# American Institute of Mathematical Sciences

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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show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 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), 282-285. 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), 2253-2268. Google Scholar [4] L. Boccardo, A. Dall'Aglio, Th. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. 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), 4213-4218. 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), 1795-1859. 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), 4524-4532. 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), Viii+114 pp. 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), 193-224. 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, New York, 2000.  Google Scholar [11] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Wiley, New York, 1999.  Google Scholar [12] D. J. Griffiths, Introduction to Electrodynamics, 3rd edition, Pearson Education, 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), p04025. 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), 423-448. 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), 351-371. 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), 610-638. doi: 10.1016/j.neuron.2010.09.039.  Google Scholar [17] J. D. Jackson, Classical Electrodynamics, Second edition, John Wiley and Sons, New York-London-Sydney, 1975.  Google Scholar [18] H. M. Jäger and S. R. Nagel, Physics of the granular state, Science, 255 (1982), 1523-1531. Google Scholar [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.  Google Scholar [20] I. Lasiecka, Unified theory for abstract boundary problems-a semigroup approach, Appl. Math. Optim., 6 (1980), 287-333. 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), 125-156. 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), 384-390. 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), 749-781. 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, Dordrecht, 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, New York, 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), 1-9. doi: 10.1093/jxb/ers358.  Google Scholar [28] M. Riesz, Sur les fonction conjuguées, Math. Zeit., 27 (1928), 218-244. 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), 709-723. 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), 3-163 (Russian). Engl. Transl.: Proc. Steklov Inst. Math., 83 (1965), 1-184.  Google Scholar [32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar
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