Geometric blow-up criteria for the non-homogeneous incompressible Euler equations in 2-D
Francesco Fanelli
TL;DR
This work studies the two-dimensional density-dependent incompressible Euler equations and proves geometric blow-up/continuation criteria that rely on controlling the derivative of the velocity along the direction $X=\nabla^{\perp}\rho$. The authors introduce the new momentum-vorticity quantity $\eta = \operatorname{curl}(\rho u)$ to eliminate the pressure and obtain transport-type estimates, treating both subcritical and critical Besov regularities, including logarithmic Besov spaces, via paradifferential calculus. In the subcritical regime they apply a log-interpolation inequality and Osgood's lemma to obtain continuation criteria, while in the critical regime they derive bounds for singular integrals in $B^0_{\infty,1}$ using refined transport estimates and log-Besov interpolation. As a consequence, the paper provides geometric criteria ensuring continuation (and hence precluding blow-up) in the non-homogeneous 2D Euler setting, and recovers global well-posedness in the constant-density case, highlighting how density variations shape the dynamics.
Abstract
This paper concerns the study of the incompressible Euler equations with variable density, in the case of space dimension $d=2$. Contrarily to their homogeneous (constant density) counterpart, those equations are not known to be well-posed globally in time. A classical blow-up/continuation criterion for smooth solutions relies on the control of the Lipschitz norm of the velocity field $u$. Here we show that, for establishing blow-up or continuation of solutions, it is enough to determine a control of $\nabla u$ only along the direction $X=\nabla^\perpρ$, where $ρ$ represents the density of the fluid. Our results deal with both the subcritical regularity and critical regularity frameworks. They rely on a novel approach to study regularity of solutions for the density-dependent incompressible Euler equations. Besides, they allow to recover the global well-posedness for $ρ\equiv {\rm cst}$ as a particular case.