34 problems

Let the one-dimensional flow model have particle trajectories governed by $dx/dt=v=-H(\theta)$, where $H(\theta)$ is antisymmetric, so that a particle initially at the origin remai…

In one spatial dimension, consider a pole decomposition of a solution of the Burgers equation with more than four poles, and prescribe the initial positions of the poles. Pole-posi…

Consider the stochastic Burgers equation … on $L^2([0,1])$, with viscosity $\nu>0$, under the hypotheses on the coefficients stated for equation (4.11). Burgers stability conjectur…

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…

Seppäläinen's conjecture. Burgers' equation should govern the macroscopic propagation of disturbances of equilibria in this scaling regime for essentially all such interacting part…

Yakhot's conjecture. In the hydrodynamic limit, the probability distribution of the Kuramoto–Sivashinsky equation may be mimicked by that of the noise-driven inviscid Burgers equat…

Chain-recurrent shock-merger conjecture. In this example, the omega-limit set consists of the shock merger itself and the family of steady shocks with smaller amplitude; together t…

Exact omega-limit-set conjecture. For this initial datum, $\omega_0(u_0)$ is equal to the right-hand side of.

Realizability conjecture. For any probability measure $\mu$ on $[\alpha,\beta]$, there exist initial data $u_0\in L^\infty(\mathbb{R})$ satisfying such that $\omega(u_0)$ contains…

Asymptotic expansion conjecture. We conjecture that $U_{\ell}^{\mathrm{in}}$ admits an asymptotic expansion about infinity of the form

Let … and time horizon $T$. Their rescaled copies $(1/m)(mx)$, $x[0,1]$, for $m=2,3,$, are also local maximizers but attain smaller enstrophy. Global-maximizer conjecture. At least…

Global existence conjecture. If the prescribed integral relation exists for all times, then the regulator problem has a unique global solution.

Failure conjecture. A version of Theorem SimplyMain does not hold for the Gaussian measure generated by this Ornstein–Uhlenbeck process.

Consider the one-dimensional viscous Burgers equation … with positive viscosity coefficient, whose inviscid version admits shock and rarefaction waves for Riemann initial data. The…

Let $y^{\ast}(0,t)$ denote the minimizer associated with the solution at $x=0$ and time $t$, and let $p\in\mathbb{N}$. The preceding formal calculation also uses that $g'$ is Brown…

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 $u_{0}$ be a standard initial value for Burgers' equation, and let $u$ be the unique ultrafunction solving the natural ultrafunction formulation called Problem BEG. The source…

McKean's conjecture. The one-dimensional mean-field equation is the Burgers equation.

Let $u$ be a space-time velocity field on ${\mathbb R}\times[0,t_f]$, let $u_0$ be an initial condition on ${\mathbb R}$, and let $\tilde{x}(\tau)$ be a stochastic process satisfyi…

Let $u$ be a velocity field for inviscid Burgers equation, and let a stochastic process of generalized characteristics have the backward martingale property for velocities. Stochas…

Let $u$ be a velocity field for inviscid Burgers equation, and let a backward stochastic flow have the Burgers velocity as a martingale. Backward stochastic-flow admissibility conj…

Let $\mathbf{u}$ be the velocity field on the $d$-dimensional torus, let $\hat{\mathbf{u}}(\mathbf{n})$ denote its Fourier coefficients, and let $\nu>0$ be the viscosity. For a wav…

Consider the potential Burgers equation in the small-viscosity regime $0<\nu\ll 1$, with Fourier coefficients $\hat{u}(k)$ of the solution. For moderately large wavenumbers $k$, co…

Consider the dispersive Burgers equation … with dispersion parameter $\alpha>0$. Dispersive Burgers conjecture. The Cauchy problem is globally well-posed in a suitable functional s…

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…