# American Institute of Mathematical Sciences

January  2021, 41(1): 1-28. doi: 10.3934/dcds.2020345

## Generalized solutions to models of compressible viscous fluids

 1 Institute of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany 2 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic 3 Université de Toulon, IMATH, EA 2134, BP 20132, 83957 La Garde, France

* Corresponding author: Eduard Feireisl

Received  December 2019 Revised  August 2020 Published  January 2021 Early access  October 2020

Fund Project: The research of A.A. is supported by Einstein Foundation, Berlin. The research of E.F. leading to these results has received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 18-12719S. The work of A.N. was supported by Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea(NRF-2019H1D3A2A01101128)

We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak–strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). We consider general models of compressible viscous fluids with non–linear viscosity tensor and non–homogeneous boundary conditions, for which the problem of existence of global–in–time weak/strong solutions is largely open.

Citation: Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
##### References:
 [1] A. Abbatiello and E. Feireisl, On a class of generalized solutions to equations describing incompressible viscous fluids, Ann. Mat. Pura Appl., 199 (2020), 1183-1195.  doi: 10.1007/s10231-019-00917-x. [2] D. Breit, A. Cianchi and L. Diening, Trace-free Korn inequalities in Orlicz spaces, SIAM J. Math. Anal., 49 (2017), 2496-2526.  doi: 10.1137/16M1073662. [3] D. Breit, E. Feireisl and M. Hofmanová, Solution semiflow to the isentropic Euler system, Arch. Ration. Mech. Anal., 235 (2020), 167-194.  doi: 10.1007/s00205-019-01420-6. [4] T. Chang, B. J. Jin and A. Novotný, Compressible Navier-Stokes system with general inflow-outflow boundary data, SIAM J. Math. Anal., 51 (2019), 1238-1278.  doi: 10.1137/17M115089X. [5] G.-Q. Chen, M. Torres and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math., 62 (2009), 242-304.  doi: 10.1002/cpa.20262. [6] G. Crippa, C. Donadello and L. V. Spinolo, A note on the initial–boundary value problem for continuity equations with rough coefficients, Hyperbolic Problems: Theory, Numerics, Applications, AIMS Series in Appl. Math., Vol. 8, Springfield, MO, 2014,957–966. [7] L. Diening, C. Ebmeyer and M. Růžička, Optimal convergence for the implicit space-time discretization of parabolic systems with $p$-structure, SIAM J. Numer. Anal., 45 (2007), 457-472.  doi: 10.1137/05064120X. [8] L. Fang and Z. Li, On the existence of local classical solution for a class of one-dimensional compressible non-Newtonian fluids, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 157-181.  doi: 10.1016/S0252-9602(14)60148-X. [9] L. Fang, H. Zhu and Z. Guo, Global classical solution to a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum, Nonlinear Anal., 174 (2018), 189-208.  doi: 10.1016/j.na.2018.04.025. [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [11] E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Math. Methods Appl. Sci., 38 (2015), 3482-3494.  doi: 10.1002/mma.3432. [12] V. Girinon, Navier-Stokes equations with nonhomogenous boundary conditions in a bounded three-dimensional domain, J. Math. Fluid Mech., 13 (2011), 309-339.  doi: 10.1007/s00021-009-0018-x. [13] Y. Kwon, A. Novotný, Dissipative solutions to compressible Navier-Stokes equations with general inflow-outflow data: existence, stability and weak-strong uniqueness, accepted in J. Math. Fluid Mech. (2020). [14] P.-L. Lions, Mathematical topics in fluid dynamics. Vol. 2. Compressible models, The Clarendon Press, Oxford University Press, New York, 1998. [15] A. E. Mamontov, On the global solvability of the multidimensional Navier-Stokes equations of a nonlinearly viscous fluid, Ⅰ, Siberian Math. J., 40 (1999), 351-362. [16] A. E. Mamontov, On the global solvability of the multidimensional Navier-Stokes equations of a nonlinearly viscous fluid, Ⅱ, Siberian Math. J., 40 (1999), 541-555.  doi: 10.1007/BF02679762. [17] S. Matušů-Nečasová and A. Novotný, Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid, Acta Appl. Math., 37 (1994), 109-128.  doi: 10.1007/BF00995134. [18] P. I. Plotnikov and W. Weigant, Isothermal Navier-Stokes equations and Radon transform, SIAM J. Math. Anal., 47 (2015), 626-653.  doi: 10.1137/140960542. [19] G. Talenti, Boundedness of minimizers, Hokkaido Math. J., 19 (1990), 259-279.  doi: 10.14492/hokmj/1381517360. [20] A. Valli and M. Zajaczkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., 103 (1986), 259-296.  doi: 10.1007/BF01206939.

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##### References:
 [1] A. Abbatiello and E. Feireisl, On a class of generalized solutions to equations describing incompressible viscous fluids, Ann. Mat. Pura Appl., 199 (2020), 1183-1195.  doi: 10.1007/s10231-019-00917-x. [2] D. Breit, A. Cianchi and L. Diening, Trace-free Korn inequalities in Orlicz spaces, SIAM J. Math. Anal., 49 (2017), 2496-2526.  doi: 10.1137/16M1073662. [3] D. Breit, E. Feireisl and M. Hofmanová, Solution semiflow to the isentropic Euler system, Arch. Ration. Mech. Anal., 235 (2020), 167-194.  doi: 10.1007/s00205-019-01420-6. [4] T. Chang, B. J. Jin and A. Novotný, Compressible Navier-Stokes system with general inflow-outflow boundary data, SIAM J. Math. Anal., 51 (2019), 1238-1278.  doi: 10.1137/17M115089X. [5] G.-Q. Chen, M. Torres and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math., 62 (2009), 242-304.  doi: 10.1002/cpa.20262. [6] G. Crippa, C. Donadello and L. V. Spinolo, A note on the initial–boundary value problem for continuity equations with rough coefficients, Hyperbolic Problems: Theory, Numerics, Applications, AIMS Series in Appl. Math., Vol. 8, Springfield, MO, 2014,957–966. [7] L. Diening, C. Ebmeyer and M. Růžička, Optimal convergence for the implicit space-time discretization of parabolic systems with $p$-structure, SIAM J. Numer. Anal., 45 (2007), 457-472.  doi: 10.1137/05064120X. [8] L. Fang and Z. Li, On the existence of local classical solution for a class of one-dimensional compressible non-Newtonian fluids, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 157-181.  doi: 10.1016/S0252-9602(14)60148-X. [9] L. Fang, H. Zhu and Z. Guo, Global classical solution to a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum, Nonlinear Anal., 174 (2018), 189-208.  doi: 10.1016/j.na.2018.04.025. [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [11] E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids, Math. Methods Appl. Sci., 38 (2015), 3482-3494.  doi: 10.1002/mma.3432. [12] V. Girinon, Navier-Stokes equations with nonhomogenous boundary conditions in a bounded three-dimensional domain, J. Math. Fluid Mech., 13 (2011), 309-339.  doi: 10.1007/s00021-009-0018-x. [13] Y. Kwon, A. Novotný, Dissipative solutions to compressible Navier-Stokes equations with general inflow-outflow data: existence, stability and weak-strong uniqueness, accepted in J. Math. Fluid Mech. (2020). [14] P.-L. Lions, Mathematical topics in fluid dynamics. Vol. 2. Compressible models, The Clarendon Press, Oxford University Press, New York, 1998. [15] A. E. Mamontov, On the global solvability of the multidimensional Navier-Stokes equations of a nonlinearly viscous fluid, Ⅰ, Siberian Math. J., 40 (1999), 351-362. [16] A. E. Mamontov, On the global solvability of the multidimensional Navier-Stokes equations of a nonlinearly viscous fluid, Ⅱ, Siberian Math. J., 40 (1999), 541-555.  doi: 10.1007/BF02679762. [17] S. Matušů-Nečasová and A. Novotný, Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid, Acta Appl. Math., 37 (1994), 109-128.  doi: 10.1007/BF00995134. [18] P. I. Plotnikov and W. Weigant, Isothermal Navier-Stokes equations and Radon transform, SIAM J. Math. Anal., 47 (2015), 626-653.  doi: 10.1137/140960542. [19] G. Talenti, Boundedness of minimizers, Hokkaido Math. J., 19 (1990), 259-279.  doi: 10.14492/hokmj/1381517360. [20] A. Valli and M. Zajaczkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., 103 (1986), 259-296.  doi: 10.1007/BF01206939.
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