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On the Lagrangian structure of quantum fluid models
1. | Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria, Austria |
2. | Institute for Mathematics, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany |
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Second edition, (2008).
|
[2] |
A. Arnold, Mathematical properties of quantum evolution equations,, in, 1946 (2008), 45.
doi: 10.1007/978-3-540-79574-2_2. |
[3] |
V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319.
doi: 10.5802/aif.233. |
[4] |
J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2000), 375.
doi: 10.1007/s002110050002. |
[5] |
S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model,, Z. Angew. Math. Mech., 90 (2010), 219.
doi: 10.1002/zamm.200900297. |
[6] |
B. van Brunt, "The Calculus of Variations,'', Universitext, (2004).
|
[7] |
P. Dirac, The Lagrangian in quantum mechanics,, Phys. Z. Sowjet., 3 (1933), 64. Google Scholar |
[8] |
Dj. Djukic and B. Vujanović, Noether's theory in classical nonconservative mechanics,, Acta Mech., 23 (1975), 17.
|
[9] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids,'', Oxford Lecture Series in Mathematics and its Applications, 26 (2004).
|
[10] |
J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws,, J. Math. Pures Appl. (9), 97 (2012), 318.
doi: 10.1016/j.matpur.2011.11.004. |
[11] |
L. Brown, ed., "Feynman's Thesis. A New Approach to Quantum Theory,'', World Scientific Publishing Co. Pte. Ltd., (2005).
doi: 10.1142/9789812567635. |
[12] |
T. Frankel, "The Geometry of Physics. An Introduction,'', Cambridge University Press, (1997).
|
[13] |
G. Frederico and D. Torres, Nonconservative Noether's theorem in optimal control,, Intern. J. Tomogr. Stat., 5 (2007), 109.
|
[14] |
J.-L. Fu and L.-Q. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems,, Phys. Lett. A, 317 (2003), 255.
doi: 10.1016/j.physleta.2003.08.028. |
[15] |
A. Jüngel, "Transport Equations for Semiconductors,'', Lecture Notes in Physics, 773 (2009).
doi: 10.1007/978-3-540-89526-8. |
[16] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids,, SIAM J. Math. Anal., 42 (2010), 1025.
doi: 10.1137/090776068. |
[17] |
A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach,, J. Stat. Phys., 145 (2011), 1661.
doi: 10.1007/s10955-011-0388-3. |
[18] |
A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution,, Kinetic Related Models, 4 (2011), 785.
doi: 10.3934/krm.2011.4.785. |
[19] |
J. Lafferty, The density manifold and configuration space quantization,, Trans. Amer. Math. Soc., 305 (1988), 699.
doi: 10.1090/S0002-9947-1988-0924776-9. |
[20] |
J. Lott, Some geometric calculations on Wasserstein space,, Commun. Math. Phys., 277 (2008), 423.
doi: 10.1007/s00220-007-0367-3. |
[21] |
E. Madelung, Quantentheorie in hydrodynamischer Form,, Z. Phys., 40 (1926), 322.
doi: 10.1007/BF01400372. |
[22] |
P. Markowich, T. Paul and C. Sparber, Bohmian measures and their classical limit,, J. Funct. Anal., 259 (2010), 1542.
doi: 10.1016/j.jfa.2010.05.013. |
[23] |
R. McCann, Polar factorization of maps on Riemannian manifolds,, GAFA Geom. Funct. Anal., 11 (2001), 589.
doi: 10.1007/PL00001679. |
[24] |
E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics,, Phys. Rev., 150 (1966), 1079.
doi: 10.1103/PhysRev.150.1079. |
[25] |
F. Otto, The geometry of dissipative evolution equations: The porous-medium equation,, Commun. Part. Diff. Eqs., 26 (2001), 101.
doi: 10.1081/PDE-100002243. |
[26] |
F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality,, J. Funct. Anal., 173 (2000), 361.
doi: 10.1006/jfan.1999.3557. |
[27] |
M.-K. von Renesse, On optimal transport view on Schrödinger's equation,, Canad. Math. Bull., 55 (2012), 858.
doi: 10.4153/CMB-2011-121-9. |
[28] |
W. Sarlett and F. Cantrijn, Generalization of Noether's Theorem in classical mechanics,, SIAM Review, 23 (1981), 467.
doi: 10.1137/1023098. |
[29] |
R. Talman, "Geometric Mechanics,'', Wiley, (2000). Google Scholar |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Second edition, (2008).
|
[2] |
A. Arnold, Mathematical properties of quantum evolution equations,, in, 1946 (2008), 45.
doi: 10.1007/978-3-540-79574-2_2. |
[3] |
V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319.
doi: 10.5802/aif.233. |
[4] |
J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2000), 375.
doi: 10.1007/s002110050002. |
[5] |
S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model,, Z. Angew. Math. Mech., 90 (2010), 219.
doi: 10.1002/zamm.200900297. |
[6] |
B. van Brunt, "The Calculus of Variations,'', Universitext, (2004).
|
[7] |
P. Dirac, The Lagrangian in quantum mechanics,, Phys. Z. Sowjet., 3 (1933), 64. Google Scholar |
[8] |
Dj. Djukic and B. Vujanović, Noether's theory in classical nonconservative mechanics,, Acta Mech., 23 (1975), 17.
|
[9] |
E. Feireisl, "Dynamics of Viscous Compressible Fluids,'', Oxford Lecture Series in Mathematics and its Applications, 26 (2004).
|
[10] |
J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws,, J. Math. Pures Appl. (9), 97 (2012), 318.
doi: 10.1016/j.matpur.2011.11.004. |
[11] |
L. Brown, ed., "Feynman's Thesis. A New Approach to Quantum Theory,'', World Scientific Publishing Co. Pte. Ltd., (2005).
doi: 10.1142/9789812567635. |
[12] |
T. Frankel, "The Geometry of Physics. An Introduction,'', Cambridge University Press, (1997).
|
[13] |
G. Frederico and D. Torres, Nonconservative Noether's theorem in optimal control,, Intern. J. Tomogr. Stat., 5 (2007), 109.
|
[14] |
J.-L. Fu and L.-Q. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems,, Phys. Lett. A, 317 (2003), 255.
doi: 10.1016/j.physleta.2003.08.028. |
[15] |
A. Jüngel, "Transport Equations for Semiconductors,'', Lecture Notes in Physics, 773 (2009).
doi: 10.1007/978-3-540-89526-8. |
[16] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids,, SIAM J. Math. Anal., 42 (2010), 1025.
doi: 10.1137/090776068. |
[17] |
A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach,, J. Stat. Phys., 145 (2011), 1661.
doi: 10.1007/s10955-011-0388-3. |
[18] |
A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution,, Kinetic Related Models, 4 (2011), 785.
doi: 10.3934/krm.2011.4.785. |
[19] |
J. Lafferty, The density manifold and configuration space quantization,, Trans. Amer. Math. Soc., 305 (1988), 699.
doi: 10.1090/S0002-9947-1988-0924776-9. |
[20] |
J. Lott, Some geometric calculations on Wasserstein space,, Commun. Math. Phys., 277 (2008), 423.
doi: 10.1007/s00220-007-0367-3. |
[21] |
E. Madelung, Quantentheorie in hydrodynamischer Form,, Z. Phys., 40 (1926), 322.
doi: 10.1007/BF01400372. |
[22] |
P. Markowich, T. Paul and C. Sparber, Bohmian measures and their classical limit,, J. Funct. Anal., 259 (2010), 1542.
doi: 10.1016/j.jfa.2010.05.013. |
[23] |
R. McCann, Polar factorization of maps on Riemannian manifolds,, GAFA Geom. Funct. Anal., 11 (2001), 589.
doi: 10.1007/PL00001679. |
[24] |
E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics,, Phys. Rev., 150 (1966), 1079.
doi: 10.1103/PhysRev.150.1079. |
[25] |
F. Otto, The geometry of dissipative evolution equations: The porous-medium equation,, Commun. Part. Diff. Eqs., 26 (2001), 101.
doi: 10.1081/PDE-100002243. |
[26] |
F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality,, J. Funct. Anal., 173 (2000), 361.
doi: 10.1006/jfan.1999.3557. |
[27] |
M.-K. von Renesse, On optimal transport view on Schrödinger's equation,, Canad. Math. Bull., 55 (2012), 858.
doi: 10.4153/CMB-2011-121-9. |
[28] |
W. Sarlett and F. Cantrijn, Generalization of Noether's Theorem in classical mechanics,, SIAM Review, 23 (1981), 467.
doi: 10.1137/1023098. |
[29] |
R. Talman, "Geometric Mechanics,'', Wiley, (2000). Google Scholar |
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