In this paper we investigate the question of the existence of strong solution in finite time for the Korteweg system for small initial data provided that the initial momentum ρ 0 u 0 belongs to bmo −1 T (R N) for T > 0 and the initial density ρ 0 is in L ∞ (R N) with N ≥ 1 and far away from the vacuum. This result extends the so called Koch-Tataru theorem for the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any ρ(t, ·) with t > 0. We also prove the existence of global strong solution for initial data (ρ 0 − 1, ρ 0 u 0) ∈ (B N 2 −1 2,∞ (R N) ∩ B N 2 2,∞ (R N)∩L ∞ (R N))×(B N 2 −1 2,∞ (R N)) N. This result allows in particular to extend the notion of Oseen solution (corresponding to particular solution of the incompressible Navier Stokes system in dimension N = 2) to the Korteweg system provided that the vorticity of the momentum ρ 0 u 0 is a Dirac mass αδ 0 with α sufficiently small. IHowever unlike the Navier Stokes equations the property of self similarity is not conserved for the Korteweg system since there is no invariance by scaling because the term of pressure.

中文翻译：

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初始密度为L∞的bmo-1（RN）中Korteweg系统的强解

在本文中，我们针对Korteweg系统的小初始数据研究在有限时间内存在强解的问题，前提是初始动量ρ0 u 0属于bmo -1 T（RN），且T> 0且初始密度ρ0在L∞（RN）中，N≥1，并且远离真空。这个结果扩展了所谓的Korteweg系统的Koch-Tataru定理。还有趣的是，对于任何ρ（t，·），当t> 0时密度变为Lipschitz时，任何对密度的初始冲击都是瞬时正则化的。我们还证明了初始数据（ρ0）的全局强解的存在− 1，ρ0 u 0）∈（BN 2 -1 2，∞（RN）∩BN 2 2，∞（RN）∩L∞（RN））×（BN 2 -1 2，∞（RN））N 。如果动量ρ0 u 0的涡度是狄拉克质量αδ0，则该结果尤其允许将Oseen解（对应于N = 2的不可压缩Navier Stokes系统的特定解）的概念扩展到Korteweg系统。 α足够小。但是，与Navier Stokes方程不同，Korteweg系统不保留自相似性，因为压力项不会因比例缩放而保持不变。