8 problems

On blowup for large data. For initial data with sufficiently large energy, the solutions of equation blow up in finite time in the sense that the derivative $u_r(t,0)$ diverges as…

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.…

Fix soliton parameters $A$, $\boldsymbol{\lambda}$, $\nu$, and $\sigma$. Let $\phi_{\mathrm{rad}}$ be a smooth solution of the wave equation $\Box\phi_{\mathrm{rad}}=0$ satisfying…

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$, and let $\phi$ be a solution arising from smooth initial data in the stated Type I self-similar stability resu…

Consider a vortex patch whose boundary has a single corner of acute angle, and define the order of a cusp at $x=x_0$ by … where $f_1$ and $f_2$ represent the graphs of the two side…

Let $\boldsymbol{u}_c(t)$ denote the constructed blow-up solution from Part (a) of Theorem, and let $\boldsymbol{u}(t) \in \mathcal{H}_1$ be any other solution that blows up backwa…

Let $\{g(t)\}_{t\in [0,T)}$ be the solution of the Ricci flow on a closed Riemannian manifold $M^n$ with $n\geq 3$, and suppose that … as $t\to T$. Type I scalar-curvature conjectu…

Consider the generalized KP equation … with $\epsilon$-independent initial data $u(x,y,0;\epsilon)=u_0(x,y)$, and assume its solution is at least $C^4$ for $t>0$. Let…