64 problems

Self-similar long-time behavior conjecture. In this regime, the long-time behavior of the solution should be given by a self-similar solution of the displayed approximate equation.…

The paper constructs a weak self-similar solution of the two-dimensional incompressible Euler equations for sufficiently small non-axisymmetric perturbations of the specified perio…

Stability and quenching-profile conjecture. The following statements are conjectured:

Let solutions of the Lifshitz–Slyozov–Wagner (LSW) model evolve from admissible initial data. LSW self-similar attractor conjecture. The large-time behavior of solutions to the LSW…

Weak asymptotic decomposition conjecture. The solution admits a decomposition

Let $\rho>0$ be a mass, let the initial datum have mass $\rho$ and momentum $0$, and let $g_t$ be the associated solution in rescaled variables. Let $G_\rho$ be the steady state of…

Let $G_\rho$ denote a self-similar profile with mass $\rho>0$ and momentum $0$. Uniqueness conjecture. For any mass $\rho>0$, the self-similar profile $G_\rho$ with mass $\rho$ and…

Let $d>2$, and let ${\cal S}_1$ be the set of solutions $\varphi$ of … for $\eta>0$ that have exactly one intersection with the singular solution $\varphi_S(\eta)=2d/\eta^2$. Relax…

Nondegeneracy conjecture. Based on numerical calculations, this condition is satisfied for $\sigma>2/d$.

The solution $f_0$ denotes the fundamental self-similar wave map in the sequence of connecting-orbit solutions constructed above. Asymptotic stability conjecture. The solution…

Let $\alpha$ be the dimensionless coupling constant, and let $\alpha_0<\alpha_1$ denote the parameter values associated with the CSS solutions. Let the $n=1$ solution be the contin…

Perelman's conjecture. Every smooth rotated backward self-similar profile with $\alpha\neq 0$ and the stated decay is trivial.

Consider the focusing nonlinear wave equation $P[\phi]=0$ with $p=3$ in spatial dimension $n\geq 5$. Let $(\phi_{0,\omega},\phi_{1,\omega})$, $\omega\in[0,1]$, be a one-parameter f…

Nonexistence conjecture. The equation does not have solutions.

Uniqueness conjecture. The self-similar profile with this asymptotic behavior is unique.

Let the setting be one-dimensional, and let $\phi_{A,B}$ denote the profile referred to in the source. A graph-like self-similar solution is called trivial in the sense intended by…

Let $m$ be the number of branches in an $m$-fold symmetric algebraic spiral vortex sheet solution of the two-dimensional incompressible Euler equations, with…

Branch-counting conjecture. The supercritical CGL equation has a countable number of branches of nontrivial radial self-similar singular solutions. These branches can be labeled by…

Uniqueness conjecture. The self-similar profile with this decay is unique in every dimension $N\geq 2$.

The three-dimensional and higher-dimensional Keller–Segel system admits several known localised radial blow-up patterns, including type I self-similar blow-up, type II blow-up conc…

The three-dimensional Navier–Stokes equation admits self-similar steady states, and consider solutions of the form $\beta\bar{\omega}$, where $\beta$ is a scalar and $\bar{\omega}$…

For the Yang–Mills system in the energy-supercritical range $d\geq 5$, consider the higher-dimensional Bizoń-Biernat self-similar solution and generic large-data evolutions in the…

Consider the Yang–Mills system in temporal gauge in dimension $d\geq 5$, restricted to the equivariant solution class, and let … where $\phi$ is the Bizoń-Biernat self-similar prof…

Let $u$ solve the cubic wave equation on $[0,\infty)$ with initial data $(u_0,u_1)\in\mathcal{H}^{1/2}\cap\mathcal{H}^1$, where $\boldsymbol{u}(t)=(u(t),\partial_tu(t))$ and…

Global non-scattering solutions conjecture. Let $u$ be a solution of the cubic wave equation with the prescribed initial-data problem, defined for $t\in[0,\infty)$ and not scatteri…