## Abstract

In this paper, we deal with global weak solutions of a non-conservative viscous compressible two-phase model in one space dimension. This work extends in some sense the previous work, [Bresch et al., Arch Rat Mech Anal 196:599–629, 2010], which provides the global existence of weak solutions in the multi-dimensional framework with 1 < γ_{±} < 6 assuming non-zero surface tension. In our study, we strongly improve the results by taking advantage of the one space dimension. More precisely, we obtain global existence of weak solutions without using capillarity terms and for pressure laws with the same range of coefficients as the degenerate barotropic mono-fluid system, namely γ_{±} > 1. Then we prove that any possible vacuum state has to vanish within finite time after which densities are always away from vacuum. This allows to prove that at least one phase corresponding to the global weak solution is a locally in time and space (in a sense to be defined) strong solution after the vacuum states vanish. Our paper may be understood as a non-straightforward generalization to the two-phase flow system of a previous paper [Li et al., Commun Math Phys 281(2):401–444, 2008], which treated the usual compressible barotropic Navier-Stokes equations for mono-fluid with a degenerate viscosity. Various important mathematical difficulties occur when we want to generalize those results to the two-phase flows system since the corresponding model is non-conservative. Far from vacuum, it involves a strong coupling between a nonlinear algebraic system and a degenerate PDE system under constraint linked to fractions. Moreover, fractional densities may vanish if densities or fractions vanish: A difficulty is to find estimates on the densities from estimates on fractional densities using the algebraic system. Original approximate systems have also to be introduced compared to the works on the degenerate barotropic mono-fluid system. Note that even if our result concerns “only” the one-dimensional case, it points out possible global weak solutions (for such a non-conservative system) candidates to approach for instance shock structures and to define an appropriate a priori family of paths in the phase space (in numerical schemes) at the zero dissipation limit.

This is a preview of subscription content, access via your institution.

## References

- 1
Berthon, C., Coquel, F., Le Floch, Ph.:

*Why many theories of shock waves are necessary*: kinetic relations for non-conservative systems. Submitted, 2010 - 2
Bresch D., Desjardins B.: Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys.

**238**(1-2), 211–223 (2003) - 3
Bresch D., Desjardins B., Lin C.-K.: On some compressible fluid models Korteweg, lubrication, and shallow water systems. Comm. Part. Diff. Equ.

**28**(3–4), 843–868 (2003) - 4
Bresch D., Desjardins B., Ghidaglia J.-M., Grenier E.: Global weak solution to a generic two-fluid model. Arch. Rat. Mech. Anal.

**196**, 599–629 (2010) - 5
Bresch, D., Huang, X., Li, J.:

*On a spherically symmetric bi-phase compressible model*. In preparation, 2011 - 6
Bresch D., Renardy M.: Well-posedness of two-layer shallow water flow between two horizontal rigid plates. Nonlinearity

**29**, 1081–1088 (2011) - 7
Chanteperdrix, G., Villedieu, P., Vila, J.-P.: A compressible model for separated two-phase flows computations. In:

*ASME Fluids Engineering Division Summer Meeting*. ASME, Montreal, Canada, July 2002 - 8
Chen G.Q., Perepelitsa M.: Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible flow. Comm. Pure Appl. Math.

**63**(11), 1469–1504 (2010) - 9
Dellacherie S.: Relaxation schemes for the multicomponent Euler system. M2AN

**37**(6), 909–936 (2003) - 10
Drew, D.A., Passman, S.L.: Theory of multicomponent fluids. Applied Mathematical Sciences,

**135**. Berlin-Heidelberg-New York: Springer-Verlag, 1998 - 11
Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford Science, Oxford (2004)

- 12
Feireisl E., Novotný A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids. J. Math. Fluid Mech.

**3**, 358–392 (2001) - 13
Huang, F., Li, M., Wang, Y.: Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations. Submited, 2010

- 14
Guo Z., Jiu Q., Xin Z.: Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients. SIAM J. Math. Anal.

**39**, 1402–1427 (2008) - 15
Ishii M.: Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles, Paris (1975)

- 16
Jiang S., Xin Z., Zhang P.: Global weak solutions to 1D compressible isentropy Navier-Stokes with density-dependent viscosity. Meth. and Appl. of Anal.

**12**(3), 239–252 (2005) - 17
Keyfitz, B.L.: Mathematical properties of nonhyperbolic models for incompressible two-phase ow. In: Michaelides, E.E. (ed.)

*Proceedings of the Fourth International Conference on Multiphase Flow*. New Orleans (CD ROM), Tulane University, 2001 - 18
Keyfitz B.L., Sanders R., Sever M.: Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Disc. Cont. Dyn. Syst. B

**3**, 541–563 (2003) - 19
Keyfitz B.L., Sever M., Zhang F.: Viscous singular shock structure for a nonhyperbolic two-Fluid model. Nonlinearity

**17**(5), 1731–1747 (2009) - 20
Li H., Li J., Xin Z.: Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Commun. Math. Phys.

**281**(2), 401–444 (2008) - 21
Lions P.L.: Mathematical Topics in Fluid Dynamics, Vol. 2. Compressible Models. Oxford Science, Oxford (1998)

- 22
Mellet A., Vasseur A.: On the barotropic compressible Navier-Stokes equation. Comm. Part. Diff. Eqs.

**32**(3), 431–452 (2007) - 23
Saurel R., Abgrall R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys.

**150**(2), 425–467 (1999)

## Author information

### Affiliations

### Corresponding author

## Additional information

Communicated by P. Constantin

## Rights and permissions

## About this article

### Cite this article

Bresch, D., Huang, X. & Li, J. Global Weak Solutions to One-Dimensional Non-Conservative Viscous Compressible Two-Phase System.
*Commun. Math. Phys.* **309, **737–755 (2012). https://doi.org/10.1007/s00220-011-1379-6

Received:

Accepted:

Published:

Issue Date:

### Keywords

- Weak Solution
- Global Existence
- Strong Solution
- Fraction Density
- Global Weak Solution