# American Institute of Mathematical Sciences

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

 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.

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.

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##### 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.
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