9 problems

Turbulence is considered in the setting of the Boussinesq equations and their Lagrangian process, where deterministic and stochastic descriptions of the fluid dynamics are compared…

Convex integration relaxation conjecture. The obtained relaxation would yield the matrix inequality

Higher-spatial-regularity conjecture. The Augmented formula is well-posed in $\mathscr{C}^k\left([0,T),\mathbb{H}^n\times\mathbb{R}^4\right)$, with the corresponding weighted norm…

Let $k\in\mathbb{N}$, and let $T_{\varepsilon,\kappa,R}$ denote the time scale introduced for the Augmented formula. The solution takes values in the product of the Sobolev space…

Optimal stability-space conjecture. The optimal space for stability is given by this Fourier weight.

Minimal-dimension conjecture. Another multi-symplectic structure for this equation in $\mathds{R}^{d}$, with $d<5$, is not possible.

Consider the one-dimensional system … with Biot–Savart law … and singular steady solution … where $c>0$ is suitable. Let $\omega_0$ be a smooth profile compactly supported inside…

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

Let $u0,theta0 0text{?}$ be initial data in $H^{sigma}(mathbb{R}^2)$ with $sigma2$, and let $u,theta$ solve the 2D Boussinesq system with dissipation exponents $alpha$ and $beta$,…