Skip to main content

Existence of Dissipative (and Weak) Solutions for Models of General Compressible Viscous Fluids with Linear Pressure


In this work we will focus on the existence of dissipative solutions for a system describing a general compressible viscous fluid in the case of the pressure being a linear function of the density and the viscous stress tensor being a non-linear function of the symmetric velocity gradient. Moreover, we will study under which conditions it would be possible to get the existence of weak solutions.

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


  1. Abbatiello, A., Feireisl, E., Novotný, A.: Generalized solutions to mathematical models of compressible viscous fluids. Discrete Continu. Dyn. Syst. 41(1), 1–28 (2021)

    Article  Google Scholar 

  2. Barnes, H.A.: Shear-Thickening (“Dilatancy’’) in suspensions on nonaggreating solidparticles dispersed in Newtonian liquids. J. Rheology 33, 329–366 (1989)

    ADS  Article  Google Scholar 

  3. Basarić, D.: Semiflow selection to models of general compressible viscous fluids. J. Math. Fluid Mech. 23(2) (2021)

  4. Blechta, J., Málek, J., Rajagopal, J.R.: On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion (2019). arXiv:1902.04853

  5. Breit, D., Cianchi, A., Diening, L.: Trace-free Korn inequality in Orlicz spaces. SIAM J. Math. Anal. 49(4), 2496–2526 (2017)

    MathSciNet  Article  Google Scholar 

  6. Bulíček, M., Gwiazda, P., Málek, J., Świerczewska Gwiazda, Á.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44(4), 2756–2801 (2012)

    MathSciNet  Article  Google Scholar 

  7. Chang, T., Jin, B.J., Novotný, A.: Compressible Navier-Stokes system with inflow-outflow boundary data. SIAM J. Math. Anal. 51(2), 1238–1278 (2019)

    MathSciNet  Article  Google Scholar 

  8. Crippa, G., Donadello, C., Spinolo, L.V.: A note on the initial-boundary value problem for continuity equations with rough coefficients, HYP 2012 conference proceedings. AIMS Ser. Appl. Math. 8, 957–966 (2014)

    Google Scholar 

  9. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2003)

    Book  Google Scholar 

  10. Feireisl, E., Liao, X., Málek, J.: Global weak solutions to a class of non-Newtonian compressible fluids. Math. Meth. Appl. Sci. 38(16), 3482–3494 (2015)

    MathSciNet  Article  Google Scholar 

  11. Feireisl, E., Lukáčová-Medvid’ová, M.: Convergence of a mixed finite element-finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions. Found. Comput. Math. 18, 703–730 (2018)

    MathSciNet  Article  Google Scholar 

  12. Girinon, V.: Navier-Stokes equations with nonhomogeneous boundary conditions in a bounded three-dimensional domain. J. Math. Fluid Mech. 13, 309–339 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  13. Lions, P.L.: Mathematical Topics in Fluid Mechanics, Volume 2: Compressible Models. Oxford Science Publications, Oxford (1998)

    MATH  Google Scholar 

  14. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Berlin (1995)

    Book  Google Scholar 

  15. Mamontov, A.E.: Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. I. Sib. Math. J. 40, 351–362 (1999)

    Article  Google Scholar 

  16. Mamontov, A.E.: Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. II. Sib. Math. J. 40, 541–555 (1999)

    Article  Google Scholar 

  17. Matuš\(\mathring{\text{u}}\)-Nečasová, Š., Novotný, A.: Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid. Acta Appl. Math. 37, 109–128 (1994)

  18. Pedregal, P.: Parametrized Measures and Variational Principles. Birkhaüser, Basel (1997)

    Book  Google Scholar 

  19. Plotnikov, P.I., Weigant, W.: Isothermal Navier-Stokes equations and Radon transform. SIAM J. Math. Anal. 47(1), 626–653 (2015)

    MathSciNet  Article  Google Scholar 

  20. Rodrigues, J.-F.: On the mathematical analysis of thick fluids. J. Math. Sci. (N. Y.) 210(6), 835–848 (2015)

    MathSciNet  Article  Google Scholar 

Download references


The author wishes to thank Prof. Eduard Feireisl for the helpful discussions, and the reviewers for the careful reading and useful advice, which in her opinion contributed to improve the quality of the paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Danica Basarić.

Ethics declarations

Conflict of interest

The author declares that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of the author was funded from the Czech Science Foundation (GAČR), Grant Agreement 21-02411S. The institute of Mathematics of the Czech Academy of Sciences is supported by RVO:67985840.

Communicated by D. Bresch.

De la Vallée–Poussin Criterion

De la Vallée–Poussin Criterion

In this section, we prove a slightly modified version of the De la Vallée–Poussin criterion as we require the stronger condition, with respect to the standard formulation, that the Young function satisfies the \(\Delta _2\)-condition. We first recall the definitions of Young function and \(\Delta _2\)-condition.

Definition A.1

  1. (i)

    We say that \(\Phi \) is a Young function generated by \(\varphi \) if

    $$\begin{aligned} \Phi (t) = \int _{0}^{t} \varphi (s) \ {\mathrm{d}}s \quad \text{ for } \text{ any } t\ge 0, \end{aligned}$$

    where the real-valued function \(\varphi \) defined on \([0,\infty )\) is non-negative, non-decreasing, left-continuous and such that

    $$\begin{aligned} \varphi (0)=0, \quad \lim _{s\rightarrow \infty }\varphi (s)=\infty . \end{aligned}$$
  2. (ii)

    A Young function \(\Phi \) is said to satisfy the \(\Delta _2\)-condition if there exist a positive constant K and \(t_0 \ge 0\) such that

    $$\begin{aligned} \Phi (2t) \le K \Phi (t) \quad \text{ for } \text{ any } t\ge t_0. \end{aligned}$$

Theorem A.2

Let \(Q \subset {\mathbb {R}}^d\) be a bounded measurable set and let \(\{ f_n \}_{n\in {\mathbb {N}}}\) be a sequence in \(L^1(Q)\). Then, the following statements are equivalent.

  1. (i)

    The sequence \(\{ f_n \}_{n\in {\mathbb {N}}}\) is equi-integrable, meaning that for any \(\varepsilon >0\) there exists \(\delta =\delta (\varepsilon )>0\) such that

    $$\begin{aligned} \int _{M} |f_n (y)| \ {\mathrm{d}}y< \varepsilon \quad \text{ for } \text{ any } M\subset Q \text{ such } \text{ that } |M| < \delta , \end{aligned}$$

    independently of n.

  2. (ii)

    There exists a Young function \(\Phi \) satisfying the \(\Delta _2\)-condition (A.1) such that the sequence \(\{ f_n \}_{n\in {\mathbb {N}}}\) is uniformly bounded in the Orlicz space \(L_{\Phi }(Q)\).


(ii) \(\Rightarrow \) (i) See Pedregal [18], Chapter 6, Lemma 6.4.

(i) \(\Rightarrow \) (ii) For \(n\in {\mathbb {N}}\) and \(j\ge 1\) fixed, let

$$\begin{aligned} \mu _j(f_n):= |\{ y\in Q: \ |f_n(y)| >j \}|. \end{aligned}$$

As the sequence \(\{ f_n \}_{n\in {\mathbb {N}}}\) is equi-integrable, from the Dunford-Pettis theorem there exists a strictly increasing sequence of positive integers \(\{ C_m \}_{m\in {\mathbb {N}}}\) such that for each m

$$\begin{aligned} \sup _{n\in {\mathbb {N}}} \int _{\{ |f_n|>C_m\}} |f_n(y)| \ {\mathrm{d}}y \le \frac{1}{2^m}. \end{aligned}$$

For \(n\in {\mathbb {N}}\) and \(m\ge 1\) fixed

$$\begin{aligned} \int _{\{ |f_n|>C_m\}} |f_n(y)| \ {\mathrm{d}}y = \sum _{j=C_m}^{\infty } \int _{\{ j<|f_n|\le j+1\}} |f_n(y)| \ {\mathrm{d}}y \ge \sum _{j=C_m}^{\infty } j \ [\mu _j(f_n)-\mu _{j+1}(f_n)] \ge \sum _{j=C_m}^{\infty }\mu _j(f_n). \end{aligned}$$

In particular, we obtain

$$\begin{aligned} \sum _{m=1}^{\infty } \sum _{j=C_m}^{\infty }\mu _j(f_n) \le \sum _{m=1}^{\infty } \int _{\{ |f_n|>C_m\}} |f_n(y)| \ {\mathrm{d}}y \le \sum _{m=1}^{\infty } \frac{1}{2^m} =1. \end{aligned}$$

For \(m\ge 0\), we define

$$\begin{aligned} \alpha _m= {\left\{ \begin{array}{ll} 0 &{}\quad \text{ if } m<C_1, \\ \max \{k: \ C_k \le m\} &{}\quad \text{ if } m\ge C_1. \end{array}\right. } \end{aligned}$$

Notice that

$$\begin{aligned} \alpha _m \ge j \quad \Leftrightarrow \quad C_j \le m. \end{aligned}$$

It is straightforward that \(\alpha _m \rightarrow \infty \) as \(m\rightarrow \infty \). We define a step function \(\varphi \) on \([0,\infty )\) by

$$\begin{aligned} \varphi (s) = \sum _{m=0}^{\infty } \alpha _m \chi _{(m,m+1]}(s) \quad \text{ for } \text{ any } 0\le s<\infty . \end{aligned}$$

It is clear that \(\varphi \) is non-negative, non-decreasing, left-continuous and such that \(\varphi (0)=0\), \(\lim _{s\rightarrow \infty } \varphi (s)= \infty \). Then, we can define the Young function \(\Phi \) generated by \(\varphi \) as

$$\begin{aligned} \Phi (t)= \int _{0}^{t} \varphi (s)\ {\mathrm{d}}s, \quad \text{ for } \text{ any } 0\le t< \infty . \end{aligned}$$

At this point, notice that we have the freedom to take the constants \(C_j\), \(j\ge 1\), as large as we want and consequently, the constants \(\alpha _m\), \(m\ge 1\), will be as small as we want. More precisely, we may find a positive constant c such that

$$\begin{aligned} \alpha _{2m} \le c \ \alpha _m \quad \text{ for } \text{ any } m\ge 1. \end{aligned}$$

We then obtain, for all \(s\in [0,\infty )\),

$$\begin{aligned} \varphi (2s) = \sum _{m=0}^{\infty } \alpha _m \chi _{\left( \frac{m}{2}, \frac{m+1}{2}\right) }(s) = \sum _{k=0}^{\infty } \alpha _{2k} \chi _{\left( k, k+ \frac{1}{2}\right) }(s) \le c \sum _{k=0}^{\infty } \alpha _k \chi _{\left( k, k+ \frac{1}{2}\right) }(s) \le c\ \varphi (s); \end{aligned}$$

consequently, for all \(t\in [0,\infty )\),

$$\begin{aligned} \Phi (2t) = \int _{0}^{2t} \varphi (s) \ {\mathrm{d}}s = 2\int _{0}^{t} \varphi (2z) \ {\mathrm{d}}z \le 2c \int _{0}^{t} \varphi (z) \ {\mathrm{d}}z = 2c \ \Phi (t), \end{aligned}$$

and thus we get that the Young function \(\Phi \) satisfies the \(\Delta _2\)-condition (A.1).

Finally, for \(n\in {\mathbb {N}}\) fixed, using the fact that \(\Phi (0)=\Phi (1)=0\) and for \(j\ge 1\), noticing that \(\alpha _0=0\),

$$\begin{aligned} \Phi (j+1) = \int _{0}^{j+1} \varphi (s) \ {\mathrm{d}} s = \sum _{m=0}^{j} \int _{m}^{m+1} \varphi (s) \ {\mathrm{d}} s \le \sum _{m=0}^{j} \varphi (m+1) = \sum _{m=0}^{j} \alpha _m= \sum _{m=1}^{j} \alpha _m, \end{aligned}$$

we get

$$\begin{aligned} \int _{Q} \Phi (|f_n(y)|) \ {\mathrm{d}} y&= \int _{\{ |f_n|=0 \}} \Phi (|f_n(y)|) \ {\mathrm{d}} y + \sum _{j=0}^{\infty } \int _{\{ j<|f_n|\le j+1\}} \Phi (|f_n(y)|) \ {\mathrm{d}}y \\&\le \sum _{j=1}^{\infty } [\mu _j(f_n)-\mu _{j+1}(f_n)] \ \Phi (j+1) \\&\le \sum _{j=1}^{\infty } [\mu _j(f_n)-\mu _{j+1}(f_n)] \sum _{m=1}^{j} \alpha _m \\&= \sum _{m=1}^{\infty } \alpha _m \sum _{j=m}^{\infty } [\mu _j(f_n)-\mu _{j+1}(f_n)] \\&= \sum _{m=1}^{\infty } \alpha _m \mu _m(f_n)= \sum _{m=1}^{\infty } \mu _m(f_n) \sum _{j=1}^{\alpha _m} 1 = \sum _{j=1}^{\infty } \sum _{m=C_j}^{\infty } \mu _m(f_n) \le 1 \end{aligned}$$

where we used (A.2) in the last line. In particular, we obtain that the sequence \(\{ f_n \}_{n\in {\mathbb {N}}}\) is uniformly bounded in the Orlicz space \(L_{\Phi }(Q)\). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Basarić, D. Existence of Dissipative (and Weak) Solutions for Models of General Compressible Viscous Fluids with Linear Pressure. J. Math. Fluid Mech. 24, 56 (2022).

Download citation

  • Accepted:

  • Published:

  • DOI:


  • Compressible viscous fluid
  • Dissipative solution
  • Linear pressure
  • Non-linear viscosity

Mathematics Subject Classification

  • 35A01
  • 35Q35
  • 76N10