September  2019, 8(3): 633-661. doi: 10.3934/eect.2019030

Waves and diffusion on metric graphs with general vertex conditions

1. 

University of L'Aquila, Department of Information Engineering, Computer Science and Mathematics, Via Vetoio, Coppito, I-67100 L'Aquila (AQ), Italy

2. 

University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, SI-1000 Ljubljana, Slovenia

3. 

Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia

* Corresponding author: Marjeta Kramar Fijavž

Received  October 2018 Revised  October 2018 Published  September 2019 Early access  May 2019

Fund Project: The second author is supported in part by the Slovenian Research Agency, Grant No. P1-0222.

We prove well-posedness for general linear wave- and diffusion equations on compact or non-compact metric graphs allowing various conditions in the vertices. More precisely, using the theory of strongly continuous operator semigroups we show that a large class of (not necessarily self-adjoint) second order differential operators with general (possibly non-local) boundary conditions generate cosine families, hence also analytic semigroups, on $ {\mathrm{L}}^p({\mathbb{R}_+}, {\mathbb{C}}^{\ell})\times{\mathrm{L}}^p([0, 1], {\mathbb{C}}^m) $ for $ 1\le p<+\infty $.

Citation: Klaus-Jochen Engel, Marjeta Kramar Fijavž. Waves and diffusion on metric graphs with general vertex conditions. Evolution Equations and Control Theory, 2019, 8 (3) : 633-661. doi: 10.3934/eect.2019030
References:
[1]

M. Adler, M. Bombieri and K.-J. Engel, On perturbations of generators of $C_0$-semigroups, Abstr. Appl. Anal., (2014), Art. ID 213020, 13pp. doi: 10.1155/2014/213020.

[2]

M. AdlerM. Bombieri and K.-J. Engel, Perturbation of analytic semigroups and applications to partial differential equations, J. Evol. Equ., 17 (2017), 1183-1208.  doi: 10.1007/s00028-016-0377-8.

[3]

F. Ali Mehmeti, Problèmes de transmission pour des équations des ondes linéaires et quasilinéaires, in Hyperbolic and Holomorphic Partial Differential Equations, Travaux en Cours, Hermann, Paris, (1984), 75–96.

[4]

F. Ali Mehmeti, Nonlinear Waves in Networks, Akademie-Verlag, Berlin, 1994.

[5]

F. Ali Mehmeti, J. von Below and S. Nicaise, eds., Partial Differential Equations on Multistructures, Marcel Dekker Inc., New York, 2001. doi: 10.1201/9780203902196.

[6]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser/Springer Basel AG, 2nd edition, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[7]

J. BanasiakA. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. Models Methods Appl. Sci., 26 (2016), 215-247.  doi: 10.1142/S0218202516400017.

[8]

J. BanasiakA. Falkiewicz and P. Namayanja, Semigroup approach to diffusion and transport problems on networks, Semigroup Forum, 93 (2016), 427-443.  doi: 10.1007/s00233-015-9730-4.

[9]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, 2013.

[10]

M. Bombieri and K.-J. Engel, A semigroup characterization of well-posed linear control systems, Semigroup Forum, 88 (2014), 366-396.  doi: 10.1007/s00233-013-9545-0.

[11]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.

[12]

C. Cattaneo, The spectrum of the continuous Laplacian on a graph, Monatsh. Math., 124 (1997), 215-235.  doi: 10.1007/BF01298245.

[13]

C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 284 (2003), 403-424.  doi: 10.1016/S0022-247X(02)00392-X.

[14]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[15]

K.-J. Engel, Generator property and stability for generalized difference operators, J. Evol. Equ., 13 (2013), 311-334.  doi: 10.1007/s00028-013-0179-1.

[16]

K.-J. EngelM. Kramar FijavžB. KlössR. Nagel and E. Sikolya, Maximal controllability for boundary control problems, Appl. Math. Optim., 62 (2010), 205-227.  doi: 10.1007/s00245-010-9101-1.

[17]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.

[18]

P. Exner, A model of resonance scattering on curved quantum wires, Annalen der Physik, 47 (1990), 123-138.  doi: 10.1002/andp.19905020207.

[19]

W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.  doi: 10.1090/S0002-9947-1954-0063607-6.

[20]

B. Gaveau, M. Okada and T. Okada, Explicit heat kernels on graphs and spectral analysis, in Several Complex Variables (Stockholm, 1987/1988), Princeton Univ. Press, Princeton, NJ, (1993), 364–388.

[21]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.  doi: 10.1007/s00233-011-9361-3.

[22]

S. Hadd, Unbounded perturbations of $C_0$-semigroups on Banach spaces and applications, Semigroup Forum, 70 (2005), 451-465.  doi: 10.1007/s00233-004-0172-7.

[23]

S. HaddR. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discrete Contin. Dyn. Syst., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703.

[24]

A. HusseinD. Krejčiřík and P. Siegl, Non-self-adjoint graphs, Trans. Amer. Math. Soc., 367 (2015), 2921-2957.  doi: 10.1090/S0002-9947-2014-06432-5.

[25]

B. JacobK. Morris and H. Zwart, $C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.  doi: 10.1007/s00028-014-0271-1.

[26]

B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128.  doi: 10.7153/oam-06-08.

[27]

V. Kostrykin, J. Potthoff and R. Schrader, Contraction semigroups on metric graphs, in: Analysis on Graphs and its Applications, Proc. Sympos. Pure Math., . Amer. Math. Soc., Providence, RI, 77 (2008), 423–458. doi: 10.1090/pspum/077/2459885.

[28]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.

[29]

T. Kottos and U. Smilansky, Quantum chaos on graphs, Physical Review Letters, 79 (1997), 4794-4797.  doi: 10.1103/physrevlett.79.4794.

[30]

M. Kramar FijavžD. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240.  doi: 10.1007/s00245-006-0887-9.

[31]

P. Kuchment, Graph models for waves in thin structures, Waves Random Media, 12 (2002), R1-R24.  doi: 10.1088/0959-7174/12/4/201.

[32]

P. Kuchment, Quantum graphs: An introduction and a brief survey, in Analysis on Graphs and Its Applications, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 77 (2008), 291–312. doi: 10.1090/pspum/077/2459876.

[33]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8.

[34]

Y. Le GorrecH. Zwart and B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators, SIAM J. Control Optim., 44 (2005), 1864-1892.  doi: 10.1137/040611677.

[35]

G. Lumer, Espaces ramifiés, et diffusions sur les réseaux topologiques, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A627–A630.

[36]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems. Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[37]

D. Mugnolo and S. Nicaise, Well-posedness and spectral properties of heat and wave equations with non-local conditions, J. Differential Equations, 256 (2014), 2115-2151.  doi: 10.1016/j.jde.2013.12.016.

[38]

S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission, in Orthogonal Polynomials and Applications (Bar-le-Duc, 1984), Springer, Berlin, 1171 (1985), 532–541. doi: 10.1007/BFb0076584.

[39]

J.-P. Roth, Le spectre du laplacien sur un graphe, in Théorie du Potentiel (Orsay, 1983), Springer, Berlin, 1096 (1984), 521–539. doi: 10.1007/BFb0100128.

[40]

C. SchubertC. SeifertJ. Voigt and M. Waurick, Boundary systems and (skew-)self-adjoint operators on infinite metric graphs, Mathematische Nachrichten, 288 (2015), 1776-1785.  doi: 10.1002/mana.201500054.

[41]

J. von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks, Linear Algebra Appl., 71 (1985), 309-325.  doi: 10.1016/0024-3795(85)90258-7.

show all references

References:
[1]

M. Adler, M. Bombieri and K.-J. Engel, On perturbations of generators of $C_0$-semigroups, Abstr. Appl. Anal., (2014), Art. ID 213020, 13pp. doi: 10.1155/2014/213020.

[2]

M. AdlerM. Bombieri and K.-J. Engel, Perturbation of analytic semigroups and applications to partial differential equations, J. Evol. Equ., 17 (2017), 1183-1208.  doi: 10.1007/s00028-016-0377-8.

[3]

F. Ali Mehmeti, Problèmes de transmission pour des équations des ondes linéaires et quasilinéaires, in Hyperbolic and Holomorphic Partial Differential Equations, Travaux en Cours, Hermann, Paris, (1984), 75–96.

[4]

F. Ali Mehmeti, Nonlinear Waves in Networks, Akademie-Verlag, Berlin, 1994.

[5]

F. Ali Mehmeti, J. von Below and S. Nicaise, eds., Partial Differential Equations on Multistructures, Marcel Dekker Inc., New York, 2001. doi: 10.1201/9780203902196.

[6]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser/Springer Basel AG, 2nd edition, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[7]

J. BanasiakA. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. Models Methods Appl. Sci., 26 (2016), 215-247.  doi: 10.1142/S0218202516400017.

[8]

J. BanasiakA. Falkiewicz and P. Namayanja, Semigroup approach to diffusion and transport problems on networks, Semigroup Forum, 93 (2016), 427-443.  doi: 10.1007/s00233-015-9730-4.

[9]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, 2013.

[10]

M. Bombieri and K.-J. Engel, A semigroup characterization of well-posed linear control systems, Semigroup Forum, 88 (2014), 366-396.  doi: 10.1007/s00233-013-9545-0.

[11]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.

[12]

C. Cattaneo, The spectrum of the continuous Laplacian on a graph, Monatsh. Math., 124 (1997), 215-235.  doi: 10.1007/BF01298245.

[13]

C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 284 (2003), 403-424.  doi: 10.1016/S0022-247X(02)00392-X.

[14]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[15]

K.-J. Engel, Generator property and stability for generalized difference operators, J. Evol. Equ., 13 (2013), 311-334.  doi: 10.1007/s00028-013-0179-1.

[16]

K.-J. EngelM. Kramar FijavžB. KlössR. Nagel and E. Sikolya, Maximal controllability for boundary control problems, Appl. Math. Optim., 62 (2010), 205-227.  doi: 10.1007/s00245-010-9101-1.

[17]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.

[18]

P. Exner, A model of resonance scattering on curved quantum wires, Annalen der Physik, 47 (1990), 123-138.  doi: 10.1002/andp.19905020207.

[19]

W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.  doi: 10.1090/S0002-9947-1954-0063607-6.

[20]

B. Gaveau, M. Okada and T. Okada, Explicit heat kernels on graphs and spectral analysis, in Several Complex Variables (Stockholm, 1987/1988), Princeton Univ. Press, Princeton, NJ, (1993), 364–388.

[21]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.  doi: 10.1007/s00233-011-9361-3.

[22]

S. Hadd, Unbounded perturbations of $C_0$-semigroups on Banach spaces and applications, Semigroup Forum, 70 (2005), 451-465.  doi: 10.1007/s00233-004-0172-7.

[23]

S. HaddR. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discrete Contin. Dyn. Syst., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703.

[24]

A. HusseinD. Krejčiřík and P. Siegl, Non-self-adjoint graphs, Trans. Amer. Math. Soc., 367 (2015), 2921-2957.  doi: 10.1090/S0002-9947-2014-06432-5.

[25]

B. JacobK. Morris and H. Zwart, $C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.  doi: 10.1007/s00028-014-0271-1.

[26]

B. Klöss, The flow approach for waves in networks, Oper. Matrices, 6 (2012), 107-128.  doi: 10.7153/oam-06-08.

[27]

V. Kostrykin, J. Potthoff and R. Schrader, Contraction semigroups on metric graphs, in: Analysis on Graphs and its Applications, Proc. Sympos. Pure Math., . Amer. Math. Soc., Providence, RI, 77 (2008), 423–458. doi: 10.1090/pspum/077/2459885.

[28]

V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.

[29]

T. Kottos and U. Smilansky, Quantum chaos on graphs, Physical Review Letters, 79 (1997), 4794-4797.  doi: 10.1103/physrevlett.79.4794.

[30]

M. Kramar FijavžD. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240.  doi: 10.1007/s00245-006-0887-9.

[31]

P. Kuchment, Graph models for waves in thin structures, Waves Random Media, 12 (2002), R1-R24.  doi: 10.1088/0959-7174/12/4/201.

[32]

P. Kuchment, Quantum graphs: An introduction and a brief survey, in Analysis on Graphs and Its Applications, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 77 (2008), 291–312. doi: 10.1090/pspum/077/2459876.

[33]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8.

[34]

Y. Le GorrecH. Zwart and B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators, SIAM J. Control Optim., 44 (2005), 1864-1892.  doi: 10.1137/040611677.

[35]

G. Lumer, Espaces ramifiés, et diffusions sur les réseaux topologiques, C. R. Acad. Sci. Paris Sér. A-B, 291 (1980), A627–A630.

[36]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems. Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[37]

D. Mugnolo and S. Nicaise, Well-posedness and spectral properties of heat and wave equations with non-local conditions, J. Differential Equations, 256 (2014), 2115-2151.  doi: 10.1016/j.jde.2013.12.016.

[38]

S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission, in Orthogonal Polynomials and Applications (Bar-le-Duc, 1984), Springer, Berlin, 1171 (1985), 532–541. doi: 10.1007/BFb0076584.

[39]

J.-P. Roth, Le spectre du laplacien sur un graphe, in Théorie du Potentiel (Orsay, 1983), Springer, Berlin, 1096 (1984), 521–539. doi: 10.1007/BFb0100128.

[40]

C. SchubertC. SeifertJ. Voigt and M. Waurick, Boundary systems and (skew-)self-adjoint operators on infinite metric graphs, Mathematische Nachrichten, 288 (2015), 1776-1785.  doi: 10.1002/mana.201500054.

[41]

J. von Below, A characteristic equation associated to an eigenvalue problem on $c^2$-networks, Linear Algebra Appl., 71 (1985), 309-325.  doi: 10.1016/0024-3795(85)90258-7.

Figure 1.  Star-shaped graph from Example 2.15
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