We prove global existence of weak solutions for a version of the one velocity Baer--Nunziato system with dissipation describing a mixture of two noninteracting viscous compressible fluids in a piecewise regular Lipschitz domain with general inflow/outflow boundary conditions. The geometrical setting is general enough to comply with most current domains important for applications, such as (curved) pipes of piecewise regular and axis-dependent cross-sections. For the existence proof, we adapt to the system the classical Lions--Feireisl approach to the compressible Navier--Stokes equations which is combined with a generalization of the theory of renormalized solutions to the transport equations in the spirit of Vasseur, Wen, and Yu [J. Math. Pures Appl. (9), 125 (2019), pp. 247--282]. The results related to the families of transport equations presented in this paper extend/improve some statements of the theory of renormalized solutions and are therefore of independent interest.


  1. bifluid system
  2. Baer--Nunziato system
  3. compressible Navier--Stokes equations
  4. transport equation
  5. continuity equation
  6. renormalized solutions

MSC codes

  1. 76N10
  2. 35Q30

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A. Abbatiello, E. Feireisl, A. Novotný, Generalized solutions to models of compressible viscous fluids, Discrete Contin. Dyn. Syst., 41 (2020), pp. 1--28.
G. Allaire, S. Clerc, and S. Kokh, A five-equation model for the numerical simulation of interfaces in two-phase flows, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), pp. 1017--1022.
G. Allaire, S. Clerc, and S. Kokh, A five-equation model for the numerical simulation of interfaces between compressible fluids, J. Comput. Phys., 181 (2002), pp. 577--616.
L. Ambrosio and G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, in Transport Equations and Multi-D Hyperbolic Conservation Laws, Lect. Notes Unione Mat. Ital. 5, Springer, Berlin, 2008, pp. 3--57.
M. R. Baer and J. W. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flow, 12 (1986), pp. 861--889.
S. Bianchini and P. Bonicatto, A uniqueness result for the decomposition of vector fields in $R^d$, Invent. Math., 220 (2020), pp. 255--393.
M. E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad, in Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics, Trudy Sem. S. L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, pp. 5--40, 149 (in Russian).
F. Boyer, Trace theorems and spatial continuity properties for the solutions of the transport equation, Differential Integral Equations, 18 (2005), pp. 891--934.
D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier, and M. Hilliairet, Multifluid models including compressible fluids, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Y. Giga and A. Novotný, eds., Springer, Cham, 2018, pp. 2927--2978.
D. Bresch and P.-E. Jabin, Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. of Math. (2), 188 (2018), pp. 577--684.
D. Bresch, P. B. Mucha, and E. Zatorska, Finite-energy solutions for compressible two-fluid Stokes system, Arch. Ration. Mech. Anal., 232 (2019), pp. 987--1029.
T. Chang, B. J. Jin, and A. Novotný, Compressible Navier--Stokes system with general inflow-outflow boundary data, SIAM J. Math. Anal., 51 (2019), pp. 1238--1278, https://doi.org/10.1137/17M115089X.
H. J. Choe, A. Novotný, and M. Yang, Compressible Navier-Stokes system with general inflow-outflow boundary data on piecewise regular domains, ZAMM Z. Angew. Math. Mech., 98 (2018), pp. 1447--1471.
G. Crippa, C. Donadello, and L. V. Spinolo, A note on the initial--boundary value problem for continuity equations with rough coefficients, in Hyperbolic Problems: Theory, Numerics, Applications, AIMS Ser. Appl. Math. 8, Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2014, pp. 957--966.
G. Crippa, C. Donadello, and L. V. Spinolo, Initial-boundary value problems for the continuity equations with BV coefficients, J. Math. Pures Appl. (9), 102 (2014), pp. 79--98.
S. Dallet, A comparative study of numerical schemes for the Baer-Nunziato model, Int. J. Finite Vol., EDF Special Workshop (2015), http://www.ijfv.org/; complete preprint version available at https://hal.archives-ouvertes.fr/hal-01412148.
J. P. Demailly, Analyse Numérique et équations différentielles, Presse Universitaire de Grenoble, 1991.
R. Denk, M. Hieber, and J. Prüss, Optimal $Lp$-$Lq$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), pp. 193--224.
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), pp. 511--547.
D. Drew and S. L. Passman, Theory of Multicomponent Fluids, Appl. Math. Sci. 135, Springer, New York, 1999.
S. Evje, An integrative multiphase model for cancer cell migration under influence of physical cues from the tumor microenvironment, Chem. Engrg. Sci., 165 (2017), pp. 240--259.
S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Differential Equations, 245 (2008), pp. 2660--2703.
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Ser. Math. Appl. 26, Oxford University Press, Oxford, UK, 2004.
E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Adv. Math. Fluid Mech., Birkhäuser Verlag, Basel, 2009.
E. Feireisl, A. Novotný, and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids, J. Math. Fluid Mech., 3 (2001), pp. 358--392.
R. L. Foote, Regularity of distance function, Proc. AMS, 92 (1984), pp. 154--156.
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, 2nd ed., Springer Monogr. Math., Springer, New York, 2011.
M. Geissert, H. Heck, and M. Hieber, On the equation ${div}$ $u=g$ and Bogovskii operator in Sobolev spaces of negative order, in Partial Differential Equations and Functional Analysis, Oper. Theory Adv. Appl. 168, Birkhäuser, Basel, pp. 113--121.
A. Gray, Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, Topology, 21 (1982), pp. 201--228.
H. Guillard and A. Murrone, A five equation reduced model for compressible two phase flow problems, J. Comput. Phys., 202 (2005), pp. 664--698.
V. Guillemaud, Modélisation et simulation numérique des écoulements diphasiques par une approche bifluide à deux pressions, Ph.D. thesis, Université de Provence-Aix-Marseille I, 2007, https://tel.archives-ouvertes.fr/tel-00169178.
M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer, New York, 2006.
Y.-S. Kwon and A. Novotný, Dissipative solutions to compressible Navier--Stokes equations with general inflow--outflow data: Existence, stability and weak-strong uniqueness, J. Math. Fluid Mech., 23 (2021), 23, https://doi.org/10.1007/s00021-020-00553-z.
P.-L. Lions, Mathematical Topics in Fluid Mechanics: Vol. 2: Compressible Models, Oxford Lecture Ser. Math. Appl. 10, Oxford Science Publications, Oxford, UK, 1998.
D. Maltese, M. Michálek, P. B. Mucha, A. Novotný, M. Pokorný, and E. Zatorska, Existence of weak solutions for compressible Navier-Stokes equations with entropy transport, J. Differential Equations, 261 (2016), pp. 4448--4485.
F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), pp. 489--507.
A. Novotný, Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids, Sci. China Math., 63 (2020), pp. 2399--2414.
A. Novotný and M. Pokorný, Weak solutions for some compressible multicomponent fluid models, Arch. Ration. Mech. Anal., 235 (2020), pp. 355--403.
A. Novotný and M. Pokorný, Continuity equation and vacuum regions in compressible flows, J. Evol. Eq., 21 (2021), pp. 2891--2922, https://doi.org/10.1007/s00028-021-00704-3.
A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Ser. Math. Appl. 27, Oxford University Press, Oxford, UK, 2004.
Y. Qiao, H. Wen, and S. Evje, Viscous two-phase flow in porous media driven by source terms: Analysis and numerics, SIAM J. Math. Anal., 51 (2019), pp. 5103--5140, https://doi.org/10.1137/19M1252491.
P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations: Theory and Shape Optimization, Monogr. Mat., Birkhäuser, Basel, 2012.
A. Vasseur, H. Wen, and C. Yu, Global weak solution to the viscous two-fluid model with finite energy, J. Math. Pures Appl. (9), 125 (2019), pp. 247--282.
W. P. Ziemer, Weakly Differentiable Functions, Grad. Texts Math., Springer, New York, 1989.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 818 - 871
ISSN (online): 1095-7154


Submitted: 11 May 2021
Accepted: 25 August 2021
Published online: 7 February 2022


  1. bifluid system
  2. Baer--Nunziato system
  3. compressible Navier--Stokes equations
  4. transport equation
  5. continuity equation
  6. renormalized solutions

MSC codes

  1. 76N10
  2. 35Q30



Funding Information

Distinguished Edurad Cech visiting program
Grantová Agentura České Republiky https://doi.org/10.13039/501100001824 : GA19-04243S
National Research Foundation of Korea https://doi.org/10.13039/501100003725 : NRF2020R1F1A1A01049805

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