This note is based on a presentation done at the Séminaire Laurent Schwartz in November 2023. The goal is to review our recent result [20], obtained in collaboration with Daniel Han-Kwan. We consider the equations of thick sprays, modeling the evolution of a gas-particles mixture through a Vlasov-Boltzmann equation for the particles and compressible Navier-Stokes equations for the gas. In the so-called thick spray regime, the volume occupied by the cloud of particles is not negligible in front of that of the gas. Unlike other fluid-kinetic models, the mathematical analysis of such dense sprays is still in its infancy and raises several challenges, the Cauchy theory being one of the first. Inspired by recent works on singular Vlasov equations, we show that the thick spray equations are locally well-posed in Sobolev regularity, provided that the initial data satisfies a suitable Penrose stability condition.
@article{SLSEDP_2023-2024____A2_0, author = {Lucas Ertzbischoff}, title = {On thick spray equations}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:4}, pages = {1--10}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2023-2024}, doi = {10.5802/slsedp.165}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.165/} }
TY - JOUR AU - Lucas Ertzbischoff TI - On thick spray equations JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:4 PY - 2023-2024 SP - 1 EP - 10 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.165/ DO - 10.5802/slsedp.165 LA - en ID - SLSEDP_2023-2024____A2_0 ER -
%0 Journal Article %A Lucas Ertzbischoff %T On thick spray equations %J Séminaire Laurent Schwartz — EDP et applications %Z talk:4 %D 2023-2024 %P 1-10 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.165/ %R 10.5802/slsedp.165 %G en %F SLSEDP_2023-2024____A2_0
Lucas Ertzbischoff. On thick spray equations. Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 4, 10 p. doi : 10.5802/slsedp.165. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.165/
[1] A. Baradat. Nonlinear instability in Vlasov type equations around rough velocity profiles. Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 37(3):489–547, 2020. | DOI | MR | Zbl
[2] A. Baradat, L. Ertzbischoff, and D. Han-Kwan. On ill-posedness issues for thick spray equations (working title). In preparation.
[3] C. Baranger and L. Desvillettes. Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions. J. Hyperbolic Differ. Equ., 3(01):1–26, 2006. | DOI | MR | Zbl
[4] C. Bardos. About a variant of the 1d Vlasov equation, dubbed “Vlasov-Dirac-Benney equation”. Sémin. Laurent Schwartz, EDP Appl., 15:ex, 2012-2013. | DOI | Zbl
[5] C. Bardos and N. Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinet. Relat. Models, 6(4):893–917, 2013. | DOI | MR | Zbl
[6] C. Bardos and N. Besse. Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-Benney equation. In Hamiltonian partial differential equations and applications, volume 75 of Fields Inst. Commun., pages 1–30. Fields Inst. Res. Math. Sci., Toronto, ON, 2015. | DOI | Zbl
[7] C. Bardos and A. Nouri. A Vlasov equation with Dirac potential used in fusion plasmas. J. Math. Phys., 53(11):115621, 16, 2012. | DOI | MR | Zbl
[8] S. Benjelloun, L. Desvillettes, J. Ghidaglia, and K. Nielsen. Modeling and simulation of thick sprays through coupling of a finite volume Euler equation solver and a particle method for a disperse phase. Note Mat., 32(1):63–85, 2012. | Zbl
[9] L. Boudin, L. Desvillettes, and R. Motte. A modeling of compressible droplets in a fluid. Commun. Math. Sci., 1(4):657–669, 2003. | DOI | MR | Zbl
[10] D. Bresch, B. Desjardins, J.-M. Ghidaglia, and E. Grenier. Global weak solutions to a generic two-fluid model. Arch. Ration. Mech. Anal., 196(2):599–629, 2010. | DOI | MR | Zbl
[11] C. Buet, B. Després, and L. Desvillettes. Linear stability of thick sprays equations. J. Stat. Phys., 190(3):53, 2023. | DOI | MR | Zbl
[12] C. Buet, B. Després, and V. Fournet. Analog of linear landau damping in a coupled vlasov-euler system for thick sprays, 2023. | HAL
[13] K. Carrapatoso, D. Han-Kwan, and F. Rousset. Wellposedness of singular Vlasov equations under optimal stability conditions (working title), 2023. in preparation.
[14] T. Chaub. Local well-posedness for a class of singular Vlasov equations. Kinet. Relat. Models, 16(2):187–206, 2023. | DOI | MR | Zbl
[15] L. Desvillettes. Some aspects of the modeling at different scales of multiphase flows. Comput. Methods Appl. Mech. Eng., 199(21-22):1265–1267, 2010. | DOI | MR | Zbl
[16] L. Desvillettes and L. Mathiaud. Some aspects of the asymptotics leading from gas-particles equations towards multiphase flows equations. J. Stat. Phys., 141(1):120–141, 2010. | DOI | MR | Zbl
[17] M. Doi and S. Edwards. The theory of polymer dynamics, volume 73. Oxford university press, 1988.
[18] J. Dukowicz. A particle-fluid numerical model for liquid sprays. J. Comput. Phys., 35(2):229–253, 1980. | DOI | MR | Zbl
[19] L. Ertzbischoff. Mathematical analysis of some fluid-kinetic systems of equations. PhD thesis - Institut Polytechnique de Paris, 2023.
[20] L. Ertzbischoff and D. Han-Kwan. On well-posedness for thick spray equations, 2023. | arXiv
[21] V. Fournet, C. Buet, and B. Després. Local-in-time existence of strong solutions to an averaged thick sprays model, 2022. | HAL
[22] F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 76(1):110–125, 1988. | DOI | MR | Zbl
[23] F. Golse, B. Perthame, and R. Sentis. Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport. C. R. Acad. Sci., Paris, Sér. I, 301:341–344, 1985. | Zbl
[24] M. Griffin-Pickering and M. Iacobelli. Recent developments on quasineutral limits for Vlasov-type equations. Recent Advances in Kinetic Equations and Applications, pages 211–231, 2021. | DOI | Zbl
[25] E. Guazzelli and J. Morris. A physical introduction to suspension dynamics, volume 45. Cambridge University Press, 2011. | DOI | Zbl
[26] D. Han-Kwan. On propagation of higher space regularity for nonlinear Vlasov equations. Anal. PDE, 12(1):189–244, 2019. | DOI | MR | Zbl
[27] D. Han-Kwan and T. Nguyen. Ill-posedness of the hydrostatic Euler and singular Vlasov equations. Arch. Ration. Mech. Anal., 221(3):1317–1344, 2016. | DOI | MR | Zbl
[28] D. Han-Kwan and F. Rousset. Quasineutral limit for Vlasov-Poisson with Penrose stable data. Ann. Sci. Éc. Norm. Supér. (4), 49(6):1445–1495, 2016. | DOI | MR | Zbl
[29] D. Han-Kwan and F. Rousset. From Vlasov-Poisson to the kinetic incompressible Euler equation (working title), 2023. in preparation.
[30] R. Höfer. Sedimentation of particle suspensions in Stokes flows. Universitäts-und Landesbibliothek Bonn, 2020.
[31] R. Höfer and R. Schubert. Sedimentation of particles with very small inertia II: Derivation, Cauchy problem and hydrodynamic limit of the Vlasov-Stokes equation, 2023. | arXiv
[32] M. Ishii and T. Hibiki. Thermo-fluid dynamics of two-phase flow. Springer Science & Business Media, 2010.
[33] P.-E. Jabin and A. Nouri. Analytic solutions to a strongly nonlinear Vlasov equation. C. R., Math., Acad. Sci. Paris, 349(9-10):541–546, 2011. | DOI | MR | Zbl
[34] D. Koch. Kinetic theory for a monodisperse gas–solid suspension. Physics of Fluids A: Fluid Dynamics, 2(10):1711–1723, 1990. | DOI | Zbl
[35] F. Laurent, M. Massot, and P. Villedieu. Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays. J. Comput. Phys., 194(2):505–543, 2004. | DOI | MR | Zbl
[36] N. Masmoudi and T. Wong. On the theory of hydrostatic Euler equations. Arch. Ration. Mech. Anal., 204(1):231–271, 2012. | DOI | MR | Zbl
[37] J. Mathiaud. Local smooth solutions of a thin spray model with collisions. Math. Models Methods Appl. Sci., 20(02):191–221, 2010. | DOI | MR | Zbl
[38] G. Métivier. Stability of multidimensional shocks. In Advances in the theory of shock waves, pages 25–103. Boston, MA: Birkhäuser, 2001. | DOI | Zbl
[39] C. Mouhot and C. Villani. On Landau damping. Acta Math., 207(1):29–201, 2011. | DOI | MR | Zbl
[40] A. Moussa. Étude mathématique de modèles cinétiques, fluides et paraboliques issus de la biologie. Habilitation à Diriger les Recherches - Sorbonne Université, 2018.
[41] P. O’Rourke. Collective drop effects on vaporizing liquid sprays. Technical report, Los Alamos National Lab., NM (USA), 1981.
[42] P. Pakseresht and S. V. Apte. Modeling the dense spray regime using an Euler-Lagrange approach with volumetric displacement effects, 2019. | arXiv
[43] O. Penrose. Electrostatic instabilities of a uniform non-Maxwellian plasma. The Physics of Fluids, 3(2):258–265, 1960. | DOI | Zbl
[44] D. Ramos. Quelques résultats mathématiques et simulations numériques d’écoulements régis par des modèles bifluides. PhD thesis, ENS Cachan, (in French), 2000.
[45] R. Reitz. Computer modeling of sprays. Spray Technology Short Course, Pittsburgh, PA, 1996.
[46] L. Saint-Raymond. Hydrodynamic limits of the Boltzmann equation, volume 1971 of Lect. Notes Math. Berlin: Springer, 2009. | DOI | Zbl
[47] W. Sirignano. Fluid dynamics and transport of droplets and sprays. Cambridge university press, 2010.
Cité par Sources :