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On the Weak Solutions to the Multicomponent Reactive Flows Driven by Non-conservative Boundary Conditions

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Abstract

We propose a new concept of weak solutions to the multicomponent reactive flows driven by large boundary data. When the Gibbs’ equation incorporates the species mass fractions, we establish the global-in-time existence of weak solutions for any finite energy initial data. Moreover, if the classical solutions exist, the weak solutions coincide with them in the same time interval.

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Notes

  1. We denote \(\textbf{Y}:=(Y_1,...,Y_n)\).

  2. We deonte \([\textbf{u}_b\cdot \textbf{n}]^+=\max \{\textbf{u}_b\cdot \textbf{n}, 0\} [\textbf{u}_b\cdot \textbf{n}]^-=\min \{\textbf{u}_b\cdot \textbf{n}, 0\}\).

  3. We denote \({\tilde{\textbf{Y}}}:=({\tilde{Y}}_1,...,{\tilde{Y}}_n)\), \({\tilde{e}}_M:=e_M({\tilde{\varrho }},{\tilde{\vartheta }})\), \({\tilde{e}}_R:=e_R({\tilde{\varrho }},{\tilde{\vartheta }})\), etc.

References

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

    Article  MathSciNet  Google Scholar 

  2. Bechtel, S.E., Rooney, F.J., Forest, M.G.: Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005)

    Article  ADS  Google Scholar 

  3. Chaudhuri, N., Feireisl, E.: Navier–Stokes–Fourier system with Dirichlet boundary conditions. Appl. Anal. 101, 4076–4094 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Chen, G.-Q., Hoff, D., Trivisa, K.: Global solutions to a model for exothermically reacting compressible flows with large discontinuous initial data. Arch. Ration. Mech. Anal. 166, 321–358 (2003)

    Article  MathSciNet  Google Scholar 

  6. Chen, G.-Q., Torres, M., Ziemer, W.P.: Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Commun. Pure Appl. Math. 62, 242–304 (2009)

    Article  MathSciNet  Google Scholar 

  7. Donatelli, D., Trivisa, K.: A multidimensional model for the combustion of compressible fluids. Arch. Ration. Mech. Anal. 185, 379–408 (2007)

    Article  MathSciNet  Google Scholar 

  8. Donatelli, D., Trivisa, K.: On the motion of a viscous compressible radiative-reacting gas. Commun. Math. Phys. 265, 463–491 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  9. Feireisl, E.: Relative entropies in thermodynamics of complete fluid systems. Discrete Contin. Dyn. Syst. Ser. A 32, 3059–3080 (2012)

    Article  MathSciNet  Google Scholar 

  10. Feireisl, E., Gwiazda, P., Kwon, Y.S., Świerczewska-Gwiazda, A.: Mathematical theory of compressible magnetohydrodynamics driven by non-conservative boundary conditions. J. Math. Fluid Mech. 25(4), 27 (2023)

    Article  MathSciNet  Google Scholar 

  11. Feireisl, E., Novotný, A.: Mathematics of Open Fluid Systems. Birkhäuser-Verlag, Basel (2022)

    Book  Google Scholar 

  12. Feireisl, E., Novotný, A.: Navier–Stokes–Fourier system with general boundary conditions. Commun. Math. Phys. 386, 975–1010 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  13. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser-Verlag, Basel (2009)

    Book  Google Scholar 

  14. Feireisl, E., Novotný, A.: Weak-strong uniqueness property for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)

    Article  MathSciNet  Google Scholar 

  15. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics, 2nd edn. Birkhäuser, Cham (2017)

    Book  Google Scholar 

  16. Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  17. Feireisl, E., Novotný, A., Sun, Y.: Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60, 611–631 (2011)

    Article  MathSciNet  Google Scholar 

  18. Feireisl, E., Petzeltová, H.: On the long time behaviour of solutions to the Navier–Stokes–Fourier system with a time dependent driving force. J. Dyn. Differ. Equ. 19, 685–707 (2007)

    Article  MathSciNet  Google Scholar 

  19. Feireisl, E., Petzeltová, H., Trivisa, K.: Multicomponent reactive flows: global-in-time existence for large data. Commun. Pure Appl. Anal. 7, 1017–1047 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  21. Giovangigli, V.: Multicomponent flow modeling. Sci. China Math. 55, 285–308 (2012)

    Article  MathSciNet  Google Scholar 

  22. Giovangigli, V.: Multicomponent Flow Modeling. Birkhäuser, Boston (1999)

    Book  Google Scholar 

  23. Hoff, D.: Asymptotic behavior of solutions to a model for the flow of a reacting fluid. Arch. Ration. Mech. Anal. 196, 951–979 (2010)

    Article  MathSciNet  Google Scholar 

  24. Kwon, Y.S., Novotný, A.: Dissipative solutions to compressible Navier–Stokes equations with general inflow-outflow data: existence, stability and weak-strong uniqueness. J. Math. Fluid Mech. 23, 1–27 (2021)

    Article  MathSciNet  Google Scholar 

  25. Kwon, Y.S., Trivisa, K.: Stability and large time behaviour for multicomponent reactive flows. Nonlinearity 22, 2443–2471 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  26. Klein, R., Botta, N., Schneider, T., Munz, C.D., Roller, S., Meister, A., Hoffmann, L., Sonar, T.: Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Eng. Math. 39, 261–343 (2001)

    Article  MathSciNet  Google Scholar 

  27. Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Compressible Models, vol. 2. Oxford Science Publication, Oxford (1998)

    Google Scholar 

  28. Mucha, P.B., Pokorný, M., Zatorska, E.: Approximate solutions to a model of two-component reactive flow. Discrete Contin. Dyn. Syst. Ser. S 7, 1079–1099 (2014)

    Article  MathSciNet  Google Scholar 

  29. Mucha, P.B., Pokorný, M., Zatorska, E.: Heat-conducting, compressible mixtures with multicomponent diffusion: construction of a weak solution. SIAM J. Math. Anal. 47, 3747–3797 (2015)

    Article  MathSciNet  Google Scholar 

  30. Norman, D.E.: Chemically reacting fluid flows: weak solutions and global attractors. J. Differ. Equ. 152, 75–135 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  31. Piasecki, T., Pokorný, M.: Weak and variational entropy solutions to the system describing steady flow of a compressible reactive mixture. Nonlinear Anal. 159, 365–392 (2017)

    Article  MathSciNet  Google Scholar 

  32. Zatorska, E.: Mixtures: sequential stability of variational entropy solutions. J. Math. Fluid Mech. 17, 437–461 (2015)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The research of B.-K. Huang is supported by the grant from National Natural Science Foundation of China (grant No.11901148), the Fundamental Research Funds for the Central Universities (grant No.JZ2022HGTB0257), and the China Scholarship Council (grant No.202106695016). This work was done when B.- K. Huang visited the Institute of Mathematics of the Czech Academy of Sciences, and he would like to thank Prof. Eduard Feireisl for fruitful discussions.

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  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, China

    Bingkang Huang

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Huang, B. On the Weak Solutions to the Multicomponent Reactive Flows Driven by Non-conservative Boundary Conditions. J. Math. Fluid Mech. 26, 19 (2024). https://doi.org/10.1007/s00021-024-00856-5

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