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February
2021, 26(2): 875-905.
doi: 10.3934/dcdsb.2020146
On the rigid-lid approximation of shallow water Bingham
| 1. | Sorbonne University, Laboratory of Jacques-Louis Lions, 4 Place Jussieu 75005, Paris, France, INRIA Paris, ANGE Team, 2 Rue Simone IFF, 75012 Paris, France |
| 2. | Savoie Mont Blanc University, LAMA UMR5127 CNRS, 73376 Le Bourget du Lac, France, Lebanese University, Faculty of sciences 1, Laboratory of Mathematics-EDST, Hadath, Lebanon |
This paper discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham's constitutive law, we follow a similar approach to that introduced in [G. DUVAUT and J.-L. LIONS, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.
Mathematics Subject Classification:
Primary: 35Q30, 76N10, 35B65; Secondary: 35D35.
Citation:
Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete and Continuous Dynamical Systems - B,
2021, 26
(2)
: 875-905.
doi: 10.3934/dcdsb.2020146
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References:
| [1] |
N. Aïssiouene, M.-O. Bristeau, E. Godlewski, A. Mangeney, C. Parés and J. Sainte-Marie, A Two-dimensional Method for a Dispersive Shallow Water Model, https://hal.archives-ouvertes.fr/hal-01632522, Working paper or preprint, 2017. |
| [2] |
B. Al Taki,
Viscosity effect on the degenerate lake equations, Nonlinear Anal., 148 (2017), 30-60.
doi: 10.1016/j.na.2016.09.017. |
| [3] |
E. Bingham, Fluidity and Plasticity, McGraw-Hill, 1922. |
| [4] |
F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/b93802. |
| [5] |
C. Bourdarias and S. Gerbi,
A finite volume scheme for a model coupling free surface and pressurised flows in pipes, J. Comput. Appl. Math., 209 (2007), 109-131.
doi: 10.1016/j.cam.2006.10.086. |
| [6] |
C. Bourdarias, S. Gerbi and M. Gisclon,
A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes, J. Comput. Appl. Math., 218 (2008), 522-531.
doi: 10.1016/j.cam.2007.09.009. |
| [7] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183 of Applied Mathematical Sciences, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
| [8] |
D. Bresch and B. Desjardins,
Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.
doi: 10.1007/s00220-003-0859-8. |
| [9] |
D. Bresch, E. D. Fernández-Nieto, I. R. Ionescu and P. Vigneaux, Augmented Lagrangian method and compressible visco-plastic flows: Applications to shallow dense avalanches, in New Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser Verlag, Basel, (2010), 57–89. |
| [10] |
M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska Gwiazda,
On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44 (2012), 2756-2801.
doi: 10.1137/110830289. |
| [11] |
A. J. Chorin,
Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762.
doi: 10.1090/S0025-5718-1968-0242392-2. |
| [12] |
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976. |
| [13] |
D. E. Edmunds and R. Hurri-Syrjänen,
Weighted Hardy inequalities, J. Math. Anal. Appl., 310 (2005), 424-435.
doi: 10.1016/j.jmaa.2005.01.066. |
| [14] |
R. Farwig and H. Sohr,
Weighted $L^q$-theory for the Stokes resolvent in exterior domains, J. Math. Soc. Japan, 49 (1997), 251-288.
doi: 10.2969/jmsj/04920251. |
| [15] |
E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Cham, 2017, Second edition of [MR2499296]. |
| [16] |
E. Fernández-Nieto, P. Noble and J. Vila,
Shallow water equations for power law and Bingham fluids, Sci. China Math., 55 (2012), 277-283.
doi: 10.1007/s11425-011-4358-7. |
| [17] |
A. Fröhlich,
The Stokes operator in weighted $L^q$-spaces. Ⅱ. Weighted resolvent estimates and maximal $L^p$-regularity, Math. Ann., 339 (2007), 287-316.
doi: 10.1007/s00208-007-0114-2. |
| [18] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, NY, 2006, Unabridged republication of the 1993 original. |
| [19] |
A. Kał amajska,
Coercive inequalities on weighted Sobolev spaces, Colloq. Math., 66 (1994), 309-318.
|
| [20] |
C. D. Levermore and M. Sammartino,
A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14 (2001), 1493-1515.
doi: 10.1088/0951-7715/14/6/305. |
| [21] |
J.-L. Lions, Remarks on some nonlinear evolution problems arising in Bingham flows, Israel J. Math., 13 (1972), 155–172 (1973). |
| [22] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, vol. 10 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998.
|
| [23] |
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207–226, URL https://doi-org.camphrier-1.grenet.fr/10.2307/1995882.
doi: 10.1090/S0002-9947-1972-0293384-6. |
| [24] |
M. Naaim and A. Bouchet, Etude Expérimentale des Écoulements D'avalanches de Neige Dense, Mesures et interprétations des profils de vitesse en écoulements quasi permanents et pleinement développés. Rapport scientifique. UR ETNA, Grenoble. |
| [25] |
A. Nekvinda,
Characterization of traces of the weighted Sobolev space $W^{1, p}(\Omega, d^\epsilon_M)$ on $M$, Czechoslovak Math. J., 43 (1993), 695-711.
|
| [26] |
K. Nishimura and N. Maeno,
Contribution of viscous forces to avalanche dynamics, Annals of Glaciology, 13 (1989), 202-206.
|
| [27] |
R. Perla, T. Cheng and D. McClung,
A two–parameter model of snow–avalanche motion, Ann. Glaciol., 26 (1980), 197-207.
|
| [28] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979. |
| [29] |
B. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, vol. 1736 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. URL https://doi-org.camphrier-1.grenet.fr/10.1007/BFb0103908. |
| [30] |
A. Vasseur and C. Yu,
Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.
doi: 10.1007/s00222-016-0666-4. |
show all references
References:
| [1] |
N. Aïssiouene, M.-O. Bristeau, E. Godlewski, A. Mangeney, C. Parés and J. Sainte-Marie, A Two-dimensional Method for a Dispersive Shallow Water Model, https://hal.archives-ouvertes.fr/hal-01632522, Working paper or preprint, 2017. |
| [2] |
B. Al Taki,
Viscosity effect on the degenerate lake equations, Nonlinear Anal., 148 (2017), 30-60.
doi: 10.1016/j.na.2016.09.017. |
| [3] |
E. Bingham, Fluidity and Plasticity, McGraw-Hill, 1922. |
| [4] |
F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/b93802. |
| [5] |
C. Bourdarias and S. Gerbi,
A finite volume scheme for a model coupling free surface and pressurised flows in pipes, J. Comput. Appl. Math., 209 (2007), 109-131.
doi: 10.1016/j.cam.2006.10.086. |
| [6] |
C. Bourdarias, S. Gerbi and M. Gisclon,
A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes, J. Comput. Appl. Math., 218 (2008), 522-531.
doi: 10.1016/j.cam.2007.09.009. |
| [7] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183 of Applied Mathematical Sciences, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
| [8] |
D. Bresch and B. Desjardins,
Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.
doi: 10.1007/s00220-003-0859-8. |
| [9] |
D. Bresch, E. D. Fernández-Nieto, I. R. Ionescu and P. Vigneaux, Augmented Lagrangian method and compressible visco-plastic flows: Applications to shallow dense avalanches, in New Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser Verlag, Basel, (2010), 57–89. |
| [10] |
M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska Gwiazda,
On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44 (2012), 2756-2801.
doi: 10.1137/110830289. |
| [11] |
A. J. Chorin,
Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762.
doi: 10.1090/S0025-5718-1968-0242392-2. |
| [12] |
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976. |
| [13] |
D. E. Edmunds and R. Hurri-Syrjänen,
Weighted Hardy inequalities, J. Math. Anal. Appl., 310 (2005), 424-435.
doi: 10.1016/j.jmaa.2005.01.066. |
| [14] |
R. Farwig and H. Sohr,
Weighted $L^q$-theory for the Stokes resolvent in exterior domains, J. Math. Soc. Japan, 49 (1997), 251-288.
doi: 10.2969/jmsj/04920251. |
| [15] |
E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Cham, 2017, Second edition of [MR2499296]. |
| [16] |
E. Fernández-Nieto, P. Noble and J. Vila,
Shallow water equations for power law and Bingham fluids, Sci. China Math., 55 (2012), 277-283.
doi: 10.1007/s11425-011-4358-7. |
| [17] |
A. Fröhlich,
The Stokes operator in weighted $L^q$-spaces. Ⅱ. Weighted resolvent estimates and maximal $L^p$-regularity, Math. Ann., 339 (2007), 287-316.
doi: 10.1007/s00208-007-0114-2. |
| [18] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, NY, 2006, Unabridged republication of the 1993 original. |
| [19] |
A. Kał amajska,
Coercive inequalities on weighted Sobolev spaces, Colloq. Math., 66 (1994), 309-318.
|
| [20] |
C. D. Levermore and M. Sammartino,
A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14 (2001), 1493-1515.
doi: 10.1088/0951-7715/14/6/305. |
| [21] |
J.-L. Lions, Remarks on some nonlinear evolution problems arising in Bingham flows, Israel J. Math., 13 (1972), 155–172 (1973). |
| [22] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, vol. 10 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998.
|
| [23] |
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207–226, URL https://doi-org.camphrier-1.grenet.fr/10.2307/1995882.
doi: 10.1090/S0002-9947-1972-0293384-6. |
| [24] |
M. Naaim and A. Bouchet, Etude Expérimentale des Écoulements D'avalanches de Neige Dense, Mesures et interprétations des profils de vitesse en écoulements quasi permanents et pleinement développés. Rapport scientifique. UR ETNA, Grenoble. |
| [25] |
A. Nekvinda,
Characterization of traces of the weighted Sobolev space $W^{1, p}(\Omega, d^\epsilon_M)$ on $M$, Czechoslovak Math. J., 43 (1993), 695-711.
|
| [26] |
K. Nishimura and N. Maeno,
Contribution of viscous forces to avalanche dynamics, Annals of Glaciology, 13 (1989), 202-206.
|
| [27] |
R. Perla, T. Cheng and D. McClung,
A two–parameter model of snow–avalanche motion, Ann. Glaciol., 26 (1980), 197-207.
|
| [28] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979. |
| [29] |
B. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, vol. 1736 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. URL https://doi-org.camphrier-1.grenet.fr/10.1007/BFb0103908. |
| [30] |
A. Vasseur and C. Yu,
Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.
doi: 10.1007/s00222-016-0666-4. |
Figure 2.
(a) variations of the velocity $ u $ and (b) variations of the pressure $ p $ in the fluid domain at time $ t = T/2 $
Figure 3.
(a) variations of the quantity $ \tilde\sigma $ and (b) variations of $ \,\partial_x u $ at time $ t = T/2 $
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