $(u,p)$ be a smooth solution of the incompressible Euler equations on $M$. Tao's conjecture.…

Let $D$ be an open disk centered at the origin. Let $z\in D$ and let $\mathbf{v}\in C^2(\overline{D}\backslash \\{z\\})$ be a bounded flow solving the Euler equations on…

Sharpness conjecture. For every $\beta<\frac{1}{3}$, there exists a non-conservative weak solution

Isett–Oh conjecture. For every $\theta<\frac{1}{3}$, there exists a weak Euler solution $v\in C^\theta(\mathbb R\times\mathbb T^n)$ such that

Finite-time blowup conjecture. There exists such a compact Riemannian manifold $(M,g)$, of some dimension $d>2$, and such a smooth solution that cannot be smoothly continued to the…

Let $\omega_0\in H^1(\mathbb{T}^2)\cap L^\infty(\mathbb{T}^2)$ be mean-zero, and let the corresponding solution of the two-dimensional incompressible Euler equations be a Yudovich…