We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value $γ_*>0$ for the surface tension coefficient $γ$, such that in the case $γ>γ_*$ this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable.
A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differenti…
This paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent contributions arXiv:1401.5943 [math.AP] and arXiv:1501.07057 [math.AP] from the viewpoint of well-posedness, long time behavior and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in arXiv:1501.07057 [math.AP] by showing two indepen…
We study a moving boundary problem describing the growth of nonnecrotic tumors in different regimes of vascularisation. This model consists of two decoupled Dirichlet problem, one for the rate at which nutrient is added to the tumor domain and one for the pressure inside the tumor. These variables are coupled by a relation which describes the dynamic of the boundary. By re-expressing the problem as an abstract evolution equation, we prove local well-posedness in the small Hoelder spaces context. Further on, we use the principle of linearised stability to characterise the stability properties o…
Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short time.
In this paper we study asymptotic behavior of solutions for a multidimensional free boundary problem modelling the growth of nonnecrotic tumors. We first establish a general result for differential equations in Banach spaces possessing a local Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either $D_A(θ)$ or $D_A(θ,\infty)$ type. By using this general result and making delicate analysis of the spectrum of …
A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of $L^2$-based Sobolev spaces. The system is transformed into a generalized Stokes problem, and the proper conditions on the parameters of the model that guarantee that the system is uniformly elliptic are given. Under those conditions, existence, uniqueness and low-order regularity are obtained under classical regularity conditions on the data, while higher-order regularity is proved under less stringent conditions than classical ones. Explicit est…
Within the framework of variational modelling we derive a two-phase moving boundary problem that describes the motion of a semipermeable membrane separating two viscous liquids in a fixed container. The model includes the effects of osmotic pressure and surface tension of the membrane. For this problem we prove that the manifold of steady states is locally exponentially attractive.
In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalization of a phase field system of Cahn-Hilliard type modelling tumor growth that has been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24) and investigated in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn-Hilliard equation for the tumor cell fraction, coupled to a reaction-diffusion equation for a f…
A new construction technique of multiple solutions of the Euler equa- tion in strong spaces is introduced which reveals the relationship to multi- ple Navier Stokes equation solutions with special force terms while avoid- ing viscosity limit constructions. This shows that severe restrictions have to be imposed on time dependent external force terms in order to ob- tain uniqueness for the Navier Stokes equation Cauchy problem. Such restrictions are imposed in the statement of the so-called millenium prob- lem. Minimal turbulence models should arguably incorporate weaker force terms in order to …
This paper is concerned with the boundary layer problem on a chemotaxis-Navier-Stokes system modelling boundary layer formation of aerobic bacteria in fluid. Completing the system with physical Robin-type boundary conditions for oxygen, no-flux and Dirichlet boundary conditions for bacteria and fluid velocity, we show that the gradients of its radial solutions in a region between two concentric spheres possessing boundary layer effects as the oxygen diffusion rate $\varepsilon$ goes to zero and the boundary-layer thickness is of order $\mathcal{O}(\varepsilon^α)$ with $0<α<\frac{1}{2}$.
In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-system and a fixed point argument. In addition, stability results of the critical points are given under some constraints on parameters. Finally, we design a fully discrete finite element scheme for the model which preserves the pointwise and energy es…
We consider a chemotaxis-Navier-Stokes system modelling cellular swimming in fluid drops where an exchange of oxygen between the drop and its environment is taken into account. This phenomenon results in an inhomogeneous Robin-type boundary condition. Moreover, the system is studied without the logistic growth of the bacteria population. We prove that in two dimensions, the system has a unique global classical solution, while the existence of a global weak solution is shown in three dimensions. In the latter case, we show that the energy is bounded uniformly in time. A key idea is to utilise a…
Navier-Stokes Modelling of Non-Newtonian Blood Flow in Cerebral Arterial Circulation and its Dynamic Impact on Electrical Conductivity in a Realistic Multi-Compartment Head Model
本文提出基于 Navier-Stokes 方程和 Carreau-Yasuda 模型的非牛顿脑血流动态建模方法，结合 Fick 定律与 Archie 定律，在真实多隔室头部模型中近似血流及电导率分布，适用于先验信息有限的任意可区分动脉的头部模型。
We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier-Stokes-Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an L p-L q setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the R-sectoriality of the co…
In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn-Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a C…
Analyticity of the data-to-solution map for a stationary Navier-Stokes fluid-structure interaction problem
University of Zagreb Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程学院与计算学院）；University of Zagreb Faculty of Science, Department of Mathematics（萨格勒布大学理学院数学系）
The Nernst-Planck-Stokes (NPS) system models electroconvection of ions in a fluid. We consider the system, for two oppositely charged ionic species, on three dimensional bounded domains with Dirichlet boundary conditions for the ionic concentrations (modelling ion selectivity), Dirichlet boundary conditions for the electrical potential (modelling an applied potential), and no-slip boundary conditions for the fluid velocity. In this paper, we obtain quantitative bounds on solutions of the NPS system in the long time limit, which we use to prove 1) the existence of a compact global attractor wit…
We study a nonlinear coupled fluid-structure system modelling the blood flow through arteries. The fluid is described by the incompressible Navier-Stokes equations in a 2D rectangular domain where the upper part depends on a structure satisfying a damped Euler-Bernoulli beam equation. The system is driven by time-periodic source terms on the inflow and outflow boundaries. We prove the existence of time-periodic strong solutions for this problem under smallness assumptions for the source terms.
We study a dissipative nonlinear equation modelling certain features of the Navier-Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions $n\leq 4$. For dimensions $n>4$ we present strong numerical evidence supporting existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding existence of self-similar singular solutions to a semi-linear heat equation.
