Lower bounds for the lifespan of solutions to the Euler--Korteweg equations

We consider the initial value problem for the Euler–Korteweg equations in the class of irrotational flows. In every space dimension $d⩾2$, we establish a quantitative lower bound for the lifespan of solutions that explicitly exhibits its dependence on the Weber number, the Mach number, and the Sobolev norm of the initial perturbation. In particular, this lower bound can be taken arbitrarily large in a low-Weber-number regime under a suitable algebraic relation between the Mach and Weber numbers.