# American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - A, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [14] P.-L. Lions, Mathematical topics in fluid dynamics. Vol. 2. Compressible models, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] G. Talenti, Boundedness of minimizers, Hokkaido Math. J., 19 (1990), 259-279.  doi: 10.14492/hokmj/1381517360.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [14] P.-L. Lions, Mathematical topics in fluid dynamics. Vol. 2. Compressible models, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] G. Talenti, Boundedness of minimizers, Hokkaido Math. J., 19 (1990), 259-279.  doi: 10.14492/hokmj/1381517360.  Google Scholar [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.  Google Scholar
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