August  2019, 39(8): 4471-4486. doi: 10.3934/dcds.2019183

Geophysical internal equatorial waves of extreme form

Department of Computing & Mathematics, Waterford Institute of Technology, Waterford, Ireland

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received  August 2018 Revised  October 2018 Published  May 2019

The existence of internal geophysical waves of extreme form is confirmed and an explicit solution presented. The flow is confined to a layer lying above an eastward current while the mean horizontal flow of the solutions is westward, thus incorporating flow reversal in the fluid.

Citation: Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183
References:
[1]

C. J. AmickL. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), 193-214.  doi: 10.1007/BF02392728.  Google Scholar

[2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511734939.  Google Scholar
[3] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton University Press, 2003.  doi: 10.1515/9781400884339.  Google Scholar
[4]

A. Compelli and R. Ivanov, On the dynamics of internal waves interacting with the equatorial undercurrent, J. Nonlin. Math. Phys., 22 (2015), 531-539.  doi: 10.1080/14029251.2015.1113052.  Google Scholar

[5]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[6]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.  Google Scholar

[7]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012). doi: 10.1029/2012JC007879.  Google Scholar

[8]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012). doi: 10.1029/2012GL051169.  Google Scholar

[9]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.  Google Scholar

[10]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanog., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[11]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789.  doi: 10.1175/JPO-D-13-0174.1.  Google Scholar

[12]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219.  Google Scholar

[13]

A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids, 29 (2017), 056604. Google Scholar

[14]

A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528.  doi: 10.1017/jfm.2017.223.  Google Scholar

[15]

R. Courant, Differential and Integral Calculus, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[16]

B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101, Academic Press, 2011.  Google Scholar

[17]

M. L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides, Atti. Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat., 15 (1932), 814-819.   Google Scholar

[18]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16 (2014), 661-667.  doi: 10.1007/s00021-014-0175-4.  Google Scholar

[19]

F. Gerstner, Theorie der wellen samt einer daraus abgeleiteten theorie der deichprofile, Ann. Phys., 2 (1809), 412-445.   Google Scholar

[20]

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, Pearson International Edition, Addison Wesley, 2002. Google Scholar

[21]

D. Henry, On Gerstner's water wave, J. Nonlin. Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[22]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21.  doi: 10.1016/j.euromechflu.2012.10.001.  Google Scholar

[23]

D. Henry, On three-dimensional Gerstner-like equatorial water waves, Phil. Trans. R. Soc. A, 376 (2018), 20170088, 16pp. doi: 10.1098/rsta.2017.0088.  Google Scholar

[24]

D. Henry and H. C. Hsu, Instability of internal equatorial water waves, J. Diff. Eqn., 258 (2015), 1015-1024.  doi: 10.1016/j.jde.2014.08.019.  Google Scholar

[25]

D. Henry and S. Sastre-Gomez, Mean flow velocities and mass transport for equatorially-trapped water waves with an underlying current, J. Math. Fluid Mech., 18 (2016), 795-804.  doi: 10.1007/s00021-016-0262-9.  Google Scholar

[26]

D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phil. Trans. R. Soc. A, 376 (2018), 20170090, 21pp. doi: 10.1098/rsta.2017.0090.  Google Scholar

[27]

M. Kluczek, Exact and explicit internal equatorial water waves with underlying currents, J. Math. Fluid Mech., 19 (2017), 305-314.  doi: 10.1007/s00021-016-0281-6.  Google Scholar

[28]

T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Disc. Contin. Dyn. Sys. Ser. A, 34 (2014), 3095-3107.  doi: 10.3934/dcds.2014.34.3095.  Google Scholar

[29]

T. Lyons, The pressure distribution in extreme Stokes waves, Nonlin. Anal. Real World Appl., 31 (2016), 77-87.  doi: 10.1016/j.nonrwa.2016.01.008.  Google Scholar

[30]

T. Lyons, The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209–218. doi: 10.1007/s00021-016-0249-6.  Google Scholar

[31]

T. Lyons, The dynamic pressure in deep-water extreme Stokes waves, Phil. Trans. R. Soc. A, 376 (2018), 20170095, 13pp. doi: 10.1098/rsta.2017.0095.  Google Scholar

[32]

C. I. Martin, Dynamics of the thermocline in the equatorial region of the pacific ocean, J. Nonlin. Math. Phys., 22 (2015), 516-522.  doi: 10.1080/14029251.2015.1113049.  Google Scholar

[33]

A. V. Matioc, Exact geophysical waves in stratified fluids, Appl, Anal., 92 (2013), 2254-2261.  doi: 10.1080/00036811.2012.727987.  Google Scholar

[34]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. R. Soc. London A, 153 (1863), 127-138.   Google Scholar

[35]

A. Rodriguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Equatorially-trapped internal water waves, Nonlin. Anal.: Theor., Meth. Appl., 149 (2017), 156-164.  doi: 10.1016/j.na.2016.10.022.  Google Scholar

[36]

A. Rodriguez-Sanjurjo and M. Kluczek, Mean flow properties for equatorially trapped internal water wave–current interactions, Appl. Anal., 96 (2017), 2333-2345.  doi: 10.1080/00036811.2016.1221943.  Google Scholar

[37]

K. W. S and M. M. J, Oceaninc equatorial waves and the 1991–1993 El Niño, J. Climate, 8 (1995), 1757-1774.   Google Scholar

[38]

S. Sastre-Gomez, Global diffeomorphism of the Lagrangian flow-map defining equatorially trapped water waves, Nonlin. Anal., 125 (2015), 725-731.  doi: 10.1016/j.na.2015.06.017.  Google Scholar

[39]

G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, Mathematical and Physical Papers, 1 (1880), 225-228.   Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Meth. Nonlin. Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001.  Google Scholar

[41]

G. K. VallisD. J. Parker and S. M. Tobias, A simple system for moist convection: The rainy-Benard model, J. Fluid Mech., 862 (2019), 162-199.  doi: 10.1017/jfm.2018.954.  Google Scholar

[42]

S. H. C. Wacogne, Dynamics of the Equatorial Undercurrent and Its Termination, PhD thesis, Massachusetts Institute of Technology, 1988. Google Scholar

show all references

References:
[1]

C. J. AmickL. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), 193-214.  doi: 10.1007/BF02392728.  Google Scholar

[2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511734939.  Google Scholar
[3] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton University Press, 2003.  doi: 10.1515/9781400884339.  Google Scholar
[4]

A. Compelli and R. Ivanov, On the dynamics of internal waves interacting with the equatorial undercurrent, J. Nonlin. Math. Phys., 22 (2015), 531-539.  doi: 10.1080/14029251.2015.1113052.  Google Scholar

[5]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[6]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.  Google Scholar

[7]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012). doi: 10.1029/2012JC007879.  Google Scholar

[8]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012). doi: 10.1029/2012GL051169.  Google Scholar

[9]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.  Google Scholar

[10]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanog., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[11]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789.  doi: 10.1175/JPO-D-13-0174.1.  Google Scholar

[12]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219.  Google Scholar

[13]

A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids, 29 (2017), 056604. Google Scholar

[14]

A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528.  doi: 10.1017/jfm.2017.223.  Google Scholar

[15]

R. Courant, Differential and Integral Calculus, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[16]

B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101, Academic Press, 2011.  Google Scholar

[17]

M. L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides, Atti. Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat., 15 (1932), 814-819.   Google Scholar

[18]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16 (2014), 661-667.  doi: 10.1007/s00021-014-0175-4.  Google Scholar

[19]

F. Gerstner, Theorie der wellen samt einer daraus abgeleiteten theorie der deichprofile, Ann. Phys., 2 (1809), 412-445.   Google Scholar

[20]

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, Pearson International Edition, Addison Wesley, 2002. Google Scholar

[21]

D. Henry, On Gerstner's water wave, J. Nonlin. Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[22]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21.  doi: 10.1016/j.euromechflu.2012.10.001.  Google Scholar

[23]

D. Henry, On three-dimensional Gerstner-like equatorial water waves, Phil. Trans. R. Soc. A, 376 (2018), 20170088, 16pp. doi: 10.1098/rsta.2017.0088.  Google Scholar

[24]

D. Henry and H. C. Hsu, Instability of internal equatorial water waves, J. Diff. Eqn., 258 (2015), 1015-1024.  doi: 10.1016/j.jde.2014.08.019.  Google Scholar

[25]

D. Henry and S. Sastre-Gomez, Mean flow velocities and mass transport for equatorially-trapped water waves with an underlying current, J. Math. Fluid Mech., 18 (2016), 795-804.  doi: 10.1007/s00021-016-0262-9.  Google Scholar

[26]

D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phil. Trans. R. Soc. A, 376 (2018), 20170090, 21pp. doi: 10.1098/rsta.2017.0090.  Google Scholar

[27]

M. Kluczek, Exact and explicit internal equatorial water waves with underlying currents, J. Math. Fluid Mech., 19 (2017), 305-314.  doi: 10.1007/s00021-016-0281-6.  Google Scholar

[28]

T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Disc. Contin. Dyn. Sys. Ser. A, 34 (2014), 3095-3107.  doi: 10.3934/dcds.2014.34.3095.  Google Scholar

[29]

T. Lyons, The pressure distribution in extreme Stokes waves, Nonlin. Anal. Real World Appl., 31 (2016), 77-87.  doi: 10.1016/j.nonrwa.2016.01.008.  Google Scholar

[30]

T. Lyons, The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209–218. doi: 10.1007/s00021-016-0249-6.  Google Scholar

[31]

T. Lyons, The dynamic pressure in deep-water extreme Stokes waves, Phil. Trans. R. Soc. A, 376 (2018), 20170095, 13pp. doi: 10.1098/rsta.2017.0095.  Google Scholar

[32]

C. I. Martin, Dynamics of the thermocline in the equatorial region of the pacific ocean, J. Nonlin. Math. Phys., 22 (2015), 516-522.  doi: 10.1080/14029251.2015.1113049.  Google Scholar

[33]

A. V. Matioc, Exact geophysical waves in stratified fluids, Appl, Anal., 92 (2013), 2254-2261.  doi: 10.1080/00036811.2012.727987.  Google Scholar

[34]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. R. Soc. London A, 153 (1863), 127-138.   Google Scholar

[35]

A. Rodriguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Equatorially-trapped internal water waves, Nonlin. Anal.: Theor., Meth. Appl., 149 (2017), 156-164.  doi: 10.1016/j.na.2016.10.022.  Google Scholar

[36]

A. Rodriguez-Sanjurjo and M. Kluczek, Mean flow properties for equatorially trapped internal water wave–current interactions, Appl. Anal., 96 (2017), 2333-2345.  doi: 10.1080/00036811.2016.1221943.  Google Scholar

[37]

K. W. S and M. M. J, Oceaninc equatorial waves and the 1991–1993 El Niño, J. Climate, 8 (1995), 1757-1774.   Google Scholar

[38]

S. Sastre-Gomez, Global diffeomorphism of the Lagrangian flow-map defining equatorially trapped water waves, Nonlin. Anal., 125 (2015), 725-731.  doi: 10.1016/j.na.2015.06.017.  Google Scholar

[39]

G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, Mathematical and Physical Papers, 1 (1880), 225-228.   Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Meth. Nonlin. Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001.  Google Scholar

[41]

G. K. VallisD. J. Parker and S. M. Tobias, A simple system for moist convection: The rainy-Benard model, J. Fluid Mech., 862 (2019), 162-199.  doi: 10.1017/jfm.2018.954.  Google Scholar

[42]

S. H. C. Wacogne, Dynamics of the Equatorial Undercurrent and Its Termination, PhD thesis, Massachusetts Institute of Technology, 1988. Google Scholar

Figure 1.  The rotating $ \left(x,y,z\right) $-coordinate system fixed to the surface of the Earth. The $ x $-axis points due-east, the $ y $-axis points due-north and the $ z $-axis points vertically upwards from the surface
Figure 2.  A cross section of the flow in the equatorial plane $ y = 0 $ with a flow wavelength of $ L = 200\,\mathrm{m} $. The average depth of the near surface layer is $ 60\,\mathrm{m} $, while the thermocline lies at an average depth of $ 120\,\mathrm{m} $. The transitional layer begins at $ 160\, \mathrm{m} $ approximately, while the motionless layer begins at $ 200\,\mathrm{m} $ beneath the free surface
Figure 3.  The circular paths in the equatorial plane $ y = 0 $ traced by particles in the layer $ \mathcal{M}(t) $ are circles with centre $ (q,r-d_0) $ whose radius increases with depth. The wavelength of the solution shown is $ L = 200\ \mathrm{m} $. The particle trajectories are counterclockwise and the particle positions shown in the figure are at time $ t = 0 $
Figure 4.  A pictorial outline for the proof of Theorem 3.2, confirming the existence of a solution of the implicit relation $ G(s,r) = G(0,0) $ of extreme form
Figure 5.  The left-hand panel shows the graph $ (s,r_0(s)) $ when $ \kappa = 0.5 $ and $ L = 200 $ meters. The parameter $ d_0 = 120 $ ensures a mean equatorial depth of $ 120\, $m. In the right-hand panel the corresponding thermocline profile in the fluid domain, at the equatorial locations indicated in the figure
Figure 6.  The thermocline surface when $ L = 200\, $m, $ r_0^* = 0\, $m and $ d_0 = 120\, $m, with $ \kappa = 0.5 $. The surface is viewed from beneath and the value of $ \kappa $ has been chosen to emphasise the extreme behaviour of the wave along the equator at the wave-troughs
[1]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[2]

Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020364

[3]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[4]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[5]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[6]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[7]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[8]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561

[9]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328

[10]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021008

[11]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156

[12]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147

[13]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[14]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[15]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[16]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328

[17]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

[18]

Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49

[19]

Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265

[20]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (104)
  • HTML views (151)
  • Cited by (1)

Other articles
by authors

[Back to Top]