Variational principles for nonlinear PDE systems via duality
Author:
Amit Acharya
Journal:
Quart. Appl. Math. 81 (2023), 127-140
MSC (2020):
Primary 49S05, 76D05, 49N15
DOI:
https://doi.org/10.1090/qam/1631
Published electronically:
September 26, 2022
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Additional Information
Abstract: A formal methodology for developing variational principles corresponding to a given nonlinear PDE system is discussed. The scheme is demonstrated in the context of the incompressible Navier-Stokes equations, systems of first-order conservation laws, and systems of Hamilton-Jacobi equations.
References
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- Scott Armstrong, Tuomo Kuusi, and Jean-Christophe Mourrat, Quantitative stochastic homogenization and large-scale regularity, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 352, Springer, Cham, 2019. MR 3932093, DOI 10.1007/978-3-030-15545-2
- Rajat Arora and Amit Acharya, A unification of finite deformation $J_2$ von-Mises plasticity and quantitative dislocation mechanics, J. Mech. Phys. Solids 143 (2020), 104050, 30. MR 4118895, DOI 10.1016/j.jmps.2020.104050
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- A. J. Beekman, J. Nissinen, K. Wu, and J. Zaanen, Dual gauge field theory of quantum liquid crystals in three dimensions, Physical Review B 96 (2017), no. 16, 165115.
- Yann Brenier, The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem, Comm. Math. Phys. 364 (2018), no. 2, 579–605. MR 3869437, DOI 10.1007/s00220-018-3240-7
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 4th ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2016. MR 3468916, DOI 10.1007/978-3-662-49451-6
- Nassif Ghoussoub and Abbas Moameni, Anti-symmetric Hamiltonians. II. Variational resolutions for Navier-Stokes and other nonlinear evolutions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 1, 223–255 (English, with English and French summaries). MR 2483820, DOI 10.1016/j.anihpc.2007.11.002
- Sean A. Hartnoll, Andrew Lucas, and Subir Sachdev, Holographic quantum matter, MIT Press, Cambridge, MA, 2018. MR 3823229
- Ryan Hynd, The Hamilton-Jacobi equation, then and now, Notices Amer. Math. Soc. 68 (2021), no. 9, 1457–1467. MR 4323817, DOI 10.1090/noti2352
- R. R. Kerswell, Variational principle for the Navier-Stokes equations, Phys. Rev. E (3) 59 (1999), no. 5, 5482–5494. MR 1690929, DOI 10.1103/PhysRevE.59.5482
- H. Kleinert, Gauge fields in condensed matter. Vol I: Superflow and Vortex lines, World Scientific, Singapore Teaneck, N.J, 1989.
- G. L. Liu, Dual variational principles for 3-d Navier-Stokes equations, New Trends in Fluid Mechanics Research, Springer, 2007, pp. 734–735.
- Léo Morin and Amit Acharya, Analysis of a model of field crack mechanics for brittle materials, Comput. Methods Appl. Mech. Engrg. 386 (2021), Paper No. 114061, 39. MR 4299715, DOI 10.1016/j.cma.2021.114061
- Michael Ortiz, Bernd Schmidt, and Ulisse Stefanelli, A variational approach to Navier-Stokes, Nonlinearity 31 (2018), no. 12, 5664–5682. MR 3876558, DOI 10.1088/1361-6544/aae722
- R. Tyrrell Rockafellar, Conjugate duality and optimization, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973. MR 0373611, DOI 10.1137/1.9781611970524
- R. L. Seliger and G. B. Whitham, Variational principles in continuum mechanics, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 305 (1968), no. 1480, 1–25.
- A. Visintin, Minimization vs. null-minimization: a note about the Fitzpatrick theory, https://www.mathematik.tu-darmstadt.de/media/isimm/ISIMM-Forum_Visintin_1409.pdf, 2014.
- J. Zaanen, Y. Liu, Y-W Sun, and K. Schalm, Holographic duality in condensed matter physics, Cambridge University Press, 2015.
- Xiaohan Zhang, Amit Acharya, Noel J. Walkington, and Jacobo Bielak, A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations, J. Mech. Phys. Solids 84 (2015), 145–195. MR 3413434, DOI 10.1016/j.jmps.2015.07.004
References
- Amit Acharya, An action for nonlinear dislocation dynamics, J. Mech. Phys. Solids 161 (2022), Paper No. 104811, 14. MR 4387914, DOI 10.1016/j.jmps.2022.104811
- Scott Armstrong, Tuomo Kuusi, and Jean-Christophe Mourrat, Quantitative stochastic homogenization and large-scale regularity, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 352, Springer, Cham, 2019. MR 3932093, DOI 10.1007/978-3-030-15545-2
- Rajat Arora and Amit Acharya, A unification of finite deformation $J_2$ von-Mises plasticity and quantitative dislocation mechanics, J. Mech. Phys. Solids 143 (2020), 104050, 30. MR 4118895, DOI 10.1016/j.jmps.2020.104050
- Aron J. Beekman, Jaakko Nissinen, Kai Wu, Ke Liu, Robert-Jan Slager, Zohar Nussinov, Vladimir Cvetkovic, and Jan Zaanen, Dual gauge field theory of quantum liquid crystals in two dimensions, Phys. Rep. 683 (2017), 1–110. MR 3654715, DOI 10.1016/j.physrep.2017.03.004
- A. J. Beekman, J. Nissinen, K. Wu, and J. Zaanen, Dual gauge field theory of quantum liquid crystals in three dimensions, Physical Review B 96 (2017), no. 16, 165115.
- Yann Brenier, The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem, Comm. Math. Phys. 364 (2018), no. 2, 579–605. MR 3869437, DOI 10.1007/s00220-018-3240-7
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 4th ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2016. MR 3468916, DOI 10.1007/978-3-662-49451-6
- Nassif Ghoussoub and Abbas Moameni, Anti-symmetric Hamiltonians. II. Variational resolutions for Navier-Stokes and other nonlinear evolutions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 1, 223–255 (English, with English and French summaries). MR 2483820, DOI 10.1016/j.anihpc.2007.11.002
- Sean A. Hartnoll, Andrew Lucas, and Subir Sachdev, Holographic quantum matter, MIT Press, Cambridge, MA, 2018. MR 3823229
- Ryan Hynd, The Hamilton-Jacobi equation, then and now, Notices Amer. Math. Soc. 68 (2021), no. 9, 1457–1467. MR 4323817, DOI 10.1090/noti2352
- R. R. Kerswell, Variational principle for the Navier-Stokes equations, Phys. Rev. E (3) 59 (1999), no. 5, 5482–5494. MR 1690929, DOI 10.1103/PhysRevE.59.5482
- H. Kleinert, Gauge fields in condensed matter. Vol I: Superflow and Vortex lines, World Scientific, Singapore Teaneck, N.J, 1989.
- G. L. Liu, Dual variational principles for 3-d Navier-Stokes equations, New Trends in Fluid Mechanics Research, Springer, 2007, pp. 734–735.
- Léo Morin and Amit Acharya, Analysis of a model of field crack mechanics for brittle materials, Comput. Methods Appl. Mech. Engrg. 386 (2021), Paper No. 114061, 39. MR 4299715, DOI 10.1016/j.cma.2021.114061
- Michael Ortiz, Bernd Schmidt, and Ulisse Stefanelli, A variational approach to Navier-Stokes, Nonlinearity 31 (2018), no. 12, 5664–5682. MR 3876558, DOI 10.1088/1361-6544/aae722
- R. Tyrrell Rockafellar, Conjugate duality and optimization, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973. MR 0373611
- R. L. Seliger and G. B. Whitham, Variational principles in continuum mechanics, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 305 (1968), no. 1480, 1–25.
- A. Visintin, Minimization vs. null-minimization: a note about the Fitzpatrick theory, https://www.mathematik.tu-darmstadt.de/media/isimm/ISIMM-Forum_Visintin_1409.pdf, 2014.
- J. Zaanen, Y. Liu, Y-W Sun, and K. Schalm, Holographic duality in condensed matter physics, Cambridge University Press, 2015.
- Xiaohan Zhang, Amit Acharya, Noel J. Walkington, and Jacobo Bielak, A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations, J. Mech. Phys. Solids 84 (2015), 145–195. MR 3413434, DOI 10.1016/j.jmps.2015.07.004
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Additional Information
Amit Acharya
Affiliation:
Department of Civil & Environmental Engineering, and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
MR Author ID:
368246
ORCID:
0000-0002-6184-3357
Email:
acharyaamit@cmu.edu.
Received by editor(s):
August 1, 2022
Received by editor(s) in revised form:
August 20, 2022
Published electronically:
September 26, 2022
Additional Notes:
This work was supported by the grant NSF OIA-DMR #2021019.
Article copyright:
© Copyright 2022
Brown University
