Skip to main content
SpringerLink
Log in

Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, we study the three-dimensional non-isentropic compressible fluid–particle flows. The system involves coupling between the Vlasov–Fokker–Planck equation and the non-isentropic compressible Navier–Stokes equations through momentum and energy exchanges. For the initial data near the given equilibrium we prove the global well-posedness of strong solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential. New ideas and techniques are developed to establish the well-posedness and large-time behavior. For the global well-posedness our methods are based on the new macro–micro decomposition which involves less dependence on the spectrum of the linear Fokker–Plank operator and fine energy estimates; while the proofs of the optimal large-time behavior rely on the Fourier analysis of the linearized Cauchy problem and the energy-spectrum method, where we provide some new techniques to deal with the nonlinear terms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations. Commun. Partial Differ. Equ. 26, 43–100 (2001)

    MathSciNet  MATH  Google Scholar 

  2. Baranger, C., Baudin, G., Boudin, L., Després, B., Lagoutière, F., Lapébie, E., Takahashi, T.: Liquid jet generation and break-up. In: Cordier, S., Goudon, T., Gutnic, M., Sonnendrucker, E. (eds.) Numerical Methods for Hyperbolic and Kinetic Equations. IRMA Lectures in Mathematics and Theoretical Physics, vol. 7, pp. 149–176. EMS Publ. House (2005)

  3. Boudin, L., Boutin, B., Fornet, B., Goudon, T., Lafitte, P., Lagoutiére, F., Merlet, B.: Fluidparticles flows: a thin spray model with energy exchanges. ESAIM: Proc. 28, 195–210 (2009)

    MATH  Google Scholar 

  4. Baranger, C., Boudin, L., Jabin, P.-E., Mancini, S.: A modeling of biospray for the upper airways. ESAIM: Proc. 14, 41–47 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Baranger, C., Desvillettes, L.: Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions. J. Hyperbolic Differ. Equ. 3, 1–26 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Benjelloun, S., Desvillettes, L., Moussa, A.: Existence theory for the kinetic-fluid coupling when small droplets are treated as part of the fluid. J. Hyperbolic Differ. Equ. 11, 109–133 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Berres, S., Bürger, R., Tory, E.M.: Mathematical model and numerical simulation of the liquid fluidization of polydisperse solid particle mixtures. Comput. Vis. Sci. 6, 67–74 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Boudin, L., Boutin, B., Fornet, B., Goudon, T., Lafitte, P., Lagoutire, F., Merlet, B.: Fluid-particles flows: a thin spray model with energy exchanges. ESAIM: Proc. 28, 195–210 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Boudin, L., Desvillettes, L., Grandmont, C., Moussa, A.: Global existence of solutions for the coupled Vlasov and Navier–Stokes equations. Differ. Integal Equ. 22, 1247–1271 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Bürger, R., Wendland, W.L., Concha, F.: Model equations for gravitational sedimentation-consolidation processes. Z. Angew. Math. Mech. 80, 79–92 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Caflisch, R., Papanicolaou, G.C.: Dynamic theory of suspensions with Brownian effects. SIAM J. Appl. Math. 43, 885–906 (1983)

    MathSciNet  MATH  Google Scholar 

  12. Carrillo, J.A., Goudon, T.: Stability and asymptotic analysis of a fluid–particle interaction model. Commun. Partial Differ. Equ. 31, 1349–1379 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Carrillo, J.A., Goudon, T., Lafitte, P.: Simulation of fluid and particles flows Asymptotic preserving schemes for bubbling and flowing regimes. J. Comput. Phys. 227, 7929–7951 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Carrillo, J.A., Duan, R., Moussa, A.: Global classical solution close to equillibrium to the Vlasov–Euler–Fokker–Planck system. Kinet. Relat. Models 4, 227–258 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Chae, M., Kang, K., Lee, J.: Global existence of weak and classical solutions for the Navier–Stokes–Vlasov–Fokker–Planck equations. J. Differ. Equ. 251, 2431–2465 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Chae, M., Kang, K., Lee, J.: Global classical solutions for a compressible fluid-particle interaction model. J. Hyperbolic Differ. Equ. 10, 537–562 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Duan, R., Fornasier, M., Toscani, G.: A kinetic flocking model with diffusions. Commun. Math. Phys. 300, 95–145 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Duan, R., Ukai, S., Yang, T., Zhao, H.: Optimal decay estimates on the linearized Boltzmann equation with time-dependent forces and their applications. Commun. Math. Phys. 277(1), 189–236 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Falkovich, G., Fouxon, A., Stepanov, M.G.: Acceleration of rain initiation by cloud turbulence. Nature 219, 151–154 (2002)

    Google Scholar 

  20. Goudon, T., He, L.-B., Moussa, A., Zhang, P.: The Navier–Stokes–Vlasov–Fokker–Planck system near equilibrium. SIAM J. Math. Anal. 42, 2177–2202 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Goudon, T., Jabin, P.-E., Vasseur, A.: Hydrodynamic limit for the Vlasov–Navier–Stokes equations. I. Light particles regime. Indiana Univ. Math. J. 53, 1495–1515 (2004)

    MathSciNet  MATH  Google Scholar 

  22. Goudon, T., Jabin, P.-E., Vasseur, A.: Hydrodynamic limit for the Vlasov–Navier–Stokes equations. II. Fine particles regime. Indiana Univ. Math. J. 53, 1517–1536 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Goudon, T., Jin, S., Yan, B.: Simulation of fluid-particles flows: heavy particles, flowing regime, and asymptotic-preserving schemes. Commun. Math. Sci. 10, 355–385 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Guo, Y.: The Boltzmann equation in the whole space. Indian Univ. Math. J. 53, 1081–1094 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Guo, Y.: The Vlasov–Maxwell–Boltzmann system near Maxwellians. Invent. Math. 153, 593–630 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231, 391–434 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Guo, Y.: The Vlasov–Poisson–Boltzmann system near Maxwellians. Commun. Pure Appl. Math. 55, 1104–1135 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Hamdache, K.: Global existence and large time behaviour of solutions for the Vlasov–Stokes equations. Jpn J. Ind. Appl. Math. 15, 51–74 (1998)

    MathSciNet  MATH  Google Scholar 

  29. Li, F., Mu, Y., Wang, D.: Strong solution to the compressible Navier–Stokes–Vlasov–Fokker–Planck equations: global existence near the equilibrium and large time behavior. SIAM J. Math. Anal. 49, 984–1026 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Lin, F.-H., Liu, C., Zhang, P.: On a micro–macro model for polymeric fluids near equilibrium. Commun. Pure Appl. Math. 60, 838–866 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Lions, P., Masmoudi, N.: Global existence of weak solutions to some micro–macro models. Partial Differ. Equ. 345, 15–20 (2007)

    MathSciNet  MATH  Google Scholar 

  32. Masmoudi, N.: Global existence of weak solutions to macroscopic models of polymeric flows. J. Math. Pures Appl. 96, 502–520 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Mathiaud, J.: Etude de syst‘emes de type gaz-particules. Ph.D. Thesis, ENS Cachan (2006)

  34. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)

    MathSciNet  MATH  Google Scholar 

  35. Matsumura, A., Nishida, T.: Initial value problem for the equations of motion of viscous and heat conductive gases. Proc. Jpn. Acad. Ser. A Math. Sci. 55, 337–342 (1979)

    MATH  Google Scholar 

  36. Mellet, A., Vasseur, A.: Global weak solutions for a Vlasov–Fokker–Planck/compressible Navier–Stokes system of equations. Math. Models Methods Appl. Sci. 17, 1039–1063 (2007)

    MathSciNet  MATH  Google Scholar 

  37. Mellet, A., Vasseur, A.: Asymptotic analysis for a Vlasov–Fokker–Planck/ compressible Navier–Stokes system of equations. Commun. Math. Phys. 281, 573–596 (2008)

    MathSciNet  MATH  Google Scholar 

  38. O’Rourke, P.: Collective drop effects on vaporizing liquid sprays. Ph.D. Thesis, Princeton University, Princeton, NJ (1981)

  39. Ranz, W.E., Marshall, W.R.: Evaporation from drops, part I. Chem. Eng. Prog. 48(3), 141–146 (1952)

    Google Scholar 

  40. Ranz, W.E., Marshall, W.R.: Evaporation from drops, part II. Chem. Eng. Prog. 48(4), 173–180 (1952)

    Google Scholar 

  41. Ukai, S., Yang, T.: Mathematical Theory of Boltzmann Equation. Lecture Notes Series-No. 8. Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, Hong Kong (2006)

  42. Wang, D., Yu, C.: Global weak solution to the inhomogeneous Navier–Stokes–Vlasov equations. J. Differ. Equ. 259(8), 3976–4008 (2015)

    MathSciNet  MATH  Google Scholar 

  43. Williams, F.A.: Spray combustion and atomization. Phys. Fluid 1, 541–555 (1958)

    MATH  Google Scholar 

  44. Williams, F.A.: Combustion Theory, 2nd edn. Westview Press, Boulder (1994)

    Google Scholar 

  45. Yu, C.: Global weak solutions to the incompressible Navier–Stokes–Vlasov equations. J. Math. Pures Appl. (9) 100, 275–293 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Y. Mu was partially supported by NSFC (Grant No. 11701268), Natural Science Foundation of Jiangsu Province of China (BK20171040) and Chinese Postdoctoral Science Foundation (2018M642277). D. Wang’s research was supported in part by the NSF Grants DMS-1613213 and DMS-1907519. The authors thank the referee for valuable comments and suggestions.

Author information

Authors and Affiliations

  1. School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210046, China

    Yanmin Mu

  2. School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing, 210023, China

    Yanmin Mu

  3. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Dehua Wang

Authors
  1. Yanmin Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dehua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dehua Wang.

Additional information

Communicated by P. Rabinowitz.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mu, Y., Wang, D. Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows. Calc. Var. 59, 110 (2020). https://doi.org/10.1007/s00526-020-01776-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-020-01776-8

Mathematics Subject Classification

Search

Navigation