Euler-Lagrange equation for a minimization problem over monotone transport maps
Author:
Michael Westdickenberg
Journal:
Quart. Appl. Math. 75 (2017), 267-285
MSC (2010):
Primary 35L65, 49J40, 82C40
DOI:
https://doi.org/10.1090/qam/1459
Published electronically:
November 14, 2016
MathSciNet review:
3614498
Full-text PDF
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Additional Information
Abstract: A variational time discretization for the compressible Euler equations has been introduced recently. It involves a minimization problem over the cone of monotone transport maps in each timestep. A matrix-valued measure field appears in the minimization as a Lagrange multiplier for the monotonicity constraint. We show that the absolutely continuous part of this measure field vanishes in the support of the density.
References
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- Fabio Cavalletti and Michael Westdickenberg, The polar cone of the set of monotone maps, Proc. Amer. Math. Soc. 143 (2015), no. 2, 781–787. MR 3283664, DOI https://doi.org/10.1090/S0002-9939-2014-12332-X
- F. Cavalletti, M. Sedjro, and M. Westdickenberg, A variational time discretization for compressible Euler equations (2014), available at arXiv:1411.1012v3.
- Fabio Cavalletti, Marc Sedjro, and Michael Westdickenberg, A simple proof of global existence for the 1D pressureless gas dynamics equations, SIAM J. Math. Anal. 47 (2015), no. 1, 66–79. MR 3296602, DOI https://doi.org/10.1137/130945296
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- Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR 1681462
- Wilfrid Gangbo and Michael Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations 34 (2009), no. 7-9, 1041–1073. MR 2560310, DOI https://doi.org/10.1080/03605300902892345
- N. Ghoussoub, A variational theory for monotone vector fields, J. Fixed Point Theory Appl. 4 (2008), no. 1, 107–135.
- Pierre-Louis Lions, Identification du cône dual des fonctions convexes et applications, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 12, 1385–1390 (French, with English and French summaries). MR 1649179, DOI https://doi.org/10.1016/S0764-4442%2898%2980397-2
- Luca Natile and Giuseppe Savaré, A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal. 41 (2009), no. 4, 1340–1365. MR 2540269, DOI https://doi.org/10.1137/090750809
- John Riedl, Partially ordered locally convex vector spaces and extensions of positive continuous linear mappings, Math. Ann. 157 (1964), 95–124. MR 169033, DOI https://doi.org/10.1007/BF01362669
- Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043
- S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. MR 1619545
References
- Giovanni Alberti and Luigi Ambrosio, A geometrical approach to monotone functions in $\textbf {R}^n$, Math. Z. 230 (1999), no. 2, 259–316. MR 1676726, DOI https://doi.org/10.1007/PL00004691
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. MR 2401600
- Roméo Awi and Wilfrid Gangbo, A polyconvex integrand; Euler-Lagrange equations and uniqueness of equilibrium, Arch. Ration. Mech. Anal. 214 (2014), no. 1, 143–182. MR 3237884, DOI https://doi.org/10.1007/s00205-014-0754-9
- Heinz H. Bauschke and Xianfu Wang, The kernel average for two convex functions and its application to the extension and representation of monotone operators, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5947–5965. MR 2529920, DOI https://doi.org/10.1090/S0002-9947-09-04698-4
- Heinz H. Bauschke and Xianfu Wang, Firmly nonexpansive and Kirszbraun-Valentine extensions: a constructive approach via monotone operator theory, Nonlinear analysis and optimization I. Nonlinear analysis, Contemp. Math., vol. 513, Amer. Math. Soc., Providence, RI, 2010, pp. 55–64. MR 2668238, DOI https://doi.org/10.1090/conm/513/10075
- V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655
- Jonathan M. Borwein and Adrian S. Lewis, Convex analysis and nonlinear optimization, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 3, Springer-Verlag, New York, 2000. Theory and examples. MR 1757448
- Guy Bouchitté and Giuseppe Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 2, 139–168. MR 1831873, DOI https://doi.org/10.1007/s100970000027
- Y. Brenier, W. Gangbo, G. Savaré, and M. Westdickenberg, Sticky particle dynamics with interactions, J. Math. Pures Appl. (9) 99 (2013), no. 5, 577–617 (English, with English and French summaries). MR 3039208, DOI https://doi.org/10.1016/j.matpur.2012.09.013
- Fabio Cavalletti and Michael Westdickenberg, The polar cone of the set of monotone maps, Proc. Amer. Math. Soc. 143 (2015), no. 2, 781–787. MR 3283664, DOI https://doi.org/10.1090/S0002-9939-2014-12332-X
- F. Cavalletti, M. Sedjro, and M. Westdickenberg, A variational time discretization for compressible Euler equations (2014), available at arXiv:1411.1012v3.
- Fabio Cavalletti, Marc Sedjro, and Michael Westdickenberg, A simple proof of global existence for the 1D pressureless gas dynamics equations, SIAM J. Math. Anal. 47 (2015), no. 1, 66–79. MR 3296602, DOI https://doi.org/10.1137/130945296
- Ennio De Giorgi, New problems on minimizing movements, Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 81–98. MR 1260440
- Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR 1681462
- Wilfrid Gangbo and Michael Westdickenberg, Optimal transport for the system of isentropic Euler equations, Comm. Partial Differential Equations 34 (2009), no. 7-9, 1041–1073. MR 2560310, DOI https://doi.org/10.1080/03605300902892345
- N. Ghoussoub, A variational theory for monotone vector fields, J. Fixed Point Theory Appl. 4 (2008), no. 1, 107–135.
- Pierre-Louis Lions, Identification du cône dual des fonctions convexes et applications, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 12, 1385–1390 (French, with English and French summaries). MR 1649179, DOI https://doi.org/10.1016/S0764-4442%2898%2980397-2
- Luca Natile and Giuseppe Savaré, A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal. 41 (2009), no. 4, 1340–1365. MR 2540269, DOI https://doi.org/10.1137/090750809
- John Riedl, Partially ordered locally convex vector spaces and extensions of positive continuous linear mappings, Math. Ann. 157 (1964), 95–124. MR 0169033
- Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043
- S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. MR 1619545
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Additional Information
Michael Westdickenberg
Affiliation:
Lehrstuhl für Mathematik (Analysis), RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany
MR Author ID:
654309
Email:
mwest@instmath.rwth-aachen.de
Keywords:
Compressible gas dynamics,
optimal transport
Received by editor(s):
March 16, 2016
Received by editor(s) in revised form:
October 14, 2016
Published electronically:
November 14, 2016
Article copyright:
© Copyright 2016
Brown University