Existence of Global Weak Solution for Compressible Fluid Models of Korteweg Type

Arnold, V.I., Khesin, B.A.: Topological methods in hydrodynamics, vol. 125. In: Applied Mathematical Sciences. Springer, New-York (1998)

Anderson, D.-M., McFadden, G.-B., Wheller, A.-A.: Diffuse-interface methods in fluid mech. In: Annal Review of Fluid Mechanics, vol. 30, pp. 139–165. Annual Reviews, Palo Alto (1998)

Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin (to appear)

Benzoni-Gavage S., Danchin R., Descombes S.: Well-posedness of one-dimensional Korteweg models. Electron. J. Differ. Equ. 59, 1–35 (2006)

Benzoni-Gavage S., Danchin R., Descombes S.: On the well-posedness of the Euler-Korteweg model in several space dimensions. Indiana Univ. Math. J. 56(4), 1499–1579 (2007)

Benzoni-Gavage S., Danchin R., Descombes S., Jamet D.: Structure of Korteweg models and stability of diffuse interfaces. interfaces and Free Boundaries. Model. Anal. Comput. 7(4), 371–414 (2005)

Bresch D., Desjardins B.: On the construction of approximate solutions for the 2d viscous shallow water model and for compressible Navier–Stokes models. J. Math Pures et Appl. (86) 4, 262–268 (2006)

Bresch D., Desjardins B.: Existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids. J. de Mathématiques Pures et Appliqués 87(1), 57–90 (2007)

Bresch D., Desjardins B., Lin C.-K.: On some compressible fluid models: Korteweg, lubrication and shallow water systems. Comm. Partial Differ. Equ. 28(3-4), 843–868 (2003)

Cahn J.-W., Hilliard J.-E.: Free energy of a nonuniform system, I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1998)

Danchin R., Desjardins B.: Existence of solutions for compressible fluid models of Korteweg type. Annales de l’IHP, Analyse non linéaire 18, 97–133 (2001)

Dunn J.-E., Serrin J.: On the thermomechanics of interstitial working. Arch. Rational Mech. Anal. 88(2), 95–133 (1985)

Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford (2004)

Gurtin M.-E., Poligone D., Vinals J.: Two-phases binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(6), 815–831 (1996)

Haspot, B.: Existence of strong solution for non isothermal Korteweg model. to appear in Annales Blaise Pascal. 16 (2009)

Hattori H., Li D.: The existence of global solutions to a fluid dynamic model for materials for Korteweg type. J. Partial Differ. Equ. 9(4), 323–342 (1996)

Hattori H., Li D.: Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198(1), 84–97 (1996)

Jamet D., Lebaigue O., Coutris N., Delhaye J.-M.: The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change. J. Comput. Phys. 169(2), 624–651 (2001)

Korteweg D.-J.: Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires par des variations de densité. Arch. Néer. Sci. Exactes Sér. II 6, 1–24 (1901)

Kotschote M.: Strong solutions for a compressible fluid model of Korteweg type. Annales de l’Institut Henri Poincare (C) Non Linear Anal. 25(4), 679–696 (2008)

Lions, P.-L.: Mathematical topics in fluid mechanics, vol. 2. In: Compressible Models. Oxford University Press, New York (1996)

Mellet A., Vasseur A.: On the barotropic compressible Navier–Stokes equations. Commun. Partial Differ. Equ. 32, 431–452 (2007)

Rowlinson J.-S.: Translation of J.D van der Waals “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density”. J. Stat. Phys. 20(2), 197–244 (1979)

Runst, T., Sickel, W.: Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. De Gruyter series in nonlinear analysis and applications, vol. 3. Berlin (1996)

Simon J.: Compact sets in the space L p (0, T;B). Ann. Mat. Pura Appl. 146, 65–96 (1987)

Truedelland C., Noll W.: The nonlinear field theories of mechanics, 2nd edn. Springer, Berlin (1992)