September  2019, 39(9): 5105-5124. doi: 10.3934/dcds.2019207

On absolutely continuous curves of probabilities on the line

WVU, Morgantown, WV 26506, USA

Received  June 2018 Revised  March 2019 Published  May 2019

In recent collaborative work we studied existence and uniqueness of a Lagrangian description for absolutely continuous curves in spaces of Borel probabilities on the real line with finite moments of given order. Of course, a measurable velocity driving the evolution in Eulerian coordinates is necessary to define the Eulerian and Lagrangian descriptions of fluid flow; here we prove that in this case it is also sufficient for a Lagrangian description. More precisely, we argue that the existence of the integrable velocity along an absolutely continuous curve in the set of Borel probabilities on the line is enough to produce a canonical Lagrangian description for the curve; this is given by the family of optimal maps between the uniform distribution on the unit interval and the measures on the curve. Moreover, we identify a necessary and sufficient condition on said family of optimal maps which ensures that the measurable velocity along the curve exists.

Citation: Adrian Tudorascu. On absolutely continuous curves of probabilities on the line. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5105-5124. doi: 10.3934/dcds.2019207
References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.

[2]

L. Ambrosio and W. Gangbo., Hamiltonian ODE in the Wasserstein spaces of probability measures, Comm. Pure Appl. Math., 61 (2008), 18-53.  doi: 10.1002/cpa.20188.

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and the Wasserstein Spaces of Probability Measures, Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005.

[4]

M. Amsaad and A. Tudorascu, On the Lagrangian description of absolutely continuous curves in the Wasserstein space on the line; well-posedness for the Continuity Equation, Indiana U. Math. J., 64 (2015), 1835-1877.  doi: 10.1512/iumj.2015.64.5727.

[5]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417.  doi: 10.1002/cpa.3160440402.

[6]

H. Brezis, Convergence in $\mathcal{D}'$ and $L^1$ under strict convexity, in Boundary Value Problems for Partial Differential Equations and Applications, dedicated to E. Magenes, (C. Baiocchi et J.L. Lions ed.), Masson, 29 (1993), 43–52.

[7]

J. A. CarrilloM. Di FrancescoA. FigalliT. Laurent and D. Slepčev, Global-in-time weak measure solutions, finite-time aggregation and confinement for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.  doi: 10.1215/00127094-2010-211.

[8]

G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields, Edizioni della Normale, Pisa, 2009.

[9]

M. Fréchet, Sur la distance de deux lois de probabilite, Comptes Rendus Acad. Sci., 244 (1957), 689-692. 

[10]

W. GangboT. Nguyen and A. Tudorascu, Euler-Poisson systems as action minimizing paths in the Wasserstein space, Arch. Rational Mech. Anal., 192 (2009), 419-452.  doi: 10.1007/s00205-008-0148-y.

[11]

W. GangboT. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Meth. Appl. Anal., 15 (2008), 155-183.  doi: 10.4310/MAA.2008.v15.n2.a4.

[12]

U. GianazzaG. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.  doi: 10.1007/s00205-008-0186-5.

[13]

G. Loeper, On the regularity of the polar factorization of time dependent maps, Calc. Var. PDE, 22 (2005), 343-374.  doi: 10.1007/s00526-004-0280-y.

[14]

T. Nguyen and A. Tudorascu, Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM J. Math. Anal., 40 (2008), 754-775.  doi: 10.1137/070704459.

[15]

J. V. Ryff, Orbits of $L^1$ functions under doubly stochastic transformations, Trans. AMS, 117 (1965), 92-100.  doi: 10.2307/1994198.

[16]

A. Tudorascu, On the velocities of flows consisting of cyclically monotone maps, Indiana U. Math. J., 59 (2010), 929-956.  doi: 10.1512/iumj.2010.59.3955.

[17]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, 2003. doi: 10.1007/b12016.

[18]

A. Visintin, Strong convergence results related to strict convexity, Commun. Part. Diff. Eq., 9 (1984), 439-466.  doi: 10.1080/03605308408820337.

show all references

References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.

[2]

L. Ambrosio and W. Gangbo., Hamiltonian ODE in the Wasserstein spaces of probability measures, Comm. Pure Appl. Math., 61 (2008), 18-53.  doi: 10.1002/cpa.20188.

[3]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and the Wasserstein Spaces of Probability Measures, Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005.

[4]

M. Amsaad and A. Tudorascu, On the Lagrangian description of absolutely continuous curves in the Wasserstein space on the line; well-posedness for the Continuity Equation, Indiana U. Math. J., 64 (2015), 1835-1877.  doi: 10.1512/iumj.2015.64.5727.

[5]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417.  doi: 10.1002/cpa.3160440402.

[6]

H. Brezis, Convergence in $\mathcal{D}'$ and $L^1$ under strict convexity, in Boundary Value Problems for Partial Differential Equations and Applications, dedicated to E. Magenes, (C. Baiocchi et J.L. Lions ed.), Masson, 29 (1993), 43–52.

[7]

J. A. CarrilloM. Di FrancescoA. FigalliT. Laurent and D. Slepčev, Global-in-time weak measure solutions, finite-time aggregation and confinement for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.  doi: 10.1215/00127094-2010-211.

[8]

G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields, Edizioni della Normale, Pisa, 2009.

[9]

M. Fréchet, Sur la distance de deux lois de probabilite, Comptes Rendus Acad. Sci., 244 (1957), 689-692. 

[10]

W. GangboT. Nguyen and A. Tudorascu, Euler-Poisson systems as action minimizing paths in the Wasserstein space, Arch. Rational Mech. Anal., 192 (2009), 419-452.  doi: 10.1007/s00205-008-0148-y.

[11]

W. GangboT. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Meth. Appl. Anal., 15 (2008), 155-183.  doi: 10.4310/MAA.2008.v15.n2.a4.

[12]

U. GianazzaG. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.  doi: 10.1007/s00205-008-0186-5.

[13]

G. Loeper, On the regularity of the polar factorization of time dependent maps, Calc. Var. PDE, 22 (2005), 343-374.  doi: 10.1007/s00526-004-0280-y.

[14]

T. Nguyen and A. Tudorascu, Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM J. Math. Anal., 40 (2008), 754-775.  doi: 10.1137/070704459.

[15]

J. V. Ryff, Orbits of $L^1$ functions under doubly stochastic transformations, Trans. AMS, 117 (1965), 92-100.  doi: 10.2307/1994198.

[16]

A. Tudorascu, On the velocities of flows consisting of cyclically monotone maps, Indiana U. Math. J., 59 (2010), 929-956.  doi: 10.1512/iumj.2010.59.3955.

[17]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, 2003. doi: 10.1007/b12016.

[18]

A. Visintin, Strong convergence results related to strict convexity, Commun. Part. Diff. Eq., 9 (1984), 439-466.  doi: 10.1080/03605308408820337.

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