27 problems

Let $\omega$ and $u$ solve the De Gregorio model … where $H$ is the Hilbert transform and $\Lambda^{-1}$ is defined by the displayed relation. De Gregorio's global regularity conje…

Scale-invariant solutions are solutions of fluid equations whose spatial behavior is unchanged under the relevant scaling; the surrounding discussion concerns solutions with scale-…

The Adkins–Nappi model is a semilinear modification of the wave map equation describing a repulsive interaction involving meson fields. By finite energy smooth initial data, we mea…

Let ${\mathcal S}$ be the space of classical data $(\phi_0,\phi_1)$, where $\phi_0:{\mathbf R}^2\to{\mathbf H}$ is Schwartz modulo constants and…

Let $n\geqslant 4$ and consider the Yang–Mills Cauchy problem with regular initial data and critical Sobolev norm $\dot{H}_A^{\frac{n-4}{2}}$. Let the conserved norm be the norm as…

Let $a\in\mathbb{R}$, and consider the generalized Constantin-Lax-Majda-DeGregorio equation on $\mathbb{S}^1=\mathbb{R}/(2\pi\mathbb{Z})$, … where $\mathcal{H}$ is the Hilbert tran…

The primitive equations are considered with fractional horizontal dissipation exponent $\alpha$, and the currently established global well-posedness result for general initial data…

Let $\theta$ be a solution of the anisotropic quasi-geostrophic equation on $\mathbb{R}^2$, with dissipation parameters $\alpha$ and $\beta$ satisfying the same conditions as in Ye…

Consider the two-and-a-half-dimensional electron MHD system for scalar functions $\psi$ and $\phi$: … Here $\nabla^\perp = (-\partial_{x^{2}},\partial_{x^{1}})$, and a resistivity…

Let $d=2$, and consider the energy-critical wave maps equation with initial data whose energy is below the energy of any non-trivial finite-energy harmonic map into the target mani…

Global regularity conjecture. Retaining the advection term $u\partial_x\omega$ leads to global regularity.

Consider the Euler alignment system with a convolution kernel $\phi\notin L^1$ in its dissipation term, as in the system under discussion. Global regularity conjecture. Such a diss…

Okamoto–Sakajo–Wunsch threshold conjecture. For smooth initial data, solutions are globally regular when $a\geq2$, while finite-time singularity formation occurs when $a<2$.

Okamoto–Sakajo–Wunsch threshold conjecture. When $a<1$, finite-time singularities occur, whereas when $a\geq1$, solutions are globally regular.

Threshold conjecture. Global regularity should hold for energy-critical wave maps whose initial data have energy less than the energy of any non-trivial harmonic map into the targe…

Consider the two-dimensional Boussinesq system with fractional dissipation exponents $\alpha$ and $\beta$, and let…

Let $\operatorname{AdS}_3$ denote three-dimensional Anti-de Sitter space, and consider small smooth perturbations whose spatial dependence admits a radius of analyticity $\rho(t)$,…

The generalized Navier–Stokes equations are considered with the specially constructed initial data introduced in Subsection 3.2. These data have finite energy and are essentially t…

Let $(u_0,0,T)$ be a smooth periodic set of data. Global regularity for homogeneous periodic data with normalised pressure. There exists a smooth periodic solution $(u,p,u_0,0,T)$…

Let $(u_0,f,T)$ be an $H^1$ set of data such that, for every multi-index $\alpha$ and every compact $K\subset\mathbf{R}^3$, … and … Global well-posedness from spatially smooth…

Let $(u_0,f,T)$ be $H^1$ data and let an $H^1$ mild solution be a solution with $u\in L^\infty_tH^1_x\cap L^2_tH^2_x$ and the corresponding pressure and Duhamel formulation. Global…

Let $(u_0,0,T)$ be a smooth homogeneous $H^1$ set of data. An almost smooth finite energy solution is smooth for positive times and has the stated continuous spatial and time-deriv…

Let $u,f:[0,T]\times\mathbf{R}^3/L\mathbf{Z}^3\to\mathbf{R}^3$, $p:[0,T]\times\mathbf{R}^3/L\mathbf{Z}^3\to\mathbf{R}$, and $u_0:\mathbf{R}^3/L\mathbf{Z}^3\to\mathbf{R}^3$, with…

Let $(u_0,f,T)$ be a smooth periodic set of data. A periodic solution has normalised pressure when its pressure satisfies the normalisation used in the periodic Navier–Stokes formu…

Let $(u_0,f,T)$ be a smooth periodic set of data. Global regularity for periodic data. There exists a smooth periodic solution $(u,p,u_0,f,T)$ with the indicated data. This is the…