Abstract
The Euler–Korteweg system is a dispersive perturbation of the usual compressible Euler equations. In dimension at least three, under a natural stability condition on the pressure, the author proved with B. Haspot that the Cauchy problem is globally well-posed for small, smooth, irrotational initial data. As a continuation of this work, we prove that if the initial velocity has a small rotational part, there exists a lower bound on the time of existence that depends only on some norm of this rotational part. In the zero vorticity limit we recover the previous global well-posedness result.
Highlights
IntroductionNote that the long wave limit for general K, g was recently studied by Benzoni and Chiron [6], the authors prove convergence to more classical equations such as Burgers, KdV or KP
The Cauchy problem for the Euler–Korteweg system reads ∂tρ + div(ρu) = 0,∂tu + u · ∇u + ∇g(ρ) = ∇ K∆ρ + 12 K (ρ)|∇ρ|2, (x, t) ∈ Rd × R+. (1.1) (ρ, u)|t=0 = (ρ0, u0).g is the pressure, K the capillary coefficient, a smooth function R+∗ → R+∗
The main focus of this paper is to describe more accurately the time of existence for small data that have a non zero rotational part
Summary
Note that the long wave limit for general K, g was recently studied by Benzoni and Chiron [6], the authors prove convergence to more classical equations such as Burgers, KdV or KP Their analysis does not require the solutions to be irrotational. The analogy with the Schrödinger equation was pushed further in [4] where the authors prove the existence of global strong solutions for small irotational data in dimension at least 3. Some difficulties arise in our case: first due to the quasi-linear nature of the problem, loss of derivatives are bound to arise This is handled by a method well-understood since the work of Klainerman–Ponce [25], where one mixes dispersive (decay) estimates with high order energy estimates (see for example the introduction of [4] for a short description).
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