Nonlinear Degenerate Parabolic System Research Articles

Weak solutions for a degenerate phase‐field model via Galerkin approximation

This paper is concerned with a phase‐field model driven by interface diffusion, which is an elliptic and nonlinear degenerate parabolic coupled system arising in a theory of interface motion in the elastically deformable materials. Spinodal decomposition is a typical example for this motion. We shall study the global existence of weak solutions to this model with a general polynomial potential function via the Galerkin method. Moreover, we perform the numerical simulations to investigate the spinodal decomposition process of binary alloy by applying this model.

Analytical Solutions with Zero Front to the Nonlinear Degenerate Parabolic System

Analytical Solutions with Zero Front to the Nonlinear Degenerate Parabolic System

Weak solutions of coupled variable-density flows and heat transfer in porous media

We consider a fully nonlinear degenerate parabolic system describing the three-dimensional variable-density Darcian flow of a heat conducting fluid through porous media. We establish the global existence of weak solutions to the relevant initial-boundary value problems in nonsmooth domains under physically reasonable hypotheses on the data and mixed boundary conditions.

The Cauchy problem for a nonlinear degenerate parabolic system in non-divergence form

We deal with degenerate quasilinear parabolic systems in the non-divergence form under positive initial conditions. An asymptotic behavior of self-similar solutions in the case of slow diffusion is established. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. In addition, the asymptotic representation of the solution is obtained.

Modeling adhesion-independent cell migration

A two-dimensional mathematical model for cells migrating without adhesion capabilities is presented and analyzed. Cells are represented by their cortex, which is modeled as an elastic curve, subject to an internal pressure force. Net polymerization or depolymerization in the cortex is modeled via local addition or removal of material, driving a cortical flow. The model takes the form of a fully nonlinear degenerate parabolic system. An existence analysis is carried out by adapting ideas from the theory of gradient flows. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for adhesion-independent motility.

Liouville-Type Theorem for a Nonlinear Degenerate Parabolic System of Inequalities

In this paper, we establish some new Liouville-type results for solutions of nonlinear degenerate parabolic system of inequalities. Nonexistence of nontrivial global solutions to initial-value problems is studied by using scaling transformations and test functions.

On the existence of weak solutions of nonlinear degenerate parabolic system with variable exponents

The main goal of the present paper is establishing the existence and uniqueness of weak solutions for the nonlinear degenerate reaction–diffusion system with variable exponents. A model also is proposed to characterize the invasion of cancer cells towards healthy cells with acidification environment. Moreover, the main results of this paper are obtained using regularization problem, the Faedo–Galerkin approximation method, some apriori estimates, compactness results and the Gronwall Lemma.

Blow-up and life span estimates for a class of nonlinear degenerate parabolic system with time-dependent coefficients

This paper deals with the singularity and global regularity for a class of nonlinear porous medium system with time-dependent coefficients under homogeneous Dirichlet boundary conditions. First, by comparison principle, some global regularity results are established. Secondly, using some differential inequality technique, we investigate the blow-up solution to the initial-boundary value problem. Furthermore, upper and lower bounds for the maximum blow-up time under some appropriate hypotheses are derived as long as blow-up occurs.

Lp Estimates for Weak Solutions to Nonlinear Degenerate Parabolic Systems

This paper is devoted to the Lp estimates for weak solutions to nonlinear degenerate parabolic systems related to Hörmander’s vector fields. The reverse Hölder inequalities for degenerate parabolic system under the controllable growth conditions and natural growth conditions are established, respectively, and an important multiplicative inequality is proved; finally, we obtain the Lp estimates for the weak solutions by combining the results of Gianazza and the Caccioppoli inequality.

Homogenization of degenerate coupled fluid flows and heat transport through porous media

We establish a homogenization result for a fully nonlinear degenerate parabolic system with critical growth arising from the heat and moisture flow through a partially saturated porous media. Existence of a global weak solution of the mesoscale problem is proven by means of a semidiscretization in time, a priori estimates and passing to the limit from discrete approximations. After that, porous material exhibiting periodic spatial oscillations is considered and the two-scale convergence (as the oscillation period vanishes) to a corresponding homogenized problem is rigorously proven.

Weak solutions of coupled dual porosity flows in fractured rock mass and structured porous media

This paper deals with a fully nonlinear degenerate parabolic system with natural (critical) growths and under non-linear boundary conditions. Such problems arise from the heat and water flow through a partially saturated fractured rock mass and structured porous media. Existence of a global weak solution of the problem (on an arbitrary interval of time) is proved by means of semidiscretization in time, deriving suitable a-priori estimates based on W1,p-regularity of the approximate solution and by passing to the limit from discrete approximations.

On doubly nonlinear elliptic-parabolic systems arising in coupled transport phenomena in unsaturated porous media

We consider an initial-boundary value problem for a fully nonlinear degenerate parabolic system. The mathematical problem is originally motivated by coupled Richards and heat equations modelling hygro-thermal flows in unsaturated porous media and fractured rock masses. We prove a global existence of a weak solution to this problem on an arbitrary interval of time.

Global existence and blow-up of solutions to a degenerate parabolic system with nonlocal sources and nonlocal boundaries

This paper deals with a nonlinear degenerate parabolic system with nonlocal source and nonlocal boundaries. By super-solution, sub-solution and auxiliary functions, a criteria for nonnegative solution of global existence and blow-up in finite time is obtained for this degenerate nonlocal problem. Finally, the blow-up rates of blow-up solutions are also estimated.

Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository

Our goal is the mathematical analysis of a two phase (liquid and gas) two components (water and hydrogen) system modeling the hydrogen displacement in a storage site for radioactive waste. We suppose that the water is only in the liquid phase and is incompressible. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water with the Henry law. The flow is described by the conservation of the mass of each components. The model is treated without simplified assumptions on the gas density. This model is degenerated due to vanishing terms. We establish an existence result for the nonlinear degenerate parabolic system based on new energy estimate on pressures.

Existence and approximation of a nonlinear degenerate parabolic system modelling acid-mediated tumour invasion

We consider a nonlinear parabolic system of reaction–diffusion equations modelling acid-mediated tumour invasion. The system couples potentially degenerate equations for the cell densities of the normal and tumour populations to a parabolic equation for the concentration of HC ions. We obtain an existence result for the system by constructing a suitable finite element approximation and analyzing its convergence. Finally, we report on corresponding numerical experiments.

New perturbation methods for nonlinear parabolic problems

We develop methods aimed at deriving regularity results for solutions to nonlinear degenerate parabolic equations and systems via local perturbation; as a consequence we obtain, in a unified way, Lipschitz continuity of solutions under weak parabolicity assumptions, and gradient continuity results in borderline cases. Nonlinear Schauder estimates as those of Misawa (2002) [29] are recovered and extended to more general settings.

On a fully nonlinear degenerate parabolic system modeling immiscible gas–water displacement in porous media

In this paper, we are interested in the simultaneous flow of two immiscible fluid phases within a porous medium. We consider a two-phase flow model where the fluids are immiscible and there is no mass transfer between the phases. The medium is saturated by compressible/incompressible phase flows. We study the gas–water displacement without simplified assumptions on the state law of gas density. We establish an existence result for the nonlinear degenerate parabolic system based on new energy estimate on pressures.

A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity

We consider a system of partial differential equationswhich describes anti-plane shear in the context of astrain-gradient theory of plasticity proposed by Gurtin in[6]. The problem couples a fully nonlineardegenerate parabolic system and an elliptic equation. Itfeatures two types of degeneracies: the first one iscaused by the nonlinear structure, the second one by thedependence of the principal part on twice the curl of aplanar vector field. Furthermore, the elliptic equationdepends on the divergence of such vector field - which isnot controlled by twice the curl - and the boundaryconditions suggested in [6] are of mixedtype. To overcome the latter complications we use a suitable,time-dependent representation of a divergence-free vectorfield which plays the role of the elastic stress.To handle the nonlinearities, by a suitable reformulationof the problem we transform the original system into onesatisfying a monotonicity property which is more 'robust'than the gradient flow structure inherited as an intrinsicfeature of the mechanical model. These two insights makeit possible to prove existence and uniqueness of a solution to theoriginal system.

Traveling wave solutions of a nonlinear degenerate parabolic system from petroleum engineering

We study existence and qualitative properties of traveling wave solutions of a new free boundary problem which describes fluid flow in diatomite rocks. Diatomites are rather fragile and are characterized by a low permeability, which can increase due to the nonlocal accumulation of damage caused by the fluid flow. The traveling wave solutions give insight in the behavior near the free boundaries and show a strong parameter dependence. In particular we find in certain parameter ranges solutions with discontinuities across the free boundaries.

Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy inequality controlling the Laplacian of the liquid heights. We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analogues of these energy inequalities. Finally, we prove convergence of this approximation, and hence existence of a solution to this nonlinear degenerate parabolic system.