Skip to main content
SpringerLink
Log in

Stochastic Variational Principles for Dissipative Equations with Advected Quantities

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

This paper presents symmetry reduction for material stochastic Lagrangian systems with advected quantities whose configuration space is a Lie group. Such variational principles yield deterministic as well as stochastic constrained variational principles for dissipative equations of motion in spatial representation. The general theory is presented for the finite-dimensional situation. In infinite dimensions we obtain partial differential equations and stochastic partial differential equations. When the Lie group is, for example, a diffeomorphism group, the general result is not directly applicable but the setup and method suggest rigorous proofs valid in infinite dimensions which lead to similar results. We apply this technique to the compressible Navier–Stokes equation and to magnetohydrodynamics for charged viscous compressible fluids. A stochastic Kelvin–Noether theorem is presented. We derive, among others, the classical deterministic dissipative equations from purely variational and stochastic principles, without any appeal to thermodynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Looking forward, we mention that the specification of the dual of \(\mathscr {M}_m\) is only for convenience. In fact, we do not need to be that precise because Eq. (3.12) in Theorem 3.5 does not depend on the expression of \(\frac{\delta p}{\delta A_j}\), \(j=1,2\), and hence does not depend on the choice of the dual of \(\mathscr {M}_m\) either. The reason is that in (3.14), with any choice for \(\frac{\delta p}{\delta A_j}\), \(j=1,2\), in a dual of \(\mathscr {M}_{m_1}\), the value of \(\left\langle \frac{\delta p}{\delta A_1}(A_1,A_2,u), B_{\omega ,1}(t,v)\right\rangle \) is the same and equals \(\lim _{\varepsilon \rightarrow 0}\frac{1}{\varepsilon }(p(A_1+ \epsilon B_{\omega ,1},A_2,u))\). The same holds for \(\frac{\delta p}{\delta A_2}\). Thus, the value of \(K(A_1,A_2,u)\) is independent of the choice of the dual.

References

  • Arnaudon, M., Cruzeiro, A.B.: Lagrangian Navier–Stokes diffusions on manifolds: variational principle and stability. Bull. Sci. Math. 136(8), 857–881 (2012)

    MathSciNet  MATH  Google Scholar 

  • Arnaudon, M., Chen, X., Cruzeiro, A.B.: Stochastic Euler–Poincaré reduction. J. Math. Phys. 55, 081507 (2014)

    MathSciNet  MATH  Google Scholar 

  • Arnold, V.I.: Conditions for nonlinear stability of the stationary plane curvilinear flows of an ideal fluid. Dokl. Mat. Nauk. 162(5), 773–777 (1965)

    Google Scholar 

  • Arnold, V.I.: On an a priori estimate in the theory of hydrodynamic stability. Engl. Transl. Am. Math. Soc. Transl. 19, 267–269 (1969)

    Google Scholar 

  • Bismut, J.-M.: Mécanique Aléatoire. Lecture Notes in Mathematics, vol. 866. Springer, Berlin (1981)

    Google Scholar 

  • Bismut, J.-M.: Mécanique aléatoire. In: Tenth Saint Flour Probability Summer School (Saint Flour, 1980). Lecture Notes in Mathematics, vol. 929, pp. 1–100. Springer, Berlin (1982)

    Google Scholar 

  • Bou-Rabee, N., Owhadi, H.: Stochastic variational integrators. IMA J. Numer. Anal. 29, 421–443 (2008)

    MathSciNet  MATH  Google Scholar 

  • Brenner, H.: Kinematics of volume transport. Physica A 349, 11–59 (2005)

    Google Scholar 

  • Brenner, H.: Navier–Stokes revisited. Physica A 349, 60–132 (2005)

    MathSciNet  Google Scholar 

  • Bruveris, M., Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: The momentum map representation of images. J. Nonlinear Sci. 21(1), 115–150 (2011)

    MathSciNet  MATH  Google Scholar 

  • Castrillón López, M., García Pérez, P., Ratiu, T.S.: Euler–Poincaré reduction on principal bundles. Lett. Math. Phys. 58(2), 167–180 (2001)

    MathSciNet  MATH  Google Scholar 

  • Castrillón López, M., Ratiu, T.S.: Reduction in principal bundles: covariant Lagrange–Poincaré equations. Commun. Math. Phys. 236(2), 223–250 (2003)

    MATH  Google Scholar 

  • Castrillón López, M., Ratiu, T.S., Shkoller, S.: Reduction in principal fiber bundles: covariant Euler–Poincaré equations. Proc. Am. Math. Soc. 128(7), 2155–2164 (2000)

    MATH  Google Scholar 

  • Cendra, H., Holm, D.D., Marsden, J.E., Ratiu, T.S.: Lagrangian reduction, the Euler–Poincaré equations, and semidirect products. In: Geometry of Differential Equations. American Mathematical Society, Translation Series 2, vol. 186, pp. 1–25. American Mathematical Society, Providence, RI (1998)

    MATH  Google Scholar 

  • Cendra, H., Marsden, J. E., Pekarsky, S., Ratiu, T.S.: Variational principles for Lie–Poisson and Hamilton–Poincaré equations. Mosc. Math. J. 3(3), 833–867, 1197–1198 (2003)

  • Cendra, H., Marsden, J.E., Ratiu, T.S.: Lagrangian reduction by stages. Mem. Am. Math. Soc. 152(722) (2001)

  • Cendra, H., Marsden, J.E., Ratiu, T.S.: Geometric Mechanics, Lagrangian Reduction, and Nonholonomic Systems. In: Mathematics Unlimited—2001 and Beyond, pp. 221–273. Springer, Berlin (2001)

  • Cipriano, F., Cruzeiro, A.B.: Navier–Stokes equation and diffusions on the group of homeomorphisms of the torus. Commun. Math. Phys. 275(1), 255–267 (2007)

    MathSciNet  MATH  Google Scholar 

  • Constantin, P., Iyer, G.: A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations. Commun. Pure Appl. Math. 61(3), 330–345 (2008)

    MathSciNet  MATH  Google Scholar 

  • Cotter, C.J., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: Numerically modelling stochastic lie transport in fluid dynamics. SIAM Multiscale Model. Simul. 17(1), 192–232 (2019). https://doi.org/10.1137/18M1167929

    Article  MathSciNet  MATH  Google Scholar 

  • Cotter, C.J., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: A particle filter for stochastic advection by Lie transport: a case study for the damped and forced incompressible two-dimensional Euler equation. SIAM/ASA J. Uncertainty Quant. 8(4), 1446–1492 (2020). https://doi.org/10.1137/19M1277606

    Article  MathSciNet  MATH  Google Scholar 

  • Cotter, C.J., Crisan, D., Holm, D.D., Pan, W., Shevchenko, I.: Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise. J. Stat. Phys. 179, 1186–1221 (2020). https://doi.org/10.1007/s10955-020-02524-0

    Article  MathSciNet  MATH  Google Scholar 

  • Cotter, C.J., Gottwald, G.A., Holm, D.D.: Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics. Proc. R. Soc. A 473, 20170388 (2020). https://doi.org/10.1098/rspa.2017.0388

    Article  MathSciNet  MATH  Google Scholar 

  • Cruzeiro, A.B., Holm, D.D., Ratiu, T.S.: Momentum maps and stochastic Clebsch action principles. Commun. Math. Phys. 357, 873–912 (2018). https://doi.org/10.1007/s00220-017-3048-x

    Article  MathSciNet  MATH  Google Scholar 

  • Demoures, F., Gay-Balmaz, F., Ratiu, T.S.: Multisymplectic variational integrators for nonsmooth Lagrangian continuum mechanics. Forum Math. Sigma 4, e19 (2016)

    MathSciNet  MATH  Google Scholar 

  • Ebin, D.G., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970)

    MathSciNet  MATH  Google Scholar 

  • Ellis, D., Gay-Balmaz, F., Holm, D.D., Putkaradze, V., Ratiu, T.S.: Symmetry reduced dynamics of charged molecular strands. Arch. Ration. Mech. Anal. 197(3), 811–902 (2010)

    MathSciNet  MATH  Google Scholar 

  • Ellis, D., Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: Lagrange–Poincaré field equations. J. Geom. Phys. 61(11), 2120–2146 (2011)

    MathSciNet  MATH  Google Scholar 

  • Emery, M.: Stochastic Calculus in Manifolds. Universitext. Springer, Berlin (1989)

    Google Scholar 

  • Eyink, G.L.: Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models. J. Math. Phys. 50, 083102 (2009)

    MathSciNet  MATH  Google Scholar 

  • Feireisl, E., Vasseur, A.: New perspectives in fluid dynamics: mathematical analysis of a model proposed by Howard Brenner. In: New Directions in Mathematical Fluid Mechanics, pp. 153–179. Birkhäuser, Basel (2010)

  • Fetecau, R.C., Marsden, J.E., Ortiz, M., West, M.: Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dyn. Syst. 2(3), 381–416 (2003)

    MathSciNet  MATH  Google Scholar 

  • Flandoli, F., Gubinelli, M., Priola, E.: Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180, 1–53 (2010)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F.: Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space. Ph.D. Thesis, Ecole Polytechnique Fédérale de Lausanne, November 21 (2008)

  • Gay-Balmaz, F., Holm, D.D., Meier, D., Ratiu, T.S., Vialard, F.-X.: Invariant higher-order variational problems. Commun. Math. Phys. 309(2), 413–458 (2012)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Holm, D.D., Meier, D., Ratiu, T.S., Vialard, F.-X.: Invariant higher-order variational problems II. J. Nonlinear Sci. 22(4), 553–597 (2012)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Holm, D.D., Putkaradze, V., Ratiu, T.S.: Exact geometric theory of dendronized polymer dynamics. Adv. Appl. Math. 48(4), 535–574 (2012)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: Variational principles for spin systems and the Kirchhoff rod. J. Geom. Mech. 1(4), 417–444 (2009)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: Higher order Lagrange–Poincaré and Hamilton–Poincaré reductions. Bull. Braz. Math. Soc. (N.S.) 42(4), 579–606 (2011)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: Geometric dynamics of optimization. Commun. Math. Sci. 11(1), 163–231 (2013)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Marsden, J.E., Ratiu, T.S.: Reduced variational formulations in free boundary continuum mechanics. J. Nonlinear Sci. 22(4), 463–497 (2012)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Monastyrsky, M., Ratiu, T.S.: Lagrangian reductions and integrable systems in condensed matter. Commun. Math. Phys. 335(2), 609–636 (2015)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Ratiu, T.S.: Reduced Lagrangian and Hamiltonian formulations of Euler–Yang–Mills fluids. J. Sympl. Geom. 6(2), 189–237 (2008)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Ratiu, T.S.: The geometric structure of complex fluids. Adv. Appl. Math. 42(2), 176–275 (2009)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Ratiu, T.S.: A new Lagrangian dynamic reduction in field theory. Ann. Inst. Fourier (Grenoble) 60(3), 1125–1160 (2010)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Ratiu, T.S.: Geometry of nonabelian charged fluids. Dyn. Partial Differ. Equ. 8(1), 5–19 (2011)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Ratiu, T.S.: Clebsch optimal control formulation in mechanics. J. Geom. Mech. 3(1), 41–79 (2011)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Ratiu, T.S., Tronci, C.: Euler–Poincaré approaches to nematodynamics. Acta Appl. Math. 120, 127–151 (2012)

    MathSciNet  MATH  Google Scholar 

  • Gay-Balmaz, F., Ratiu, T.S., Tronci, C.: Equivalent theories of liquid crystal dynamics. Arch. Ration. Mech. Anal. 210(3), 773–811 (2013)

    MathSciNet  MATH  Google Scholar 

  • Hochgerner, S., Ratiu, T.S.: Geometry of non-holonomic diffusion. J. Eur. Math. Soc. 17, 273–319 (2015)

    MathSciNet  MATH  Google Scholar 

  • Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. A 471(2176), 20140963 (2015)

    MathSciNet  MATH  Google Scholar 

  • Holm, D.D.: Stochastic parametrization of the Richardson triple. J. Nonlinear Sci. 29, 89–113 (2019)

    MathSciNet  MATH  Google Scholar 

  • Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler–Poincaré equations and semidirect products with applications to continuum mechanics. Adv. Math. 137, 1–81 (1998)

    MathSciNet  MATH  Google Scholar 

  • Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler–Poincaré equations in geophysical fluid dynamics. In: Large-Scale Atmosphere-Ocean Dynamics, vol. II, pp. 251–300. Cambridge University Press, Cambridge (2002)

  • Holm, D.D., Marsden, J.E., Ratiu, T.S., Weinstein, A.: Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics. Phys. Lett. A 98(1–2), 15–21 (1983)

    MathSciNet  Google Scholar 

  • Holm, D.D., Marsden, J.E., Ratiu, T.S., Weinstein, A.: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116 (1985)

    MathSciNet  MATH  Google Scholar 

  • Huang, Q., Zambrini, J.-C.: From second-order differential geometry to stochastic geometric mechanics. https://arxiv.org/abs/2201.03706 (2022)

  • Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. Universitext. Springer, Berlin (1981)

    MATH  Google Scholar 

  • Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49(10), 1295–1325 (2000)

    MathSciNet  MATH  Google Scholar 

  • Kobilarov, M., Marsden, J.E., Sukhatme, G.S.: Geometric discretization of nonholonomic systems with symmetries. Discrete Contin. Dyn. Syst. Ser. S 3(1), 61–84 (2010)

    MathSciNet  MATH  Google Scholar 

  • Koide, T., Kodama, T.: Navier–Stokes, Gross–Pitaevskii and generalized diffusion equations using the stochastic variational method. J. Phys. A 45(25), 255204 (2012)

    MathSciNet  MATH  Google Scholar 

  • Kunita, H.: Stochastic Flows and Stochastic Differential Equations. North-Holland Mathematical Library, Amsterdam (1990)

    MATH  Google Scholar 

  • Lázaro-Camí, J.-A., Ortega, J.-P.: Stochastic Hamiltonian dynamical systems. Rep. Math. Phys. 61(1), 65–122 (2008)

    MathSciNet  MATH  Google Scholar 

  • Lázaro-Camí, J.-A., Ortega, J.-P.: Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations. Stoch. Dyn. 9(1), 1–46 (2009)

    MathSciNet  MATH  Google Scholar 

  • Lázaro-Camí, J.-A., Ortega, J.-P.: Superposition rules and stochastic Lie–Scheffers systems. Ann. Inst. Henri Poincaré Probab. Stat. 45(4), 910–931 (2009)

    MathSciNet  MATH  Google Scholar 

  • Lázaro-Camí, J.-A., Ortega, J.-P.: The stochastic Hamilton–Jacobi equation. J. Geom. Mech. 1(3), 295–315 (2009)

    MathSciNet  MATH  Google Scholar 

  • Lew, A., Marsden, J.E., Ortiz, M., West, M.: Variational time integrators. Int. J. Numer. Methods Eng. 60(1), 153–212 (2004a)

    MathSciNet  MATH  Google Scholar 

  • Lew, A., Marsden, J.E., Ortiz, M., West, M.: An overview of variational integrators. In: Franca, L.P., Tezduyar, T.E. Masud, A. (eds.) Finite Element Methods: 1970s and Beyond, CIMNE, pp. 98–115 (2004b)

  • Leyendecker, S., Marsden, J.E., Ortiz, M.: Variational integrators for constrained dynamical systems. J. Appl. Math. Mech. 88(9), 677–708 (2008)

    MathSciNet  MATH  Google Scholar 

  • Leyendecker, S., Ober-Blöbaum, S.: A variational approach to multirate integration for constrained systems. In: Fisette, P., Samin, J.-C. (eds.) ECCOMAS Thematic Conference: Multibody Dynamics: Computational Methods and Applications, pp. 4–7. Brussels, Belgium (2011)

  • Leyendecker, S., Oberblöbaum, S., Marsden, J.E., Ortiz, M.: Discrete mechanics and optimal control for constrained systems. Optim. Control Appl. Methods 31(6), 505–528 (2010)

    MathSciNet  MATH  Google Scholar 

  • Malliavin, P.: Stochastic Analysis. Grundlehren der mathem Wissen. 313, Springer, Berlin (1997)

    MATH  Google Scholar 

  • Marsden, J.E., Ebin, D., Fischer, A.E.: Diffeomorphism groups, hydrodynamics and relativity. In: Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress, Differential Geometry and Applications, Dalhousie University, Halifax, NS, 1971, vol. 1, pp. 135–279. Canadaian Mathematical Congress, Montréal, QC (1972)

  • Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems. Texts in Applied Mathematics, vol. 17, 2nd edn. Springer, New York (1999)

    MATH  Google Scholar 

  • Marsden, J.E., Scheurle, J.: The reduced Euler–Lagrange equations. Fields Inst. Commun. 1, 139–164 (1993)

    MathSciNet  MATH  Google Scholar 

  • Marsden, J.E., Weinstein, A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Order in Chaos (Los Alamos, N.M., 1982). Physica D 7(1–3), 305–323 (1983)

    MathSciNet  Google Scholar 

  • Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)

    MathSciNet  MATH  Google Scholar 

  • Misiołek, G.: Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms. Indiana Univ. Math. J. 42(1), 215–235 (1993)

    MathSciNet  MATH  Google Scholar 

  • Moser, J., Veselov, A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 217–243 (1991)

    MathSciNet  MATH  Google Scholar 

  • Nakagomi, T., Yasue, K., Zambrini, J.-C.: Stochastic variational derivation of the Navier–Stokes equation. Lett. Math. Phys. 5(6), 545 (1981)

    MathSciNet  MATH  Google Scholar 

  • Nelson, E.: Dynamical Theories of Brownian motion. Princeton University Press, Princeton (1967)

    MATH  Google Scholar 

  • Noakes, L., Ratiu, T.S.: Bi–Jacobi fields and Riemannian cubics for left invariant \(SO(3)\). Commun. Math. Sci. 14(1), 55–68 (2016)

    MathSciNet  MATH  Google Scholar 

  • Ober-Blöbaum, S., Junge, O., Marsden, J.E.: Discrete mechanics and optimal control: an analysis. Control Optim. Calc. Var. 17(2), 322–352 (2011)

    MathSciNet  MATH  Google Scholar 

  • Poincaré, H.: Sur une forme nouvelle des équations de la mécanique. C. R. Acad. Sci. 132, 369–371 (1901)

    MATH  Google Scholar 

  • Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36(5), 635–664 (1983)

    MathSciNet  MATH  Google Scholar 

  • Suris, Y.B.: The Problem of Integrable Discretization: Hamiltonian Approach. Progress in Mathematics, 219. Birkhäuser Verlag, Basel (2003)

    Google Scholar 

  • Tao, M., Owhadi, H., Marsden, J.E.: Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8(4), 1269–1324 (2010)

    MathSciNet  MATH  Google Scholar 

  • Yasue, K.: Stochastic calculus of variations. J. Funct. Anal. 41(3), 327–340 (1981)

    MathSciNet  MATH  Google Scholar 

  • Zambrini, J.-C.: The research program of stochastic deformation (with a view toward geometric mechanics). In: Dalang, R.C., Dozzi, M., Flandoli, F., Russo, F. (eds.) Stochastic Analysis: A Series of Lectures. Progress in Probability 68. Springer, Berlin (2015)

    Google Scholar 

Download references

Acknowledgements

We thank Darryl Holm for his interest in our work and the many discussions we had about the geometric framework of stochastic mechanics. We also thank the anonymous referee for very constructive comments, additional references, and excellent suggestions for updating the introduction. We are grateful for the hospitality of the Bernoulli Center of the Swiss Federal Institute of Technology Lausanne and the Shanghai Jiao Tong University, which facilitated our collaboration.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dong Chuan Road, Minhang District, Shanghai, 200240, China

    Xin Chen & Tudor S. Ratiu

  2. Grupo de Física-Matemática and Departamento de Matemática, Instituto Superior Técnico, Univ. Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Ana Bela Cruzeiro

  3. Ministry of Education Laboratory of Scientific Computing (MOE-LSC), Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang, Shanghai, 200240, China

    Tudor S. Ratiu

  4. Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Tudor S. Ratiu

Authors
  1. Xin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana Bela Cruzeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tudor S. Ratiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ana Bela Cruzeiro.

Additional information

Communicated by Anthony Bloch.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Partially supported by National Natural Science Foundation of China (Nos. 11871338, 12122111).

Partially supported by project PTDC/MAT-STA/28812/2017 from the Portuguese F.C.T.

Partially supported by the National Natural Science Foundation of China (No. 11871334) and by the NCCR SwissMAP grant of the Swiss National Science Foundation

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Cruzeiro, A.B. & Ratiu, T.S. Stochastic Variational Principles for Dissipative Equations with Advected Quantities. J Nonlinear Sci 33, 5 (2023). https://doi.org/10.1007/s00332-022-09846-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-022-09846-1

Search

Navigation