Vortex reconnections in classical and quantum fluids

We review recent rigorous results on the phenomenon of vortex reconnection in classical and quantum fluids. In the context of the Navier–Stokes equations in $$\mathbb {T}^3$$ T 3 we show the existence of global smooth solutions that exhibit creation and destruction of vortex lines of arbitrarily complicated topologies. Concerning quantum fluids, we prove that for any initial and final configurations of quantum vortices, and any way of transforming one into the other, there is an initial condition whose associated solution to the Gross–Pitaevskii equation realizes this specific vortex reconnection scenario. Key to prove these results is an inverse localization principle for Beltrami fields and a global approximation theorem for the linear Schrödinger equation.

The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings

We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field [Formula: see text] such that the spectrum of [Formula: see text] is positive and bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends Grassin and Serre’s papers [Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique, C. R. Acad. Sci. Paris Sér. I 325 (1997) 721–726, 1997; Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998) 1397–1432; Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier Grenoble 47 (1997) 139–159] dedicated to the compressible Euler system without coupling and with integer regularity exponents.

On the Existence of Global Smooth Solutions to the Parabolic–Elliptic Keller–Segel System with Irregular Initial Data

We consider the parabolic–elliptic Keller–Segel system $$\begin{aligned} \left\{ \begin{aligned} u_t&= \Delta u - \chi \nabla \cdot (u \nabla v), \\ 0&= \Delta v - v + u \end{aligned} \right. \end{aligned}$$ u t = Δ u - χ ∇ · ( u ∇ v ) , 0 = Δ v - v + u in a smooth bounded domain $$\Omega \subseteq {\mathbb {R}}^n$$ Ω ⊆ R n , $$n\in {\mathbb {N}}$$ n ∈ N , with Neumann boundary conditions. We look at both chemotactic attraction ($$\chi > 0$$ χ > 0 ) and repulsion ($$\chi < 0$$ χ < 0 ) scenarios in two and three dimensions. The key feature of interest for the purposes of this paper is under which conditions said system still admits global classical solutions due to the smoothing properties of the Laplacian even if the initial data is very irregular. Regarding this, we show for initial data $$\mu \in {\mathcal {M}}_+({\overline{\Omega }})$$ μ ∈ M + ( Ω ¯ ) that, if either $$n = 2$$ n = 2 , $$\chi < 0$$ χ < 0 or $$n = 2$$ n = 2 , $$\chi > 0$$ χ > 0 and the initial mass is small or $$n = 3$$ n = 3 , $$\chi < 0$$ χ < 0 and $$\mu = f \in L^p(\Omega )$$ μ = f ∈ L p ( Ω ) , $$p > 1$$ p > 1 holds, it is still possible to construct global classical solutions to ($$\star $$ ⋆ ), which are continuous in $$t = 0$$ t = 0 in the vague topology on $${\mathcal {M}}_+({\overline{\Omega }})$$ M + ( Ω ¯ ) .

Global smooth solutions to 3D irrotational Euler equations for Chaplygin gases

We study here the Cauchy problem associated with the isentropic and compressible Euler equations for Chaplygin gases. Based on the new formulation of the compressible Euler equations in J. Luk and J. Speck [The hidden null structure of the compressible Euler equations and a prelude to applications, J. Hyperbolic Differ. Equ. 17 (2020) 1–60] we show that the wave system satisfied by the modified density and the velocity for Chaplygin gases satisfies the weak null condition. We then prove the global existence of smooth solutions to the irrotational and isentropic Chaplygin gases without introducing a potential function, when the initial data are small perturbations to a constant state.