8 problems

Polynomiality conjecture. The solution $U_t(z)$ must be a polynomial in both $z$ and $t$. This conjecture links the nilpotent-Hessian condition to polynomial solutions of the invis…

Let $M>0$, let $v=v(t,r)\in[-1,1]$ solve the relativistic Burgers equation on the Schwarzschild exterior $r>2M$, and let the initial data be a compactly perturbed steady shock. Asy…

Let $X=(X_x)_{x\in\mathbb R}$ be a self-similar process satisfying the growth condition needed to define the inviscid Burgers solution, and let $\mathcal L$ be the closure of the s…

Let $u$ be governed by the regularised Burgers equation with viscosity parameter $\varepsilon$, and let $\delta$ be the spatial mesh size. Write $c\in\{1,0,-1\}$ according as the d…

Multiplicative-noise correction conjecture. As $\varepsilon\to0$, its solution converges to the solution of

Multidimensional correction conjecture. As $\varepsilon\to0$, $u^{\varepsilon}$ converges to the solution of

Let $u^N$ be the solution of the finite-difference approximation with an even number $N$ of modes, and let $\nu>0$ and $\sigma$ denote the viscosity and noise parameters. Finite-di…

Let $u$ solve the stochastic Burgers equation with viscosity $u$, noise amplitude $\sigma$, and asymmetric finite-difference parameters $a,b$. Let $u^{\varepsilon}$ denote the solut…