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. BreitA. 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. BreitE. 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. ChangB. 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. ChenM. 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. DieningC. 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. FangH. 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. FeireislX. 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.

show all references

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. BreitA. 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. BreitE. 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. ChangB. 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. ChenM. 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. DieningC. 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. FangH. 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. FeireislX. 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.

[1]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[2]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. FLUID STRUCTURE INTERACTION PROBLEM WITH CHANGING THICKNESS NON-LINEAR BEAM Fluid structure interaction problem with changing thickness non-linear beam. Conference Publications, 2011, 2011 (Special) : 813-823. doi: 10.3934/proc.2011.2011.813

[3]

Ali Hadi, Saeid Mehrabian. A two-stage data envelopment analysis approach to solve extended transportation problem with non-homogenous costs. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022006

[4]

Lan Huang, Zhiying Sun, Xin-Guang Yang, Alain Miranville. Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1595-1620. doi: 10.3934/cpaa.2022033

[5]

Ahmad El Hajj, Aya Oussaily. Continuous solution for a non-linear eikonal system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3795-3823. doi: 10.3934/cpaa.2021131

[6]

Fuyi Xu, Meiling Chi, Lishan Liu, Yonghong Wu. On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2515-2559. doi: 10.3934/dcds.2020140

[7]

Shuli Chen, Zewen Wang, Guolin Chen. Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem. Inverse Problems and Imaging, 2021, 15 (4) : 619-639. doi: 10.3934/ipi.2021008

[8]

Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2121-2134. doi: 10.3934/dcdss.2020181

[9]

Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control and Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1

[10]

Muhammad Bilal Riaz, Naseer Ahmad Asif, Abdon Atangana, Muhammad Imran Asjad. Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 645-664. doi: 10.3934/dcdss.2019041

[11]

Zhi-Ying Sun, Lan Huang, Xin-Guang Yang. Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry. Electronic Research Archive, 2020, 28 (2) : 861-878. doi: 10.3934/era.2020045

[12]

Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161

[13]

Hong Cai, Zhong Tan, Qiuju Xu. Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type. Kinetic and Related Models, 2015, 8 (1) : 29-51. doi: 10.3934/krm.2015.8.29

[14]

Jin Lai, Huanyao Wen, Lei Yao. Vanishing capillarity limit of the non-conservative compressible two-fluid model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1361-1392. doi: 10.3934/dcdsb.2017066

[15]

Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure and Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010

[16]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405

[17]

Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : i-iv. doi: 10.3934/dcdss.201702i

[18]

Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165

[19]

Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33.

[20]

Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems and Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (397)
  • HTML views (140)
  • Cited by (3)

[Back to Top]