# American Institute of Mathematical Sciences

March  2016, 11(1): 1-27. doi: 10.3934/nhm.2016.11.1

## A combined finite volume - finite element scheme for a dispersive shallow water system

 1 Inria, EPC ANGE, 2 rue Simone Iff, F75012 Paris, France, France, France, France

Received  June 2015 Revised  October 2015 Published  January 2016

We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation.
The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces.
The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition.
Several numerical experiments are performed to confirm the relevance of our approach.
Citation: Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks and Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
##### References:
 [1] N. Aïssiouene, M. O. Bristeau, E. Godlewski and J. Sainte-Marie, A robust and stable numerical scheme for a depth-averaged Euler system, Submitted. [2] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4. [3] B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J., 57 (2008), 97-131. doi: 10.1512/iumj.2008.57.3200. [4] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for Shallow Water flows, SIAM J. Sci. Comput., 25 (2004), 2050-2065. doi: 10.1137/S1064827503431090. [5] E. Audusse, F. Bouchut, M.-O. Bristeau and J. Sainte-Marie, Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system, 2014, URL http://hal.inria.fr/hal-01063577, (Submitted) http://hal.inria.fr/hal-01063577/PDF/kin_hydrost.pdf. [6] J.-L. Bona, T.-B. Benjamin and J.-J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Royal Soc. London Series A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032. [7] P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes, European Journal of Mechanics - B/Fluids, 30 (2011), 589-597, URL http://www.sciencedirect.com/science/article/pii/S0997754611000185, Special Issue: Nearshore Hydrodynamics. doi: 10.1016/j.euromechflu.2011.02.005. [8] F. Bouchut, An introduction to finite volume methods for hyperbolic conservation laws, ESAIM Proc., 15 (2005), 1-17. [9] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkhäuser, 2004. doi: 10.1007/b93802. [10] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129-151. [11] M.-O. Bristeau and B. Coussin, Boundary Conditions for the Shallow Water Equations Solved by Kinetic Schemes, Rapport de recherche RR-4282, INRIA, 2001, URL http://hal.inria.fr/inria-00072305, Projet M3N. [12] M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic Shallow Water model, Computers & Fluids, 47 (2011), 51-64. doi: 10.1016/j.compfluid.2011.02.013. [13] M. O. Bristeau, A. Mangeney, J. Sainte-Marie and N. Seguin, An energy-consistent depth-averaged Euler system: Derivation and properties, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 961-988, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10801. doi: 10.3934/dcdsb.2015.20.961. [14] M.-O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733-759. doi: 10.3934/dcdsb.2008.10.733. [15] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Math., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [16] F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model, J. Sci. Comput., 48 (2011), 105-116. doi: 10.1007/s10915-010-9395-9. [17] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2. [18] M.-W. Dingemans, Wave Propagation Over Uneven Bottoms, Advanced Series on Ocean Engineering - World Scientific, 1997. doi: 10.1142/9789812796042. [19] M. Dingemans, Comparison of Computations with Boussinesq-like Models and Laboratory Measurements, Technical Report H1684-12, AST G8M Coastal Morphodybamics Research Programme, 1994. [20] A. Duran and F. Marche, Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations, Communications in Computational Physics, 17 (2015), 721-760, URL https://hal.archives-ouvertes.fr/hal-00980826. doi: 10.4208/cicp.150414.101014a. [21] W. E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057. doi: 10.1137/0732047. [22] A. Ern and S. Meunier, A posteriori error analysis of euler-galerkin approximations to coupled elliptic-parabolic problems, ESAIM Math. Model. Numer. Anal., 43 (2009), 353-375, URL http://journals.cambridge.org/abstract\_S0764583X05000166. doi: 10.1051/m2an:2008048. [23] E. Godlewski and P.-A. Raviart, Numerical Approximations of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [24] A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425. [25] P. Gresho and S. Chan, Semi-consistent mass matrix techniques for solving the incompressible Navier-Stokes equations, First Int. Conf. on Comput. Methods in Flow Analysis, Okayama University, Japan. [26] J.-L. Guermond, Some implementations of projection methods for Navier-Stokes equations, ESAIM: Mathematical Modelling and Numerical Analysis, 30 (1996), 637-667, URL http://eudml.org/doc/193818. [27] J.-L. Guermond and J. Shen, On the error estimates for the rotational pressure-correction projection methods, Math. Comput., 73 (2004), 1719-1737, URL http://dblp.uni-trier.de/db/journals/moc/moc73.html#GuermondS04. doi: 10.1090/S0025-5718-03-01621-1. [28] H. Johnston and J.-G. Liu, Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, Journal of Computational Physics, 199 (2004), 221-259, URL http://www.sciencedirect.com/science/article/pii/S002199910400083X. doi: 10.1016/j.jcp.2004.02.009. [29] D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, 21 (2009), 016601. doi: 10.1063/1.3053183. [30] O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model, J. Comput. Phys., 229 (2010), 2034-2045. doi: 10.1016/j.jcp.2009.11.021. [31] R.-J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. [32] O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, 119 (1993), 618-638. doi: 10.1061/(ASCE)0733-950X(1993)119:6(618). [33] D. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827. doi: 10.1017/S0022112067002605. [34] O. Pironneau, Méthodes Des Éléments Finis Pour Les Fluides., Masson, 1988. [35] R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations, in The Navier-Stokes Equations II - Theory and Numerical Methods (eds. G. Heywood John, K. Masuda, R. Rautmann and A. Solonnikov Vsevolod), vol. 1530 of Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1992, 167-183, URL http://dx.doi.org/10.1007/BFb0090341. doi: 10.1007/BFb0090341. [36] J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations, 11th AIAA Computational Fluid Dynamic Conference, Orlando, FL, USA. [37] J. Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J. Numer. Anal., 32 (1995), 386-403. doi: 10.1137/0732016. [38] W. C. Thacker, Some exact solutions to the non-linear shallow water wave equations, J. Fluid Mech., 107 (1981), 499-508. doi: 10.1017/S0022112081001882.

show all references

##### References:
 [1] N. Aïssiouene, M. O. Bristeau, E. Godlewski and J. Sainte-Marie, A robust and stable numerical scheme for a depth-averaged Euler system, Submitted. [2] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4. [3] B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J., 57 (2008), 97-131. doi: 10.1512/iumj.2008.57.3200. [4] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for Shallow Water flows, SIAM J. Sci. Comput., 25 (2004), 2050-2065. doi: 10.1137/S1064827503431090. [5] E. Audusse, F. Bouchut, M.-O. Bristeau and J. Sainte-Marie, Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system, 2014, URL http://hal.inria.fr/hal-01063577, (Submitted) http://hal.inria.fr/hal-01063577/PDF/kin_hydrost.pdf. [6] J.-L. Bona, T.-B. Benjamin and J.-J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Royal Soc. London Series A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032. [7] P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes, European Journal of Mechanics - B/Fluids, 30 (2011), 589-597, URL http://www.sciencedirect.com/science/article/pii/S0997754611000185, Special Issue: Nearshore Hydrodynamics. doi: 10.1016/j.euromechflu.2011.02.005. [8] F. Bouchut, An introduction to finite volume methods for hyperbolic conservation laws, ESAIM Proc., 15 (2005), 1-17. [9] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkhäuser, 2004. doi: 10.1007/b93802. [10] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129-151. [11] M.-O. Bristeau and B. Coussin, Boundary Conditions for the Shallow Water Equations Solved by Kinetic Schemes, Rapport de recherche RR-4282, INRIA, 2001, URL http://hal.inria.fr/inria-00072305, Projet M3N. [12] M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic Shallow Water model, Computers & Fluids, 47 (2011), 51-64. doi: 10.1016/j.compfluid.2011.02.013. [13] M. O. Bristeau, A. Mangeney, J. Sainte-Marie and N. Seguin, An energy-consistent depth-averaged Euler system: Derivation and properties, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 961-988, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10801. doi: 10.3934/dcdsb.2015.20.961. [14] M.-O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733-759. doi: 10.3934/dcdsb.2008.10.733. [15] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Math., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [16] F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model, J. Sci. Comput., 48 (2011), 105-116. doi: 10.1007/s10915-010-9395-9. [17] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2. [18] M.-W. Dingemans, Wave Propagation Over Uneven Bottoms, Advanced Series on Ocean Engineering - World Scientific, 1997. doi: 10.1142/9789812796042. [19] M. Dingemans, Comparison of Computations with Boussinesq-like Models and Laboratory Measurements, Technical Report H1684-12, AST G8M Coastal Morphodybamics Research Programme, 1994. [20] A. Duran and F. Marche, Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations, Communications in Computational Physics, 17 (2015), 721-760, URL https://hal.archives-ouvertes.fr/hal-00980826. doi: 10.4208/cicp.150414.101014a. [21] W. E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057. doi: 10.1137/0732047. [22] A. Ern and S. Meunier, A posteriori error analysis of euler-galerkin approximations to coupled elliptic-parabolic problems, ESAIM Math. Model. Numer. Anal., 43 (2009), 353-375, URL http://journals.cambridge.org/abstract\_S0764583X05000166. doi: 10.1051/m2an:2008048. [23] E. Godlewski and P.-A. Raviart, Numerical Approximations of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [24] A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425. [25] P. Gresho and S. Chan, Semi-consistent mass matrix techniques for solving the incompressible Navier-Stokes equations, First Int. Conf. on Comput. Methods in Flow Analysis, Okayama University, Japan. [26] J.-L. Guermond, Some implementations of projection methods for Navier-Stokes equations, ESAIM: Mathematical Modelling and Numerical Analysis, 30 (1996), 637-667, URL http://eudml.org/doc/193818. [27] J.-L. Guermond and J. Shen, On the error estimates for the rotational pressure-correction projection methods, Math. Comput., 73 (2004), 1719-1737, URL http://dblp.uni-trier.de/db/journals/moc/moc73.html#GuermondS04. doi: 10.1090/S0025-5718-03-01621-1. [28] H. Johnston and J.-G. Liu, Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, Journal of Computational Physics, 199 (2004), 221-259, URL http://www.sciencedirect.com/science/article/pii/S002199910400083X. doi: 10.1016/j.jcp.2004.02.009. [29] D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, 21 (2009), 016601. doi: 10.1063/1.3053183. [30] O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model, J. Comput. Phys., 229 (2010), 2034-2045. doi: 10.1016/j.jcp.2009.11.021. [31] R.-J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. [32] O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, 119 (1993), 618-638. doi: 10.1061/(ASCE)0733-950X(1993)119:6(618). [33] D. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827. doi: 10.1017/S0022112067002605. [34] O. Pironneau, Méthodes Des Éléments Finis Pour Les Fluides., Masson, 1988. [35] R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations, in The Navier-Stokes Equations II - Theory and Numerical Methods (eds. G. Heywood John, K. Masuda, R. Rautmann and A. Solonnikov Vsevolod), vol. 1530 of Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1992, 167-183, URL http://dx.doi.org/10.1007/BFb0090341. doi: 10.1007/BFb0090341. [36] J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations, 11th AIAA Computational Fluid Dynamic Conference, Orlando, FL, USA. [37] J. Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J. Numer. Anal., 32 (1995), 386-403. doi: 10.1137/0732016. [38] W. C. Thacker, Some exact solutions to the non-linear shallow water wave equations, J. Fluid Mech., 107 (1981), 499-508. doi: 10.1017/S0022112081001882.
 [1] Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, Nicolas Seguin. An energy-consistent depth-averaged Euler system: Derivation and properties. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 961-988. doi: 10.3934/dcdsb.2015.20.961 [2] Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715 [3] Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829 [4] Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557 [5] Eric Blayo, Antoine Rousseau. About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1565-1574. doi: 10.3934/dcdss.2016063 [6] Marcel Oliver. The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$. Communications on Pure and Applied Analysis, 2002, 1 (2) : 221-235. doi: 10.3934/cpaa.2002.1.221 [7] Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353 [8] Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315 [9] Dongho Chae, Kyungkeun Kang, Jihoon Lee. Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1181-1193. doi: 10.3934/dcds.2009.25.1181 [10] Erik Endres, Helge Kristian Jenssen, Mark Williams. Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier-Stokes shocks. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 133-169. doi: 10.3934/dcds.2010.27.133 [11] Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173 [12] Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17 [13] Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056 [14] Marie-Odile Bristeau, Jacques Sainte-Marie. Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 733-759. doi: 10.3934/dcdsb.2008.10.733 [15] Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure and Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353 [16] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 [17] Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41 [18] Vena Pearl Bongolan-walsh, David Cheban, Jinqiao Duan. Recurrent motions in the nonautonomous Navier-Stokes system. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 255-262. doi: 10.3934/dcdsb.2003.3.255 [19] Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207 [20] Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769

2020 Impact Factor: 1.213