2015, 12(2): 357-373. doi: 10.3934/mbe.2015.12.357

Modelling with measures: Approximation of a mass-emitting object by a point source

1. 

Institute for Complex Molecular Systems & Centre for Analysis, Scientific computing and Applications, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands

2. 

Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden

Received  April 2014 Revised  October 2014 Published  December 2014

We consider a linear diffusion equation on $\Omega:=\mathbb{R}^2\setminus\overline{\Omega_\mathcal{o}}$, where $\Omega_\mathcal{o}$ is a bounded domain. The time-dependent flux on the boundary $\Gamma:=∂\Omega_\mathcal{o}$ is prescribed. The aim of the paper is to approximate the dynamics by the solution of the diffusion equation on the whole of $\mathbb{R}^2$ with a measure-valued point source in the origin and provide estimates for the quality of approximation. For all time $t$, we derive an $L^2([0,t];L^2(\Gamma))$-bound on the difference in flux on the boundary. Moreover, we derive for all $t>0$ an $L^2(\Omega)$-bound and an $L^2([0,t];H^1(\Omega))$-bound for the difference of the solutions to the two models.
Citation: Joep H.M. Evers, Sander C. Hille, Adrian Muntean. Modelling with measures: Approximation of a mass-emitting object by a point source. Mathematical Biosciences & Engineering, 2015, 12 (2) : 357-373. doi: 10.3934/mbe.2015.12.357
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003.

[2]

F. Baluška, J. Šamaj and D. Menzel, Polar transport of auxin: Carrier-mediated flux across the plasma membrane or neurotransmitter-like secretion?, Trends in Cell Biology, 13 (2003), 282-285.

[3]

K. van Berkel, R. J. de Boer, B. Scheres and K. ten Tusscher, Polar auxin transport: Models and mechanisms, Development, 140 (2013), 2253-2268.

[4]

L. Boccardo, A. Dall'Aglio, Th. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040.

[5]

K. J. M. Boot, K. R. Libbenga, S. C. Hille, R. Offringa and B. van Duijn, Polar auxin transport: An early invention, Journal of Experimental Botany, 63 (2012), 4213-4218. doi: 10.1093/jxb/ers106.

[6]

E. Cancès and C. Le Bris, Mathematical modeling of point defects in materials science, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1795-1859. doi: 10.1142/S0218202513500528.

[7]

A. Chavarría-Krauser and M. Ptashnyk, Homogenization of long-range auxin transport in plant tissues, Nonlinear Analysis: Real World Applications, 11 (2010), 4524-4532. doi: 10.1016/j.nonrwa.2008.12.003.

[8]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Am. Math. Soc., 166 (2003), Viii+114 pp. doi: 10.1090/memo/0788.

[9]

R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.

[10]

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

[11]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Wiley, New York, 1999.

[12]

D. J. Griffiths, Introduction to Electrodynamics, 3rd edition, Pearson Education, 2008.

[13]

L. Gulikers, J. H. M. Evers, A. Muntean and A. V. Lyulin, The effect of perception anisotropy on particle systems describing pedestrian flows in corridors, Journal of Statistical Mechanics: Theory and Experiment, 2013 (2013), p04025. doi: 10.1088/1742-5468/2013/04/P04025.

[14]

S. C. Hille, Local well-posedness of kinetic chemotaxis models, Journal of Evolution Equations, 8 (2008), 423-448. doi: 10.1007/s00028-008-0358-7.

[15]

S. C. Hille and D. T. H. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integr. Equ. Oper. Theory, 63 (2009), 351-371. doi: 10.1007/s00020-008-1652-z.

[16]

N. Hirokawa, S. Niwa and Y. Tanaka, Molecular motors in neurons: Transport mechanisms and roles in brain function, development, and disease, Neuron, 68 (2010), 610-638. doi: 10.1016/j.neuron.2010.09.039.

[17]

J. D. Jackson, Classical Electrodynamics, Second edition, John Wiley and Sons, New York-London-Sydney, 1975.

[18]

H. M. Jäger and S. R. Nagel, Physics of the granular state, Science, 255 (1982), 1523-1531.

[19]

E. M. Kramer, Computer models of auxin transport: A review and commentary, Journal of Experimental Botany, 59 (2008), 45-53. doi: 10.1093/jxb/erm060.

[20]

I. Lasiecka, Unified theory for abstract boundary problems-a semigroup approach, Appl. Math. Optim., 6 (1980), 287-333. doi: 10.1007/BF01442900.

[21]

J. D. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Verlag, 1972. doi: 10.1007/978-3-642-65161-8.

[22]

Y. Liu and R. H. Edwards, The role of vesicular transport proteins in synaptic transmission and neural degeneration, Ann. Rev. Neurosci., 20 (1997), 125-156.

[23]

R. M. H. Merks, Y. Van de Peer, D. Inzé and G. T. S. Beemster, Canalization without flux sensors: A traveling-wave hypothesis, Trends in Plant Science, 12 (2007), 384-390. doi: 10.1016/j.tplants.2007.08.004.

[24]

P. van Meurs, A. Muntean and M. A. Peletier, Upscaling of dislocation walls in finite domains, Eur. J. Appl. Math, 25 (2014), 749-781. doi: 10.1017/S0956792514000254.

[25]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3562-7.

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlang, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

J. A. Raven, Polar auxin transport in relation to long-distance transport of nutrients in the Charales, Journal of Experimental Botany, 64 (2013), 1-9. doi: 10.1093/jxb/ers358.

[28]

M. Riesz, Sur les fonction conjuguées, Math. Zeit., 27 (1928), 218-244. doi: 10.1007/BF01171098.

[29]

U. Rüde, H. Köstler and M. Mohr, Accurate Multigrid Techniques for Computing Singular Solutions of Elliptic Problems, Eleventh Copper Mountain Conference on Multigrid Methods, 2003.

[30]

T. I. Seidman, M. K. Gobbert, D. W. Trott and M. Kružík, Finite element approximation for time-dependent diffusion with measure-valued source, Numer. Math., 122 (2012), 709-723. doi: 10.1007/s00211-012-0474-8.

[31]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Fust. Steklov, 83 (1965), 3-163 (Russian). Engl. Transl.: Proc. Steklov Inst. Math., 83 (1965), 1-184.

[32]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003.

[2]

F. Baluška, J. Šamaj and D. Menzel, Polar transport of auxin: Carrier-mediated flux across the plasma membrane or neurotransmitter-like secretion?, Trends in Cell Biology, 13 (2003), 282-285.

[3]

K. van Berkel, R. J. de Boer, B. Scheres and K. ten Tusscher, Polar auxin transport: Models and mechanisms, Development, 140 (2013), 2253-2268.

[4]

L. Boccardo, A. Dall'Aglio, Th. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040.

[5]

K. J. M. Boot, K. R. Libbenga, S. C. Hille, R. Offringa and B. van Duijn, Polar auxin transport: An early invention, Journal of Experimental Botany, 63 (2012), 4213-4218. doi: 10.1093/jxb/ers106.

[6]

E. Cancès and C. Le Bris, Mathematical modeling of point defects in materials science, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1795-1859. doi: 10.1142/S0218202513500528.

[7]

A. Chavarría-Krauser and M. Ptashnyk, Homogenization of long-range auxin transport in plant tissues, Nonlinear Analysis: Real World Applications, 11 (2010), 4524-4532. doi: 10.1016/j.nonrwa.2008.12.003.

[8]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Am. Math. Soc., 166 (2003), Viii+114 pp. doi: 10.1090/memo/0788.

[9]

R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.

[10]

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

[11]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Wiley, New York, 1999.

[12]

D. J. Griffiths, Introduction to Electrodynamics, 3rd edition, Pearson Education, 2008.

[13]

L. Gulikers, J. H. M. Evers, A. Muntean and A. V. Lyulin, The effect of perception anisotropy on particle systems describing pedestrian flows in corridors, Journal of Statistical Mechanics: Theory and Experiment, 2013 (2013), p04025. doi: 10.1088/1742-5468/2013/04/P04025.

[14]

S. C. Hille, Local well-posedness of kinetic chemotaxis models, Journal of Evolution Equations, 8 (2008), 423-448. doi: 10.1007/s00028-008-0358-7.

[15]

S. C. Hille and D. T. H. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integr. Equ. Oper. Theory, 63 (2009), 351-371. doi: 10.1007/s00020-008-1652-z.

[16]

N. Hirokawa, S. Niwa and Y. Tanaka, Molecular motors in neurons: Transport mechanisms and roles in brain function, development, and disease, Neuron, 68 (2010), 610-638. doi: 10.1016/j.neuron.2010.09.039.

[17]

J. D. Jackson, Classical Electrodynamics, Second edition, John Wiley and Sons, New York-London-Sydney, 1975.

[18]

H. M. Jäger and S. R. Nagel, Physics of the granular state, Science, 255 (1982), 1523-1531.

[19]

E. M. Kramer, Computer models of auxin transport: A review and commentary, Journal of Experimental Botany, 59 (2008), 45-53. doi: 10.1093/jxb/erm060.

[20]

I. Lasiecka, Unified theory for abstract boundary problems-a semigroup approach, Appl. Math. Optim., 6 (1980), 287-333. doi: 10.1007/BF01442900.

[21]

J. D. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Verlag, 1972. doi: 10.1007/978-3-642-65161-8.

[22]

Y. Liu and R. H. Edwards, The role of vesicular transport proteins in synaptic transmission and neural degeneration, Ann. Rev. Neurosci., 20 (1997), 125-156.

[23]

R. M. H. Merks, Y. Van de Peer, D. Inzé and G. T. S. Beemster, Canalization without flux sensors: A traveling-wave hypothesis, Trends in Plant Science, 12 (2007), 384-390. doi: 10.1016/j.tplants.2007.08.004.

[24]

P. van Meurs, A. Muntean and M. A. Peletier, Upscaling of dislocation walls in finite domains, Eur. J. Appl. Math, 25 (2014), 749-781. doi: 10.1017/S0956792514000254.

[25]

D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3562-7.

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlang, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

J. A. Raven, Polar auxin transport in relation to long-distance transport of nutrients in the Charales, Journal of Experimental Botany, 64 (2013), 1-9. doi: 10.1093/jxb/ers358.

[28]

M. Riesz, Sur les fonction conjuguées, Math. Zeit., 27 (1928), 218-244. doi: 10.1007/BF01171098.

[29]

U. Rüde, H. Köstler and M. Mohr, Accurate Multigrid Techniques for Computing Singular Solutions of Elliptic Problems, Eleventh Copper Mountain Conference on Multigrid Methods, 2003.

[30]

T. I. Seidman, M. K. Gobbert, D. W. Trott and M. Kružík, Finite element approximation for time-dependent diffusion with measure-valued source, Numer. Math., 122 (2012), 709-723. doi: 10.1007/s00211-012-0474-8.

[31]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Fust. Steklov, 83 (1965), 3-163 (Russian). Engl. Transl.: Proc. Steklov Inst. Math., 83 (1965), 1-184.

[32]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.

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