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.

中文翻译：

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全球存在2 + 1维生物网络配方模型的强大解决方案

在本文中，我们研究系统$-\ mbox {div} \ left [（I + \ mathbf {m} \ mathbf {m} ^ T）\ nabla p \ right] = s（x）的初始边值问题， \ \ \ mathbf {m} _t- \ alpha ^ 2 \ Delta \ mathbf {m} + | \ mathbf {m} | ^ {2（\ gamma-1）} \ mathbf {m} = \ beta ^ 2（\ mathbf {m} \ cdot \ nabla p）\ nabla p $在两个空间维度上。已经提出该问题作为生物运输网络的连续模型。数学上的挑战是由于系统中存在立方非线性，也称为三线性形式。我们获得了一个弱解$（\ mathbf {m}，p）$，其中两个$ | \ nabla p | $和$ | \ nabla \ mathbf {m} | $受限制。结果立即触发自举参数，可为弱解产生更高的规律性。这是通过导出$ v \ equiv（I + \ mathbf {m} \ mathbf {m} ^ T）\ nabla p \ cdot \ nabla p $的方程来实现的，