We investigate some geometric properties of the curl operator, based on its diagonalization and its expression as a non-local symmetry of the pseudo-derivative $(−\Delta)^{1/2}$ among divergence-free vector fields with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of $(−\Delta)^{−1/2}$ curl. The two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed one. Having observed that the non-linearity of Navier-Stokes has the structure of a cross-product and its weak (distributional) form is a determinant that involves the vorticity, the velocity and a test function, we revisit the conservation of energy and the balance of helicity in a geometrical fashion. We show that in the case of a finite-time blow-up, both spin-definite components of the flow will explode simultaneously and with equal rates, i.e. singularities in 3D are the result of a conflict of spin, which is impossible in the poorer geometry of 2D flows. We investigate the role of the local and non-local determinants  $$\int_0^T\int_{\mathbb{R}^3}{\rm det}({\rm curl}u,u,(-\Delta)^{\theta}u)$$ and their spin-definite counterparts, which drive the enstrophy and, more generally, are responsible for the regularity of the flow and the emergence of singularities or quasi-singularities. As such, they are at the core of turbulence phenomena.

The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients, we present the explicit estimates of the lower bound and upper bound for its Dirichlet eigenvalues.

We prove the Hölder continuity of a harmonic map from a domain of a sub-Riemannian manifold into a locally compact manifold with nonpositive curvature, and more generally into a non-positively curved metric space in the Alexandrov sense.

In this paper, we consider the first order Hardy inequalities using simple equalities. This basic setting not only permits to derive quickly many well-known Hardy inequalities with optimal constants, but also supplies improved or new estimates in miscellaneous situations, such as multipolar potential, the exponential weight, hyperbolic space, Heisenberg group, the edge Laplacian, or the Grushin type operator.

In this work, we prove an optimal global-in-time $L^p−L^q$ estimate for solutions to the Kramers-Fokker-Planck equation with short range potential in dimension three. Our result shows that the decay rate as $t→ +∞$ is the same as the heat equation in $x$-variables and the divergence rate as $t→0_+$ is related to the sub-ellipticity with loss of one third derivatives of the Kramers-Fokker-Planck operator.

In this paper, we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the diffusivity coefficient, for the solutions to the viscous incompressible Boussinesq equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible Boussinesq equations converge to that of the ideal incompressible Boussinesq equations as the viscosity and diffusivity coefficients go to zero. Moreover, the convergence rate is also given.

We study 2D and 3D Prandtl equations of degenerate hyperbolic type, and establish without any structural assumption the Gevrey well-posedness with Gevrey index ≤ 2. Compared with the classical parabolic Prandtl equations, the loss of the derivatives, caused by the hyperbolic feature coupled with the degeneracy, cannot be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart. Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature, we give in this text a straightforward proof, basing on an elementary $L^2$ energy estimate. In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.