American Institute of Mathematical Sciences

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

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

Xiangsheng Xu

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.

Keywords: $ A_q$-weights, biological network formulation, bounded mean oscillation (BMO), weakly monotone functions.
Mathematics Subject Classification: Primary: 35D30, 35Q99; Secondary: 35A01.
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

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.  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

[1]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033
[2]

Jiao Du, Longjiang Qu, Chao Li, Xin Liao. Constructing 1-resilient rotation symmetric functions over $ {\mathbb F}_{p} $ with $ {q} $ variables through special orthogonal arrays. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020018
[3]

Annalisa Cesaroni, Serena Dipierro, Matteo Novaga, Enrico Valdinoci. Minimizers of the $ p $-oscillation functional. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6785-6799. doi: 10.3934/dcds.2019231
[4]

Pak Tung Ho. Prescribing the $ Q' $-curvature in three dimension. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2285-2294. doi: 10.3934/dcds.2019096
[5]

Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029
[6]

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
[7]

Shixiong Wang, Longjiang Qu, Chao Li, Huaxiong Wang. Further improvement of factoring $ N = p^r q^s$ with partial known bits. Advances in Mathematics of Communications, 2019, 13 (1) : 121-135. doi: 10.3934/amc.2019007
[8]

Antonio Cossidente, Francesco Pavese, Leo Storme. Optimal subspace codes in $ {{\rm{PG}}}(4,q) $. Advances in Mathematics of Communications, 2019, 13 (3) : 393-404. doi: 10.3934/amc.2019025
[9]

Guangfeng Dong, Changjian Liu, Jiazhong Yang. On the maximal saddle order of $ p:-q $ resonant saddle. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5729-5742. doi: 10.3934/dcds.2019251
[10]

Claudianor O. Alves, Vincenzo Ambrosio, Teresa Isernia. Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2009-2045. doi: 10.3934/cpaa.2019091
[11]

Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056
[12]

Michael Magee, Rene Rühr. Counting saddle connections in a homology class modulo $ \boldsymbol q $ (with an appendix by Rodolfo Gutiérrez-Romo). Journal of Modern Dynamics, 2019, 15: 237-262. doi: 10.3934/jmd.2019020
[13]

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
[14]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
[15]

Haisheng Tan, Liuyan Liu, Hongyu Liang. Total $\{k\}$-domination in special graphs. Mathematical Foundations of Computing, 2018, 1 (3) : 255-263. doi: 10.3934/mfc.2018011
[16]

Ekta Mittal, Sunil Joshi. Note on a $ k $-generalised fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 797-804. doi: 10.3934/dcdss.2020045
[17]

Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. A spectral characterisation of $ t $-designs and its applications. Advances in Mathematics of Communications, 2019, 13 (3) : 477-503. doi: 10.3934/amc.2019030
[18]

Gang Wang, Yuan Zhang. $ Z $-eigenvalue exclusion theorems for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019039
[19]

Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019042
[20]

Zalman Balanov, Yakov Krasnov. On good deformations of $ A_m $-singularities. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1851-1866. doi: 10.3934/dcdss.2019122

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (39)
  • HTML views (222)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences