Bresch, Didier; Desjardins, Benoît; Lin, Chi-Kun On some compressible fluid models: Korteweg, lubrication, and shallow water systems. (English) Zbl 1106.76436 Commun. Partial Differ. Equations 28, No. 3-4, 843-868 (2003). Summary: We give some mathematical results for an isothermal model of capillary compressible fluids derived by J. E. Dunn and J. Serrin [Arch. Rat. Mech. Anal. 88, No. 2, 95–133 (1985; Zbl 0582.73004)], which can be used as a phase transition model. We consider a periodic domain \(\Omega = T^d\) (\(d=2\) or 3) or a strip domain \(\Omega = (0,1)\times T^{d-1}\). We look at the dependence of the viscosity \(\mu\) and the capillarity coefficient \(\kappa\) with respect to the density \(\rho\). Depending on the cases we consider, different results are obtained. We prove for instance for a viscosity \(\mu(\rho) = \nu\rho\) and a surface tension \(\kappa(\rho)=\tilde\kappa=\text{const}\) the global existence of weak solutions of the Korteweg system without smallness assumption on the data. This model includes a shallow water model and a lubrication model. We discuss the validity of the result for the shallow water equations since the density is less regular than in the Korteweg case. Cited in 2 ReviewsCited in 201 Documents MSC: 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q35 PDEs in connection with fluid mechanics 76D45 Capillarity (surface tension) for incompressible viscous fluids Keywords:isothermal model of capillary compressible fluids; global existence of weak solutions; Korteweg system; shallow water model; lubrication model Citations:Zbl 0582.73004 PDF BibTeX XML Cite \textit{D. Bresch} et al., Commun. Partial Differ. Equations 28, No. 3--4, 843--868 (2003; Zbl 1106.76436) Full Text: DOI OpenURL References: [1] DOI: 10.1146/annurev.fluid.30.1.139 · Zbl 1398.76051 [2] DOI: 10.1080/03605309108820752 · Zbl 0723.76033 [3] Bertozzi AL., Notices of Amer Math Soc 45 pp 689– (1998) [4] DOI: 10.1063/1.869942 · Zbl 1147.76338 [5] DOI: 10.1016/S0294-1449(00)00056-1 · Zbl 1010.76075 [6] DOI: 10.1080/03605300008821553 · Zbl 0953.35118 [7] DOI: 10.1007/BF00250907 · Zbl 0582.73004 [8] DOI: 10.1063/1.858597 · Zbl 0793.76030 [9] DOI: 10.3934/dcdsb.2001.1.89 · Zbl 0997.76023 [10] Hattori H, J Partial Differential equations 9 pp 323– (1996) [11] DOI: 10.1006/jmaa.1996.0069 · Zbl 0858.35124 [12] DOI: 10.1006/jcph.2000.6692 · Zbl 1047.76098 [13] Korteweg DJ., Archives Néerlandaises de Sciences Exactes et Naturelles pp 1– (1901) [14] DOI: 10.1088/0951-7715/14/6/305 · Zbl 0999.76033 [15] DOI: 10.1512/iumj.2000.49.1782 · Zbl 0971.76076 [16] Lions P-L., CR Acad Sci Paris Sér I. Math 316 pp 1335– (1993) [17] Lions P-L., Compressible Models 2 (1998) [18] Nadiga BT, European Journal of Mechanics B–Fluids 15 pp 885– (1996) [19] DOI: 10.1007/BF00375155 · Zbl 0839.76007 [20] DOI: 10.1103/PhysRevE.62.2480 [21] Seppecher P., Mémoire d’Habilitation à diriger des Recherches (1996) [22] DOI: 10.1006/jmaa.1996.0315 · Zbl 0863.35083 [23] DOI: 10.1007/BF02106835 · Zbl 0860.35098 [24] Vaigant VA, Diff Eqs 30 pp 935– (1994) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.