The total variation flow in metric graphs

José M. Mazón; José M. Mazón

doi:10.3934/mine.2023009

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-38. doi: 10.3934/mine.2023009

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The total variation flow in metric graphs

José M. Mazón ^,

Departamento de Análisis Matemático, Univ. Valencia, Dr. Moliner 50, 46100 Burjassot, Spain

Received: 14 October 2021 Revised: 22 December 2021 Accepted: 22 December 2021 Published: 25 January 2022

Our aim is to study the total variation flow in metric graphs. First, we define the functions of bounded variation in metric graphs and their total variation, we also give an integration by parts formula. We prove existence and uniqueness of solutions and that the solutions reach the mean of the initial data in finite time. Moreover, we obtain explicit solutions.
- metric graphs,
- functions of total variation,
- total variation flow,
- the $ 1 $-Laplacian,
- asymptotic behaviour
Citation: José M. Mazón. The total variation flow in metric graphs[J]. Mathematics in Engineering, 2023, 5(1): 1-38. doi: 10.3934/mine.2023009

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Abstract

Our aim is to study the total variation flow in metric graphs. First, we define the functions of bounded variation in metric graphs and their total variation, we also give an integration by parts formula. We prove existence and uniqueness of solutions and that the solutions reach the mean of the initial data in finite time. Moreover, we obtain explicit solutions.

References

[1]	L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, 2000.
[2]	F. Andreu, V. Caselles, J. M. Mazon, Parabolic quasilinear equations minimizing linear growth functionals, Basel: Birkhauser, 2004. http://dx.doi.org/10.1007/978-3-0348-7928-6
[3]	F. Andreu, C. Ballester, V. Caselles, J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321–360.
[4]	F. Andreu, C. Ballester, V. Caselles, J. M. Mazón, The Dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), 347–403. http://dx.doi.org/10.1006/jfan.2000.3698 doi: 10.1006/jfan.2000.3698
[5]	G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Annali di Matematica pura ed applicata, 135 (1983), 293–318. http://dx.doi.org/10.1007/BF01781073 doi: 10.1007/BF01781073
[6]	V. Barbu, Nonlinear differential equations of monotone type in Banach spaces, New York, NY: Springer, 2010. http://dx.doi.org/10.1007/978-1-4419-5542-5
[7]	G. Berkolaiko, P. Kuchment, Introduction to quantum graphs, Providence, RI: American Mathematical Society, 2013.
[8]	M. Bonforte, A. Figalli, Total variation flow and sign fast diffusion in one dimension, J. Differ. Equations, 252 (2012), 4455–4480. http://dx.doi.org/10.1016/j.jde.2012.01.003 doi: 10.1016/j.jde.2012.01.003
[9]	H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Amsterdam: North Holland, 1973.
[10]	L. Bungert, M. Burger, Asymptotic profiles of nonlinear homogeneous evolution, J. Evol. Equ., 20 (2020), 1061–1092. http://dx.doi.org/10.1007/s00028-019-00545-1 doi: 10.1007/s00028-019-00545-1
[11]	L. Bungert, M. Burger, A. Chambolle, M. Novaga, Nonlinear spectral decompositions by gradient flows of one-homogeneous functionals, Anal. PDE, 14 (2021), 823–860. http://dx.doi.org/10.2140/apde.2021.14.823 doi: 10.2140/apde.2021.14.823
[12]	R. E. Bruck Jr., Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal., 18 (1975), 15–26. http://dx.doi.org/10.1016/0022-1236(75)90027-0 doi: 10.1016/0022-1236(75)90027-0
[13]	M. G. Crandall, T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265–298. http://dx.doi.org/10.2307/2373376 doi: 10.2307/2373376
[14]	R. Dáger, E. Zuazua, Wave propagation, observation and control in 1-d flexible multistructures, Berlin: Springer, 2006. http://dx.doi.org/10.1007/3-540-37726-3
[15]	Y. Du, B. Lou, R. Peng, M. Zhou, The Fisher-KPP equation over simple graphs: varied persistence states in river networks, J. Math. Biol., 80 (2020), 1559–1616. http://dx.doi.org/10.1007/s00285-020-01474-1 doi: 10.1007/s00285-020-01474-1
[16]	Y. Jin, R. Peng, J. Shi, Population dynamics in river networks, J. Nonlinear Sci., 29 (2019), 2501–2545. http://dx.doi.org/10.1007/s00332-019-09551-6 doi: 10.1007/s00332-019-09551-6
[17]	G. Lumer, Connecting of local operators and evolution equations on networks, In: Potential theory Copenhagen 1979, Berlin: Springer, 1980,219–234. http://dx.doi.org/10.1007/BFb0086338
[18]	J. M. Mazón, M. Solera, J. Toledo, The total variation flow in metric randon walk spaces, Calc. Var., 59 (2020), 29. http://dx.doi.org/10.1007/s00526-019-1684-z doi: 10.1007/s00526-019-1684-z
[19]	Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations: The fifteenth Dean Jacqueline B. Lewis memorial lectures, Providance, RI: American Mathematical Society, 2001. http://dx.doi.org/10.1090/ulect/022
[20]	D. Mugnolo, Semigroup methods for evolution equations on networks, Cham: Springer, 2014. http://dx.doi.org/10.1007/978-3-319-04621-1
[21]	D. Noja, Nonlinear Schrödinger equation on graphs: recent results and open problems, Phil. Trans. R. Soc. A, 372 (2007), 20130002. http://dx.doi.org/10.1098/rsta.2013.0002 doi: 10.1098/rsta.2013.0002

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