    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 & Continuous Dynamical Systems - A, 2020, 40 (11) : 6289-6307. doi: 10.3934/dcds.2020280
##### References:
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##### References:
  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.  Google Scholar  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. Google Scholar  G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 229-256. Google Scholar  S. Bernstein, Sur la généralization du problème de Dirichlet, Math. Ann., 62 (1906), 253-271.  doi: 10.1007/BF01449980.  Google Scholar  E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar  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.  Google Scholar  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  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  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  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  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Math. Studies, Vol. 134, North-Holland, Amsterdam, 1987. Google Scholar  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. Google Scholar  R. P. Sperb, Maximum Principle and their Applications, Academic Press, New York, 1981. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  G. Yuan, Regularity of solutions of the thermistor problem, Appl. Anal., 53 (1994), 149-156.  doi: 10.1080/00036819408840253.  Google Scholar
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