• Previous Article
    Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case
  • DCDS Home
  • This Issue
  • Next Article
    Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure
September  2020, 40(9): 5471-5511. doi: 10.3934/dcds.2020235

Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime

IMUS & Departamento de Análisis Matemático, Universidad de Sevilla, C/Tarfia s/n, Campus Reina Mercedes, 41012 Sevilla, Spain

Received  December 2019 Revised  March 2020 Published  June 2020

Fund Project: This research is supported by the Basque Government through the BERC 2018-2021 program and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym DESFLU and by the European Research Council through the Starting Grant project H2020-EU.1.1.-639227 FLUID INTERFACE

We prove that the incompressible, density dependent, Navier-Stokes equations are globally well posed in a low Froude number regime. The density profile is supposed to be increasing in depth and linearized around a stable state. Moreover if the Froude number tends to zero we prove that such system converges (strongly) to a two-dimensional, stratified Navier-Stokes equations with full diffusivity. No smallness assumption is considered on the initial data.

Citation: Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235
References:
[1]

S. Alinhac and P. Gérard, Opérateurs Pseudo-Différentiels et Théorème de Nash-Moser, Savoirs Actuels, InterEditions, Paris; Éditions du Centre National de la Recherche Scientifique (CNRS), Meudon, 1991.  Google Scholar

[2]

A. BabinA. Mahalov and B. Nicolaenko, Regularity and integrability of $3$D Euler and Navier-Stokes equations for rotating fluids, Asymptot. Anal., 15 (1997), 103-150.  doi: 10.3233/ASY-1997-15201.  Google Scholar

[3]

A. BabinA. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48 (1999), 1133-1176.   Google Scholar

[4]

A. V. BabinA. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.  doi: 10.1512/iumj.2001.50.2155.  Google Scholar

[5]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[6]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), 209-246.  doi: 10.24033/asens.1404.  Google Scholar

[7]

Frédéric Charve, Étude de Phénomènes Dispersifs en Mécanique des Fluides Géophysiques, Ph.D. thesis, École Polytechnique, 2004. Google Scholar

[8]

F. Charve, Global well-posedness and asymptotics for a geophysical fluid system, Comm. Partial Differential Equations, 29 (2005), 1919-1940.  doi: 10.1081/PDE-200043510.  Google Scholar

[9]

F. Charve, Convergence of weak solutions for the primitive system of the quasigeostrophic equations, Asymptot. Anal., 42 (2005), 173-209.   Google Scholar

[10]

F. Charve and V.-S. Ngo, Global existence for the primitive equations with small anisotropic viscosity, Rev. Mat. Iberoam., 27 (2011), 1-38.  doi: 10.4171/RMI/629.  Google Scholar

[11]

J.-Y. Chemin, Remarques sur l'existence globale pour le système de Navier-Stokes incompressible, SIAM J. Math. Anal., 23 (1992), 20-28.  doi: 10.1137/0523002.  Google Scholar

[12]

J.-Y. Chemin, À propos d'un problème de pénalisation de type antisymétrique, J. Math. Pures Appl. (9), 76 (1997), 739-755.  doi: 10.1016/S0021-7824(97)89967-9.  Google Scholar

[13]

J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 14, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[14]

J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., 34 (2000), 315-335.  doi: 10.1051/m2an:2000143.  Google Scholar

[15]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Anisotropy and dispersion in rotating fluids, in Nonlinear Partial Differential Equations and Their Applications, Stud. Math. Appl., 31, North-Holland, Amsterdam, 2002,171–192. doi: 10.1016/S0168-2024(02)80010-8.  Google Scholar

[16]

J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Ekman boundary layers in rotating fluids, ESAIM Control Optim. Calc. Var., 8 (2002), 441-466.  doi: 10.1051/cocv:2002037.  Google Scholar

[17]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations, Oxford Lecture Series in Mathematics and its Applications, 32, The Clarendon Press, Oxford University Press, Oxford, 2006.  Google Scholar

[18]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, International Geophysics Series, 101, Elsevier/Academic Press, Amsterdam, 2011.  Google Scholar

[19]

B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271-2279.  doi: 10.1098/rspa.1999.0403.  Google Scholar

[20]

A. Dutrifoy, Examples of dispersive effects in non-viscous rotating fluids, J. Math. Pures Appl. (9), 84 (2005), 331-356.  doi: 10.1016/j.matpur.2004.09.007.  Google Scholar

[21]

P. F. Embid and A. J. Majda, Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers, Geophys. Astrophys. Fluid Dynam., 87 (1998), 1-50.  doi: 10.1080/03091929808208993.  Google Scholar

[22]

E. FeireislI. Gallagher and A. Novotný, A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192-205.  doi: 10.1137/100808010.  Google Scholar

[23]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[24]

I. Gallagher and L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, J. Anal. Math., 99 (2006), 1-34.  doi: 10.1007/BF02789441.  Google Scholar

[25]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[26]

E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations, 22 (1997), 953-975.  doi: 10.1080/03605309708821290.  Google Scholar

[27]

S. Ibrahim and T. Yoneda, Long-time solvability of the Navier-Stokes-Boussinesq equations with almost periodic initial large data, J. Math. Sci. Univ. Tokyo, 20 (2013), 1-25.   Google Scholar

[28]

O. A. Ladyženskaja, Solution "in the large" to the boundary-value problem for the Navier-Stokes equations in two space variables, Soviet Physics. Dokl., 123 (3) (1958), 1128-1131.   Google Scholar

[29]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, 1933, 82pp.  Google Scholar

[30]

J.-L. Lions and G. Prodi, Un théorème d'existence et unicité dans les équations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris, 248 (1959), 3519-3521.   Google Scholar

[31]

N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data, Comm. Pure Appl. Math., 53 (2000), 432-483.  doi: 10.1002/(SICI)1097-0312(200004)53:4<432::AID-CPA2>3.0.CO;2-Y.  Google Scholar

[32]

V.-S. Ngo, Rotating fluids with small viscosity, Int. Math. Res. Not. IMRN, 2009 (2009), 1860-1890.  doi: 10.1093/imrn/rnp004.  Google Scholar

[33]

V.-S. Ngo and S. Scrobogna, Dispersive effects of weakly compressible and fast rotating inviscid fluids, Discrete Contin. Dyn. Syst., 38 (2018), 749-789.  doi: 10.3934/dcds.2018033.  Google Scholar

[34]

V.-S. Ngo and S. Scrobogna, On the influence of gravity on density-dependent incompressible periodic fluids, J. Differential Equations, 267 (2019), 1510-1559.  doi: 10.1016/j.jde.2019.02.011.  Google Scholar

[35]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[36]

L. Schwartz, Sur l'impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris, 239 (1954), 847-848.   Google Scholar

[37]

S. Scrobogna, Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system, Discrete Contin. Dyn. Syst., 37 (2017), 5979-6034.  doi: 10.3934/dcds.2017259.  Google Scholar

[38]

S. Scrobogna, Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity, Rev. Mat. Iberoam., 34 (2018), 1-58.  doi: 10.4171/RMI/980.  Google Scholar

[39]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[40]

R. Takada, Strongly stratified limit for the 3d inviscid Boussinesq equations, Arch. Ration. Mech. Anal., 232 (2019), 1475-1503.  doi: 10.1007/s00205-018-01347-4.  Google Scholar

[41]

K. Widmayer, Convergence to stratified flow for an inviscid 3D Boussinesq system, Commun. Math. Sci., 16 (2018), 1713-1728.  doi: 10.4310/CMS.2018.v16.n6.a10.  Google Scholar

show all references

References:
[1]

S. Alinhac and P. Gérard, Opérateurs Pseudo-Différentiels et Théorème de Nash-Moser, Savoirs Actuels, InterEditions, Paris; Éditions du Centre National de la Recherche Scientifique (CNRS), Meudon, 1991.  Google Scholar

[2]

A. BabinA. Mahalov and B. Nicolaenko, Regularity and integrability of $3$D Euler and Navier-Stokes equations for rotating fluids, Asymptot. Anal., 15 (1997), 103-150.  doi: 10.3233/ASY-1997-15201.  Google Scholar

[3]

A. BabinA. Mahalov and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48 (1999), 1133-1176.   Google Scholar

[4]

A. V. BabinA. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.  doi: 10.1512/iumj.2001.50.2155.  Google Scholar

[5]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[6]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), 209-246.  doi: 10.24033/asens.1404.  Google Scholar

[7]

Frédéric Charve, Étude de Phénomènes Dispersifs en Mécanique des Fluides Géophysiques, Ph.D. thesis, École Polytechnique, 2004. Google Scholar

[8]

F. Charve, Global well-posedness and asymptotics for a geophysical fluid system, Comm. Partial Differential Equations, 29 (2005), 1919-1940.  doi: 10.1081/PDE-200043510.  Google Scholar

[9]

F. Charve, Convergence of weak solutions for the primitive system of the quasigeostrophic equations, Asymptot. Anal., 42 (2005), 173-209.   Google Scholar

[10]

F. Charve and V.-S. Ngo, Global existence for the primitive equations with small anisotropic viscosity, Rev. Mat. Iberoam., 27 (2011), 1-38.  doi: 10.4171/RMI/629.  Google Scholar

[11]

J.-Y. Chemin, Remarques sur l'existence globale pour le système de Navier-Stokes incompressible, SIAM J. Math. Anal., 23 (1992), 20-28.  doi: 10.1137/0523002.  Google Scholar

[12]

J.-Y. Chemin, À propos d'un problème de pénalisation de type antisymétrique, J. Math. Pures Appl. (9), 76 (1997), 739-755.  doi: 10.1016/S0021-7824(97)89967-9.  Google Scholar

[13]

J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 14, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[14]

J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., 34 (2000), 315-335.  doi: 10.1051/m2an:2000143.  Google Scholar

[15]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Anisotropy and dispersion in rotating fluids, in Nonlinear Partial Differential Equations and Their Applications, Stud. Math. Appl., 31, North-Holland, Amsterdam, 2002,171–192. doi: 10.1016/S0168-2024(02)80010-8.  Google Scholar

[16]

J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Ekman boundary layers in rotating fluids, ESAIM Control Optim. Calc. Var., 8 (2002), 441-466.  doi: 10.1051/cocv:2002037.  Google Scholar

[17]

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations, Oxford Lecture Series in Mathematics and its Applications, 32, The Clarendon Press, Oxford University Press, Oxford, 2006.  Google Scholar

[18]

B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, International Geophysics Series, 101, Elsevier/Academic Press, Amsterdam, 2011.  Google Scholar

[19]

B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271-2279.  doi: 10.1098/rspa.1999.0403.  Google Scholar

[20]

A. Dutrifoy, Examples of dispersive effects in non-viscous rotating fluids, J. Math. Pures Appl. (9), 84 (2005), 331-356.  doi: 10.1016/j.matpur.2004.09.007.  Google Scholar

[21]

P. F. Embid and A. J. Majda, Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers, Geophys. Astrophys. Fluid Dynam., 87 (1998), 1-50.  doi: 10.1080/03091929808208993.  Google Scholar

[22]

E. FeireislI. Gallagher and A. Novotný, A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192-205.  doi: 10.1137/100808010.  Google Scholar

[23]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[24]

I. Gallagher and L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, J. Anal. Math., 99 (2006), 1-34.  doi: 10.1007/BF02789441.  Google Scholar

[25]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[26]

E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations, 22 (1997), 953-975.  doi: 10.1080/03605309708821290.  Google Scholar

[27]

S. Ibrahim and T. Yoneda, Long-time solvability of the Navier-Stokes-Boussinesq equations with almost periodic initial large data, J. Math. Sci. Univ. Tokyo, 20 (2013), 1-25.   Google Scholar

[28]

O. A. Ladyženskaja, Solution "in the large" to the boundary-value problem for the Navier-Stokes equations in two space variables, Soviet Physics. Dokl., 123 (3) (1958), 1128-1131.   Google Scholar

[29]

J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, 1933, 82pp.  Google Scholar

[30]

J.-L. Lions and G. Prodi, Un théorème d'existence et unicité dans les équations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris, 248 (1959), 3519-3521.   Google Scholar

[31]

N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data, Comm. Pure Appl. Math., 53 (2000), 432-483.  doi: 10.1002/(SICI)1097-0312(200004)53:4<432::AID-CPA2>3.0.CO;2-Y.  Google Scholar

[32]

V.-S. Ngo, Rotating fluids with small viscosity, Int. Math. Res. Not. IMRN, 2009 (2009), 1860-1890.  doi: 10.1093/imrn/rnp004.  Google Scholar

[33]

V.-S. Ngo and S. Scrobogna, Dispersive effects of weakly compressible and fast rotating inviscid fluids, Discrete Contin. Dyn. Syst., 38 (2018), 749-789.  doi: 10.3934/dcds.2018033.  Google Scholar

[34]

V.-S. Ngo and S. Scrobogna, On the influence of gravity on density-dependent incompressible periodic fluids, J. Differential Equations, 267 (2019), 1510-1559.  doi: 10.1016/j.jde.2019.02.011.  Google Scholar

[35]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[36]

L. Schwartz, Sur l'impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris, 239 (1954), 847-848.   Google Scholar

[37]

S. Scrobogna, Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system, Discrete Contin. Dyn. Syst., 37 (2017), 5979-6034.  doi: 10.3934/dcds.2017259.  Google Scholar

[38]

S. Scrobogna, Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity, Rev. Mat. Iberoam., 34 (2018), 1-58.  doi: 10.4171/RMI/980.  Google Scholar

[39]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[40]

R. Takada, Strongly stratified limit for the 3d inviscid Boussinesq equations, Arch. Ration. Mech. Anal., 232 (2019), 1475-1503.  doi: 10.1007/s00205-018-01347-4.  Google Scholar

[41]

K. Widmayer, Convergence to stratified flow for an inviscid 3D Boussinesq system, Commun. Math. Sci., 16 (2018), 1713-1728.  doi: 10.4310/CMS.2018.v16.n6.a10.  Google Scholar

[1]

Daoyuan Fang, Ting Zhang, Ruizhao Zi. Dispersive effects of the incompressible viscoelastic fluids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5261-5295. doi: 10.3934/dcds.2018233

[2]

Miroslav Bulíček, Eduard Feireisl, Josef Málek, Roman Shvydkoy. On the motion of incompressible inhomogeneous Euler-Korteweg fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 497-515. doi: 10.3934/dcdss.2010.3.497

[3]

Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic & Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51

[4]

Tomáš Roubíček. From quasi-incompressible to semi-compressible fluids. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020414

[5]

Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001

[6]

Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021

[7]

Juliana Honda Lopes, Gabriela Planas. Well-posedness for a non-isothermal flow of two viscous incompressible fluids. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2455-2477. doi: 10.3934/cpaa.2018117

[8]

Michela Eleuteri, Elisabetta Rocca, Giulio Schimperna. On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2497-2522. doi: 10.3934/dcds.2015.35.2497

[9]

M. Bulíček, P. Kaplický. Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 41-50. doi: 10.3934/dcdss.2008.1.41

[10]

Gyungsoo Woo, Young-Sam Kwon. Incompressible limit for the full magnetohydrodynamics flows under Strong Stratification on unbounded domains. Communications on Pure & Applied Analysis, 2014, 13 (1) : 135-155. doi: 10.3934/cpaa.2014.13.135

[11]

Paolo Secchi. An alpha model for compressible fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 351-359. doi: 10.3934/dcdss.2010.3.351

[12]

Peter Constantin. Transport in rotating fluids. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 165-176. doi: 10.3934/dcds.2004.10.165

[13]

Y. Charles Li. Chaos phenotypes discovered in fluids. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1383-1398. doi: 10.3934/dcds.2010.26.1383

[14]

D. Bresch, B. Desjardins, D. Gérard-Varet. Rotating fluids in a cylinder. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 47-82. doi: 10.3934/dcds.2004.11.47

[15]

Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic & Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373

[16]

Mathieu Desbrun, Evan S. Gawlik, François Gay-Balmaz, Vladimir Zeitlin. Variational discretization for rotating stratified fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 477-509. doi: 10.3934/dcds.2014.34.477

[17]

Dominic Breit, Eduard Feireisl, Martina Hofmanová. Generalized solutions to models of inviscid fluids. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3831-3842. doi: 10.3934/dcdsb.2020079

[18]

M. Bulíček, Josef Málek, Dalibor Pražák. On the dimension of the attractor for a class of fluids with pressure dependent viscosities. Communications on Pure & Applied Analysis, 2005, 4 (4) : 805-822. doi: 10.3934/cpaa.2005.4.805

[19]

M. Berezhnyi, L. Berlyand, Evgen Khruslov. The homogenized model of small oscillations of complex fluids. Networks & Heterogeneous Media, 2008, 3 (4) : 831-862. doi: 10.3934/nhm.2008.3.831

[20]

K. R. Rajagopal. The thermo-mechanics of rate-type fluids. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1133-1145. doi: 10.3934/dcdss.2012.5.1133

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (53)
  • HTML views (68)
  • Cited by (0)

Other articles
by authors

[Back to Top]