On the motion of incompressible inhomogeneous Euler-Korteweg fluids

Pages: 497 - 515, Volume 3, Issue 3, September 2010      doi:10.3934/dcdss.2010.3.497

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Miroslav Bulíček - Mathematical Institute of Charles University, Sokolovská 83, 186 75 Prague, Czech Republic (email)
Eduard Feireisl - Institute of Mathematics AS ČR, Žitná 25, 115 67 Praha 1, Czech Republic (email)
Josef Málek - Mathematical Institute of Charles University, Sokolovská 83, 186 75 Prague, Czech Republic (email)
Roman Shvydkoy - Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan St. M/C 249 Chicago, IL 60607-7045, United States (email)

Abstract: We study a system of equations governing evolution of incompressible inhomogeneous Euler-Korteweg fluids that describe a class of incompressible elastic materials. A local well-posedness theory is developed on a bounded smooth domain with no-slip boundary condition on velocity and vanishing gradient of density. The cases of open space and periodic box are also considered, where the local existence and uniqueness of solutions is shown in Sobolev spaces up to the critical smoothness $\frac{n}{2}+1$.

Keywords:  Korteweg fluid, inhomogeneous Euler fluid, Korteweg stress, local-in-time well-posedness, smooth solution.
Mathematics Subject Classification:  76A05, 35A01.

Received: March 2010;      Revised: April 2010;      Available Online: May 2010.