Global existence of strong solutions to a biological network formulation model in 2+1 dimensions

Previous Article
Matching for a family of infinite measure continued fraction transformations
DCDS Home
This Issue
Next Article
On Morawetz estimates with time-dependent weights for the Klein-Gordon equation

November 2020, 40(11): 6289-6307. doi: 10.3934/dcds.2020280

Global existence of strong solutions to a biological network formulation model in 2+1 dimensions

Xiangsheng Xu ^,

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

^* Corresponding author: Xiangsheng Xu

Received December 2019 Revised May 2020 Published July 2020

In this paper we study the initial boundary value problem for the system $ -\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right] = s(x), \ \ \mathbf{m}_t-\alpha^2\Delta\mathbf{m}+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m} = \beta^2(\mathbf{m}\cdot\nabla p)\nabla p $ in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical challenge is due to the presence of cubic nonlinearities, also known as trilinear forms, in the system. We obtain a weak solution $ (\mathbf{m}, p) $ with both $ |\nabla p| $ and $ |\nabla\mathbf{m}| $ being bounded. The result immediately triggers a bootstrap argument which can yield higher regularity for the weak solution. This is achieved by deriving an equation for $ v\equiv(I+\mathbf{m} \mathbf{m}^T)\nabla p\cdot\nabla p $, and then suitably applying the De Giorge iteration method to the equation.

Keywords: Biological network formulation, cubic nonlinearity, De Giorgi iteration method.

Mathematics Subject Classification: Primary: 35B45, 35B65, 35M33, 35Q92.

Citation: Xiangsheng Xu. Global existence of strong solutions to a biological network formulation model in 2+1 dimensions. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6289-6307. doi: 10.3934/dcds.2020280

References:

[1]	G. Albi, M. Artina, M. Fornasier and P. A. Markowich, Biological transportation networks: Modeling and simulation, Anal. Appl. (Singap.), 14 (2016), 185-206. doi: 10.1142/S0219530515400059.
[2]	G. Albi, M. Burger, J. Haskovec, P. Markowich and M. Schlottbom, Continuum modeling of biological network formulation, Active Particles Vol.I - Advances in Theory, Models, and Applications, 1–48, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2017.
[3]	G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 229-256.
[4]	S. Bernstein, Sur la généralization du problème de Dirichlet, Math. Ann., 62 (1906), 253-271. doi: 10.1007/BF01449980.
[5]	E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[6]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.
[7]	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.
[8]	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.
[9]	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.
[10]	D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Phys. Rev. Lett., 111 (2013), 138701. doi: 10.1103/PhysRevLett.111.138701.
[11]	Q. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Tran. Math. Monographs, Vol. 23, AMS, Providence, RI, 1968.
[12]	B. Li, On the blown-up criterion and global existence of a nonlinear PDE system in biological transportation networks, Kinet. Relat. Models, 12 (2019), 1131–1162. doi: 10.3934/krm.2019043.
[13]	J.-G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, J. Differential Equations, 264 (2018), 5489–5526. doi: 10.1016/j.jde.2018.01.001.
[14]	N. G. Meyers, An $L^{p}$e-estimate for the gradient of solution of second order elliptic divergence equations, Ann. Scuola Norm. Pisa Cl. Sci. (3), 17 (1963), 189–206.
[15]	L. A. Peletier and J. Serrin, Gradient bounds and Liouville theorems for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 65–104.
[16]	J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, Vol. 134, North-Holland, Amsterdam, 1987.
[17]	J. Shen and B. Li, A Priori estimates for a nonlinear system with some essential symmetrical structures, Symmetry, 11 (2019), Article # 852. doi: 10.3390/sym11070852.
[18]	R. P. Sperb, Maximum Principle and their Applications, Academic Press, New York, 1981.
[19]	X. Xu, Partial regularity of solutions to a class of degenerate systems, Trans. Amer. Math. Soc., 349 (1997), 1973–1992. doi: 10.1090/S0002-9947-97-01734-0.
[20]	X. Xu, Regularity theorems for a biological network formulation model in two space dimensions, Kinet. Relat. Models, 11 (2018), 397-408. doi: 10.3934/krm.2018018.
[21]	X. Xu, Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model, SN Partial Differential Equations and Applications, to appear., arXiv: 1706.06057, V5, 2018.
[22]	X. Xu, Global existence of strong solutions to a groundwater flow problem, Z. angew. Math. Phys., 71 (2020), to appear. arXiv: 1912.03793 [math.AP], 2019.
[23]	G. Yuan, Regularity of solutions of the thermistor problem, Appl. Anal., 53 (1994), 149-156. doi: 10.1080/00036819408840253.

show all references

References:

[1]	G. Albi, M. Artina, M. Fornasier and P. A. Markowich, Biological transportation networks: Modeling and simulation, Anal. Appl. (Singap.), 14 (2016), 185-206. doi: 10.1142/S0219530515400059.
[2]	G. Albi, M. Burger, J. Haskovec, P. Markowich and M. Schlottbom, Continuum modeling of biological network formulation, Active Particles Vol.I - Advances in Theory, Models, and Applications, 1–48, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2017.
[3]	G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 229-256.
[4]	S. Bernstein, Sur la généralization du problème de Dirichlet, Math. Ann., 62 (1906), 253-271. doi: 10.1007/BF01449980.
[5]	E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.
[6]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.
[7]	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.
[8]	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.
[9]	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.
[10]	D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Phys. Rev. Lett., 111 (2013), 138701. doi: 10.1103/PhysRevLett.111.138701.
[11]	Q. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Tran. Math. Monographs, Vol. 23, AMS, Providence, RI, 1968.
[12]	B. Li, On the blown-up criterion and global existence of a nonlinear PDE system in biological transportation networks, Kinet. Relat. Models, 12 (2019), 1131–1162. doi: 10.3934/krm.2019043.
[13]	J.-G. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, J. Differential Equations, 264 (2018), 5489–5526. doi: 10.1016/j.jde.2018.01.001.
[14]	N. G. Meyers, An $L^{p}$e-estimate for the gradient of solution of second order elliptic divergence equations, Ann. Scuola Norm. Pisa Cl. Sci. (3), 17 (1963), 189–206.
[15]	L. A. Peletier and J. Serrin, Gradient bounds and Liouville theorems for quasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 65–104.
[16]	J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, Vol. 134, North-Holland, Amsterdam, 1987.
[17]	J. Shen and B. Li, A Priori estimates for a nonlinear system with some essential symmetrical structures, Symmetry, 11 (2019), Article # 852. doi: 10.3390/sym11070852.
[18]	R. P. Sperb, Maximum Principle and their Applications, Academic Press, New York, 1981.
[19]	X. Xu, Partial regularity of solutions to a class of degenerate systems, Trans. Amer. Math. Soc., 349 (1997), 1973–1992. doi: 10.1090/S0002-9947-97-01734-0.
[20]	X. Xu, Regularity theorems for a biological network formulation model in two space dimensions, Kinet. Relat. Models, 11 (2018), 397-408. doi: 10.3934/krm.2018018.
[21]	X. Xu, Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model, SN Partial Differential Equations and Applications, to appear., arXiv: 1706.06057, V5, 2018.
[22]	X. Xu, Global existence of strong solutions to a groundwater flow problem, Z. angew. Math. Phys., 71 (2020), to appear. arXiv: 1912.03793 [math.AP], 2019.
[23]	G. Yuan, Regularity of solutions of the thermistor problem, Appl. Anal., 53 (1994), 149-156. doi: 10.1080/00036819408840253.

[1]	Xiangsheng Xu. Regularity theorems for a biological network formulation model in two space dimensions. Kinetic and Related Models, 2018, 11 (2) : 397-408. doi: 10.3934/krm.2018018
[2]	Luis A. Caffarelli, Alexis F. Vasseur. The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 409-427. doi: 10.3934/dcdss.2010.3.409
[3]	Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017
[4]	Changfeng Gui. On some problems related to de Giorgi’s conjecture. Communications on Pure and Applied Analysis, 2003, 2 (1) : 101-106. doi: 10.3934/cpaa.2003.2.101
[5]	Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[6]	Paolo Baroni, Agnese Di Castro, Giampiero Palatucci. Intrinsic geometry and De Giorgi classes for certain anisotropic problems. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 647-659. doi: 10.3934/dcdss.2017032
[7]	Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[8]	Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[9]	Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339
[10]	Jiahui Chen, Rundong Zhao, Yiying Tong, Guo-Wei Wei. Evolutionary de Rham-Hodge method. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3785-3821. doi: 10.3934/dcdsb.2020257
[11]	Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373
[12]	Zhong-Zhi Bai. On convergence of the inner-outer iteration method for computing PageRank. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 855-862. doi: 10.3934/naco.2012.2.855
[13]	Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033
[14]	Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[15]	Chiara Zanini, Fabio Zanolin. Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4045-4067. doi: 10.3934/dcds.2012.32.4045
[16]	Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973
[17]	Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239
[18]	Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011
[19]	Kolade M. Owolabi, Kailash C. Patidar, Albert Shikongo. Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 591-613. doi: 10.3934/dcdss.2019038
[20]	Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Xiangsheng Xu
on Google Scholar
Xiangsheng Xu

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Xiangsheng Xu
on Google Scholar
Xiangsheng Xu

[Back to Top]