# 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. [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. [3] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. [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. [5] L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal., 116 (1991), 101-113. doi: 10.1007/BF00375587. [6] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton 1992. [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. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. [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. [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. [11] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993. [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. [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. [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. [15] J. Kinnunen, Higher integrability with weights, Annales Academia Scientiarum Fennica Series A.I. Mathematica, 19 (1994), 355-366. [16] J. -G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, J. Differential Equations, to appear. [17] J. J. Manfredi, Weakly monotone functions, J. Geometric Analysis, 4 (1994), 393-402. doi: 10.1007/BF02921588. [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. [19] J. R. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, 134 North-Holland, Amsterdam, 1987. [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. [21] X. Xu, Existence theorems for the quantum drift-diffusion system with mixed boundary conditions, Commun. Contemp. Math. , 18 (2016), 1550048, 21 pp.

show all references

##### 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. [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. [3] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. [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. [5] L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal., 116 (1991), 101-113. doi: 10.1007/BF00375587. [6] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton 1992. [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. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. [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. [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. [11] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993. [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. [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. [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. [15] J. Kinnunen, Higher integrability with weights, Annales Academia Scientiarum Fennica Series A.I. Mathematica, 19 (1994), 355-366. [16] J. -G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, J. Differential Equations, to appear. [17] J. J. Manfredi, Weakly monotone functions, J. Geometric Analysis, 4 (1994), 393-402. doi: 10.1007/BF02921588. [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. [19] J. R. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, 134 North-Holland, Amsterdam, 1987. [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. [21] X. Xu, Existence theorems for the quantum drift-diffusion system with mixed boundary conditions, Commun. Contemp. Math. , 18 (2016), 1550048, 21 pp.
 [1] Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159 [2] Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007 [3] Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062 [4] Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109 [5] Chengxiang Wang, Li Zeng, Wei Yu, Liwei Xu. Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction. Inverse Problems & Imaging, 2018, 12 (3) : 545-572. doi: 10.3934/ipi.2018024 [6] Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016 [7] VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 [8] Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1441-1460. doi: 10.3934/dcds.2018059 [9] Qianying Xiao, Zuohuan Zheng. $C^1$ weak Palis conjecture for nonsingular flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1809-1832. doi: 10.3934/dcds.2018074 [10] Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng. Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1889-1933. doi: 10.3934/dcds.2018077 [11] Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058 [12] Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035 [13] Renato Huzak. Cyclicity of degenerate graphic $DF_{2a}$ of Dumortier-Roussarie-Rousseau program. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1305-1316. doi: 10.3934/cpaa.2018063 [14] María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations & Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018 [15] Lianjun Zhang, Lingchen Kong, Yan Li, Shenglong Zhou. A smoothing iterative method for quantile regression with nonconvex $\ell_p$ penalty. Journal of Industrial & Management Optimization, 2017, 13 (1) : 93-112. doi: 10.3934/jimo.2016006 [16] Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅱ. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074 [17] Shu-Cherng Fang, David Y. Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅰ. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1291-1305. doi: 10.3934/jimo.2016073 [18] Adel Alahmadi, Cem Güneri, Hatoon Shoaib, Patrick Solé. Long quasi-polycyclic $t-$ CIS codes. Advances in Mathematics of Communications, 2018, 12 (1) : 189-198. doi: 10.3934/amc.2018013 [19] Joaquim Borges, Cristina Fernández-Córdoba, Roger Ten-Valls. On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 169-179. doi: 10.3934/amc.2018011 [20] Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2018087

2016 Impact Factor: 1.261