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:
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R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, 2nd edition, (2003). Google Scholar

[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. Google Scholar

[3]

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

[4]

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

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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. doi: 10.1093/jxb/ers106. Google Scholar

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

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

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

[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. doi: 10.1007/s00209-007-0120-9. Google Scholar

[10]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000). Google Scholar

[11]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications,, 2nd edition, (1999). Google Scholar

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D. J. Griffiths, Introduction to Electrodynamics,, 3rd edition, (2008). Google Scholar

[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). doi: 10.1088/1742-5468/2013/04/P04025. Google Scholar

[14]

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

[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. doi: 10.1007/s00020-008-1652-z. Google Scholar

[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. doi: 10.1016/j.neuron.2010.09.039. Google Scholar

[17]

J. D. Jackson, Classical Electrodynamics,, Second edition, (1975). Google Scholar

[18]

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

[19]

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

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.tplants.2007.08.004. Google Scholar

[24]

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

[25]

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

[26]

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

[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. doi: 10.1093/jxb/ers358. Google Scholar

[28]

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

[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). Google Scholar

[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. doi: 10.1007/s00211-012-0474-8. Google Scholar

[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. Google Scholar

[32]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, 2nd edition, (2003). Google Scholar

[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. Google Scholar

[3]

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

[4]

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

[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. doi: 10.1093/jxb/ers106. Google Scholar

[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. doi: 10.1142/S0218202513500528. Google Scholar

[7]

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

[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). doi: 10.1090/memo/0788. Google Scholar

[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. doi: 10.1007/s00209-007-0120-9. Google Scholar

[10]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000). Google Scholar

[11]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications,, 2nd edition, (1999). Google Scholar

[12]

D. J. Griffiths, Introduction to Electrodynamics,, 3rd edition, (2008). Google Scholar

[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). doi: 10.1088/1742-5468/2013/04/P04025. Google Scholar

[14]

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

[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. doi: 10.1007/s00020-008-1652-z. Google Scholar

[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. doi: 10.1016/j.neuron.2010.09.039. Google Scholar

[17]

J. D. Jackson, Classical Electrodynamics,, Second edition, (1975). Google Scholar

[18]

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

[19]

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

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.tplants.2007.08.004. Google Scholar

[24]

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

[25]

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

[26]

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

[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. doi: 10.1093/jxb/ers358. Google Scholar

[28]

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

[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). Google Scholar

[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. doi: 10.1007/s00211-012-0474-8. Google Scholar

[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. Google Scholar

[32]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar

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