# American Institute of Mathematical Sciences

April  2018, 11(2): 397-408. doi: 10.3934/krm.2018018

## Regularity theorems for a biological network formulation model in two space dimensions

 Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762, USA

Received  December 2016 Revised  May 2017 Published  January 2018

We present several regularity results for a biological network formulation model originally introduced by D. Cai and D. Hu [13]. A consequence of these results is that a stationary weak solution must be a classical one in two space dimensions. Our mathematical analysis is based upon the weakly monotone function theory and Hardy space methods.

Citation: Xiangsheng Xu. Regularity theorems for a biological network formulation model in two space dimensions. Kinetic & Related Models, 2018, 11 (2) : 397-408. doi: 10.3934/krm.2018018
##### References:
 [1] G. Albi, M. Artina, M. Fornasier and P. Markowich, Biological transportation networks: Modeling and simulation, Anal. Appl. (Singap.), 14 (2016), 185-206.  doi: 10.1142/S0219530515400059.  Google Scholar [2] S. Chanillo and R. L. Wheeden, Existence and estimates of Green's function for degenerate elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 309-340.   Google Scholar [3] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.   Google Scholar [4] G. Di Fazio, $L^p$ Estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7), 10 (1996), 409-420.   Google Scholar [5] L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal., 116 (1991), 101-113.  doi: 10.1007/BF00375587.  Google Scholar [6] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton 1992.  Google Scholar [7] F. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103 (1962), 353-393.  doi: 10.1090/S0002-9947-1962-0139735-8.  Google Scholar [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.  Google Scholar [9] J. Haskovec, P. Markowich and B. Perthame, Mathematical analysis of a PDE system for biological network formulation, Comm. Partial Differential Equations, 40 (2015), 918-956.  doi: 10.1080/03605302.2014.968792.  Google Scholar [10] J. Haskovec, P. Markowich, B. Perthame and M. Schlottbom, Notes on a PDE system for biological network formulation, Nonlinear Anal, 138 (2016), 127-155.  doi: 10.1016/j.na.2015.12.018.  Google Scholar [11] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993.  Google Scholar [12] D. Hu, Optimization, Adaptation, and Initialization of Biological Transport Networks, Workshop on multi scale problems from physics, biology, and material sciences, May 28-31,2014, Shanghai. Google Scholar [13] D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Phys. Rev. Lett., 111 (2013), 138701.  doi: 10.1103/PhysRevLett.111.138701.  Google Scholar [14] R. L. Johnson and J. C. Neugebauer, Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen, 12 (1993), 3-11.  doi: 10.4171/ZAA/583.  Google Scholar [15] J. Kinnunen, Higher integrability with weights, Annales Academia Scientiarum Fennica Series A.I. Mathematica, 19 (1994), 355-366.   Google Scholar [16] J. -G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, J. Differential Equations, to appear. Google Scholar [17] J. J. Manfredi, Weakly monotone functions, J. Geometric Analysis, 4 (1994), 393-402.  doi: 10.1007/BF02921588.  Google Scholar [18] S. Müller, A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc., 21 (1989), 245-248.  doi: 10.1090/S0273-0979-1989-15818-7.  Google Scholar [19] J. R. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, 134 North-Holland, Amsterdam, 1987.  Google Scholar [20] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations, 19 (1994), 277-319.  doi: 10.1080/03605309408821017.  Google Scholar [21] X. Xu, Existence theorems for the quantum drift-diffusion system with mixed boundary conditions, Commun. Contemp. Math. , 18 (2016), 1550048, 21 pp.  Google Scholar

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##### References:
 [1] G. Albi, M. Artina, M. Fornasier and P. Markowich, Biological transportation networks: Modeling and simulation, Anal. Appl. (Singap.), 14 (2016), 185-206.  doi: 10.1142/S0219530515400059.  Google Scholar [2] S. Chanillo and R. L. Wheeden, Existence and estimates of Green's function for degenerate elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 309-340.   Google Scholar [3] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286.   Google Scholar [4] G. Di Fazio, $L^p$ Estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7), 10 (1996), 409-420.   Google Scholar [5] L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal., 116 (1991), 101-113.  doi: 10.1007/BF00375587.  Google Scholar [6] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton 1992.  Google Scholar [7] F. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103 (1962), 353-393.  doi: 10.1090/S0002-9947-1962-0139735-8.  Google Scholar [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.  Google Scholar [9] J. Haskovec, P. Markowich and B. Perthame, Mathematical analysis of a PDE system for biological network formulation, Comm. Partial Differential Equations, 40 (2015), 918-956.  doi: 10.1080/03605302.2014.968792.  Google Scholar [10] J. Haskovec, P. Markowich, B. Perthame and M. Schlottbom, Notes on a PDE system for biological network formulation, Nonlinear Anal, 138 (2016), 127-155.  doi: 10.1016/j.na.2015.12.018.  Google Scholar [11] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993.  Google Scholar [12] D. Hu, Optimization, Adaptation, and Initialization of Biological Transport Networks, Workshop on multi scale problems from physics, biology, and material sciences, May 28-31,2014, Shanghai. Google Scholar [13] D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Phys. Rev. Lett., 111 (2013), 138701.  doi: 10.1103/PhysRevLett.111.138701.  Google Scholar [14] R. L. Johnson and J. C. Neugebauer, Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen, 12 (1993), 3-11.  doi: 10.4171/ZAA/583.  Google Scholar [15] J. Kinnunen, Higher integrability with weights, Annales Academia Scientiarum Fennica Series A.I. Mathematica, 19 (1994), 355-366.   Google Scholar [16] J. -G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, J. Differential Equations, to appear. Google Scholar [17] J. J. Manfredi, Weakly monotone functions, J. Geometric Analysis, 4 (1994), 393-402.  doi: 10.1007/BF02921588.  Google Scholar [18] S. Müller, A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc., 21 (1989), 245-248.  doi: 10.1090/S0273-0979-1989-15818-7.  Google Scholar [19] J. R. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, 134 North-Holland, Amsterdam, 1987.  Google Scholar [20] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations, 19 (1994), 277-319.  doi: 10.1080/03605309408821017.  Google Scholar [21] X. Xu, Existence theorems for the quantum drift-diffusion system with mixed boundary conditions, Commun. Contemp. Math. , 18 (2016), 1550048, 21 pp.  Google Scholar
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