A level-set method for the evolution of cells and tissue during curvature-controlled growth

Most biological tissues grow by the synthesis of new material close to the tissue's interface, where spatial interactions can exert strong geometric influences on the local rate of growth. These geometric influences may be mechanistic, or cell behavioural in nature. The control of geometry on tissue growth has been evidenced in many in-vivo and in-vitro experiments, including bone remodelling, wound healing, and tissue engineering scaffolds. In this paper, we propose a generalisation of a mathematical model that captures the mechanistic influence of curvature on the joint evolution of cell density and tissue shape during tissue growth. This generalisation allows us to simulate abrupt topological changes such as tissue fragmentation and tissue fusion, as well as three dimensional cases, through a level-set-based method. The level-set method developed introduces another Eulerian field than the level-set function. This additional field represents the surface density of tissue synthesising cells, anticipated at future locations of the interface. Numerical tests performed with this level-set-based method show that numerical conservation of cells is a good indicator of simulation accuracy, particularly when cusps develop in the tissue's interface. We apply this new model to several situations of curvature-controlled tissue evolutions that include fragmentation and fusion.Comment: 15 pages, 10 figures, 3 supplementary figure

Appropriate model use for predicting elevations and inundation extent for extreme flood events

Flood risk assessment is generally studied using flood simulation models; however, flood risk managers often simplify the computational process; this is called a “simplification strategy”. This study investigates the appropriateness of the “simplification strategy” when used as a flood risk assessment tool for areas prone to flash flooding. The 2004 Boscastle, UK, flash flood was selected as a case study. Three different model structures were considered in this study, including: (1) a shock-capturing model, (2) a regular ADI-type flood model and (3) a diffusion wave model, i.e. a zero-inertia approach. The key findings from this paper strongly suggest that applying the “simplification strategy” is only appropriate for flood simulations with a mild slope and over relatively smooth terrains, whereas in areas susceptible to flash flooding (i.e. steep catchments), following this strategy can lead to significantly erroneous predictions of the main parameters—particularly the peak water levels and the inundation extent. For flood risk assessment of urban areas, where the emergence of flash flooding is possible, it is shown to be necessary to incorporate shock-capturing algorithms in the solution procedure, since these algorithms prevent the formation of spurious oscillations and provide a more realistic simulation of the flood levels

Null injectivity estimate under an upper bound on the curvature

We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by space-like hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse function of the foliation and sup-norm bounds on the deformation tensors of the foliation. Our proof is inspired by techniques from Riemannian geometry, and it should be noted that we impose no restriction on the size of the bound satisfied by the curvature or deformation tensors, and allow for metrics that are “far” from the Minkowski one. The relevance of our estimate is illustrated with a class of plane-symmetric spacetimes which satisfy our assumptions but admit no uniform lower bound on the curvature not even in the L2 norm. The conditions we put forward, therefore, lead to a uniform control of the spacetime geometry and should be useful in the context of general relativity

Entropies and flux-splittings for the isentropic Euler equations

The authors establish the existence of a large class of mathematical entropies (the so-called weak entropies) associated with the Euler equations for an isentropic, compressible fluid governed by a general pressure law. A mild assumption on the behavior of the pressure law near the vacuum is solely required. The analysis is based on an asymptotic expansion of the fundamental solution (called here the entropy kernel) of a highly singular Euler-Poisson-Darboux equation. The entropy kernel is only Hölder continuous and its regularity is carefully investigated. Relying on a notion introduced earlier by the authors, it is also proven that, for the Euler equations, the set of entropy flux-splittings coincides with the set of entropies-entropy fluxes. These results imply the existence of a flux-splitting consistent with all of the entropy inequalities

Null injectivity estimate under an upper bound on the curvature

We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by space-like hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse function of the foliation and sup-norm bounds on the deformation tensors of the foliation. Our proof is inspired by techniques from Riemannian geometry, and it should be noted that we impose no restriction on the size of the bound satisfied by the curvature or deformation tensors, and allow for metrics that are “far” from the Minkowski one. The relevance of our estimate is illustrated with a class of plane-symmetric spacetimes which satisfy our assumptions but admit no uniform lower bound on the curvature not even in the L2 norm. The conditions we put forward, therefore, lead to a uniform control of the spacetime geometry and should be useful in the context of general relativity

Compressible Euler equations with general pressure law

We study the hyperbolic system of Euler equations for an isentropic, compressible fluid governed by a general pressure law. The existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new Euler-Poisson-Darboux equation, which is highly singular when the density of the fluid vanishes. New properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy-flux kernel are found. We prove the strong compactness of any sequence that is uniformly bounded in L∞ and whose corresponding sequence of weak entropy dissipation measures is locally H-1 compact. The existence and large-time behavior of L∞ entropy solutions of the Cauchy problem are established. This is based on a reduction theorem for Young measures, whose proof is new even for the polytropic perfect gas. The existence result also extends to the p-system of fluid dynamics in Lagrangian coordinates

Existence theory for the isentropic Euler equations

We establish an existence theorem for entropy solutions to the Euler equations modeling isentropic compressible fluids. We develop a new approach for constructing mathematical entropies for the Euler equations, which are singular near the vacuum. In particular, we identify the optimal assumption required on the singular behavior on the pressure law at the vacuum in order to validate the two-term asymptotic expansion of the entropy kernel we proposed earlier. For more general pressure laws, we introduce a new multiple-term expansion based on the Bessel functions with suitable exponents, and we also identify the optimal assumption needed to validate the multiple-term expansion and to establish the existence theory. Our results cover, as a special example, the density-pressure law p(ρ) = κ1 ργ1 + κ2 ργ2 where γ1, γ2 ∈ (1, 3) and κ1, κ2 > 0 are arbitrary constants