论文标题
标量保护定律和应用于肿瘤生长模型的多维有限体积近似的强界变化估计值
Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth model
论文作者
论文摘要
非线性标量保护定律的有限体积近似值$ \partial_tα+ \ mathrm {div}（\ boldsymbol {u} f（α）f（α））= 0 $在两个和三个空间维度中建立了界限变化的初始数据。我们假设速度$ \ mathrm {div}（\ boldsymbol {u}）$的差异是有界变化的，而不是经典假设，即$ \ mathrm {div}（\ boldsymbol {u}）$为零。本文中分析的有限体积方案设置为笛卡尔网格。保护定律的有限量解决方案$ \partial_tα+ \ mathrm {div}（\ boldsymbol {f}（t，t，\ boldsymbol {x}，α））= 0 $的统一变化估计值。 \ \ not = 0 $在非均匀的笛卡尔网格上也证明了。这样的估计值可为$ l^p $空间中的有限体积近似值提供紧凑性，这对于证明存在$α$的偏微分方程的解决方案至关重要。通过建立一个弱解决方案的存在来证明这一应用，该模型描述了S. J. Franks等人提出的乳腺癌初始阶段的演变。 [14]。该模型由四个耦合变量组成：肿瘤细胞浓度，肿瘤细胞速度 - 压力和营养浓度，分别由双曲线保护定律，粘性Stokes System和Poisson方程组成。提供了数值测试的结果，并补充了理论发现。
A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law $\partial_t α+ \mathrm{div}(\boldsymbol{u}f(α)) = 0$ in two and three spatial dimensions with an initial data of bounded variation is established. We assume that the divergence of the velocity $\mathrm{div}(\boldsymbol{u})$ is of bounded variation instead of the classical assumption that $\mathrm{div}(\boldsymbol{u})$ is zero. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law $\partial_t α+ \mathrm{div}(\boldsymbol{F}(t,\boldsymbol{x},α)) = 0$, where $\mathrm{div}_{\boldsymbol{x}}\boldsymbol{F} \not=0$ on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in $L^p$ spaces, which is essential to prove the existence of a solution for a partial differential equation with nonlinear terms in $α$, when the uniqueness of the solution is not available. This application is demonstrated by establishing the existence of a weak solution for a model that describes the evolution of initial stages of breast cancer proposed by S. J. Franks et al. [14]. The model consists of four coupled variables: tumour cell concentration, tumour cell velocity--pressure, and nutrient concentration, which are governed by a hyperbolic conservation law, viscous Stokes system, and Poisson equation, respectively. Results from numerical tests are provided and they complement theoretical findings.
