Nonlinear Degenerate Parabolic Equations Research Articles

Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory

We obtain solutions of the nonlinear degenerate parabolic equation \[ \frac{\partial \rho}{\partial t}= \mbox{div} \Big\{ \rho \nabla c^\star \left[ \nabla \left(F^\prime(\rho)+V\right) \right] \Big\} \] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires fewer uniform convexity assumptions than those imposed by Alt and Luckhaus in their pioneering work [4]. In fact, their assumptions may fail in our equation. This class of equations includes the Fokker-Planck equation, the porous-medium equation, the fast diffusion equation and the parabolic p-Laplacian equation.

Cauchy‐Dirichlet problem for the nonlinear degenerate parabolic equations

We will investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation: ∂u/∂t = ℒu + V(w)up−1 in Ω × (0, T), 1 < p < 2, u(w, 0) = u0(w) ≥ 0 in Ω, u(w, t) = 0 on ∂Ω × (0, T) where ℒ is the subelliptic p‐Laplacian and .

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.

Approximation of a nonlinear degenerate parabolic equation via a linear relaxation scheme

AbstractWe study a nonlinear degenerate parabolic equation of the type accompanied by an initial datum and mixed boundary conditions. The symbol [ · ]+ denotes the usual cutoff function. The problem represents a model of a reactive solute transport in porous media. The exponent p fulfills p ∈ (0, 1). This limits the regularity of a solution and leads to inconveniences in the error analysis. We design a new robust linear numerical scheme for the time discretization. This is based on a suitable combination of the backward Euler method and a linear relaxation scheme. We prove the convergence of relaxation iterations on each time point ti. We derive the error estimates in suitable function spaces for all values of p ∈ (0, 1). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.

$W^{2,\infty }$ regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension

In this paper we provide and analyze a nonlinear degenerate parabolic equation in one space dimension with the following smoothing property: If the initial data is only uniformly continuous, at positive times, the solution has bounded second derivatives (it belongs to W 2, ∞). We call this surprising phenomenon a W 2, ∞ regularizing effect. So far, such phenomena have only been observed in uniformly parabolic equations.

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

We prove a comparison principle for unbounded semicontinuous viscosity sub- and supersolutions of non-linear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the “geometric form” of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations (BSDEs) and (ii) pricing of European and American derivatives via BSDEs. Regarding (i), we extend previous results on BSDEs in a Lévy setting and the connection to semilinear integro-partial differential equations.

Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces

We consider a fully practical finite-element approximation of the following system of nonlinear degenerate parabolic equations: ∂u/∂t + 1/2 ⊇.(u 2 ⊇[σ(υ)]) - 1/3 ⊇.(u 3 ⊇ω) = 0, ω = -c Δu+δu -ν + au -3 , ∂υ/∂t + ⊇.(u υ ⊇[σ(υ)]) - ρΔυ - 1/2 ⊇.(u 2 υ⊇ω) = 0. The above models a surfactant-driven thin-film flow in the presence of both attractive, a > 0, and repulsive, δ > 0 with υ > 3, van der Waals forces; where u is the height of the film, υ is the concentration of the insoluble surfactant monolayer and σ(υ):= 1 - υ is the typical surface tension. Here ρ ≥ 0 and c > 0 are the inverses of the surface Peclet number and the modified capillary number. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system, (i) in one space dimension when p > 0; and, moreover, (ii) in two space dimensions if in addition υ ≥ 7. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.

An approximation scheme for a nonlinear degenerate parabolic equation with a second-order differential Volterra operator

A nonlinear degenerate parabolic initial boundary value problem with a second-order differential Volterra operator is studied. The approximate solution is found due to the semi-discretization in time which is based on Rothe's method. The convergence of the approximation scheme, existence and uniqueness of solution as well as some regularity results are proved.

Blowup properties for a class of nonlinear degenerate diffusion equation with nonlocal source

A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole region. The integral method is used to investigate the blowup properties of the solution.

An effective linear scheme to solve nonlinear degenerate parabolic differential equations

In this paper, we consider the approximate solution of the following problem. Given ω⊂ R N (1 ⪯ N ⪯ 3), find u(x, t) such that ∂ tb(u)−Δu+P e▽u=0, in Qω×(0,T) , u=0, on ∑∂ω×(0,T) , u(x,0)=u 0(x), ∀x∈ω , where the function b(s) is a monotonically increasing function satisfying 0 ⪯ b′ ⪯ ∞ To solve this problem, we introduce a new nonstandard time discretization scheme. We prove stability and convergence results.

Formation of discontinuities in flux-saturated degenerate parabolic equations

We endow the nonlinear degenerate parabolic equation used to describe propagation of thermal waves in plasma or in a porous medium, with a mechanism for flux saturation intended to correct the nonphysical gradient-flux relations at high gradients. We study both analytically and numerically the resulting equation: ut = [unQ(g(u)x)]x, n>0, where Q is a bounded increasing function. This model reveals that for n>1 the motion of the front is controlled by the saturation mechanism and instead of the typical infinite gradients resulting from the linear flux-gradients relations, Q∼ux, we obtain a sharp, shock-like front, typically associated with nonlinear hyperbolic phenomena. We prove that if the initial support is compact, independently of the smoothness of the initial datum inside the support, a sharp front discontinuity forms in a finite time, and until then the front does not expand.

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation **** equation here *** where generically **** equation here *** for any given **** equation here *** In addition to showing well-posedness of our approximation, we prove convergence in space dimensions $d \leq 3$. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented.

Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations

This paper investigates the global existence and nonexistence of positive solutions of the nonlinear degenerate parabolic equation μt = f (μu) (Δμ + a ʃ ωμ dx) with a homogeneous Dirichlet boundary condition. It is proved that there exists no global positive solution if and only if ʃ 1/(sf(s)) ds < ∞ and ʃ ωϕ(x) dx >/a , where ϕ( x) is the unique positive solution of the linear elliptic problem −Δ ϕ(x) = 1, x ∈ Ω; ϕ(x) = 0, x ∈ ∂ω

Unique Continuation for fast diffusion

We consider the non-characteristic Cauchy problem for the degenerate nonlinear parabolic equation $|u|^{\alpha}u_{t}=\triangle u$, where we assume that $1/2 \lt \alpha \lt 1$. This equation is based on the fast diffusion model. And we prove the unique continuation property for the above problem.

On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

We study nonlinear degenerate parabolic equations where the flux function $f(x,t,u)$ does not depend Lipschitz continuously on the spatial location $x$. By properly adapting the 'doubling of variables' device due to Kružkov [25] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form $k(x)f(u)$, where $k(x)$ is a vector-valued function and $f(u)$ is a scalar function.

On an Averaged Model for the 2-fluid Immiscible Flow with Surface Tension in a Thin Cylindrical Tube

We consider the two-fluid incompressible immiscible flow, with surface tension, in a thin cylindrical tube. Using the method of asymptotic expansions with respect to the thickness of the domain, effective 1D equations are derived. Supposing that the fluids are strictly separated at t=0 and that only one phase is in contact with the rigid wall, we show that the derived equations equal a nonlinear degenerate fourth order parabolic equation for the saturation. If we set the surface tension to be zero, it reduces to the Buckley–Leverett equation recently derived by Mikelic and Paoli using the same technique. In the general case we obtain the generalized Darcy's equations with extra-diagonal terms and the relative permeabilities are calculated explicitly. The capillary pressure results only from te surface tension between two fluid phases.

On a Doubly Nonlinear Parabolic Obstacle Problem Modelling Ice Sheet Dynamics

This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet flow, we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-Newtonian dynamics. An obstacle problem is then deduced and analyzed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative and numerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings.

Finite Element Approximation of Surfactant Spreading on a Thin Film

We consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations: \begin{alignat}{2} \textstyle{\frac{\partial u}{\partial t}} + \textstyle \frac{1}{2} \,\nabla . (u^2 \,\nabla [\sigma(v)]) - \textstyle \frac{1}{3}\, \nabla .(u^3 \,\nabla w) &= 0, &&%\quad\mbox{in} \;\;\Omega_T, \qquad %\hspace{2cm} \nonumber \\ w = - c \, \Delta u + a \, u^{-3} - \delta \, u^{-\nu}, \nonumber \\ \textstyle{\frac{\partial v}{\partial t}} + \nabla . (u\,v\,\nabla [\sigma(v)]) - \rho \,\Delta v - \textstyle \frac{1}{2}\, \nabla .(u^2\,v \,\nabla w) &= 0. &&%\quad\mbox{in} \;\;\Omega_T. \nonumber %\\ \end{alignat} The above models a surfactant-driven thin film flow in the presence of both attractive, a >0, and repulsive, $\delta >0$ with $\nu >3$, van der Waals forces, where u is the height of the film, v is the concentration of the insoluble surfactant monolayer, and $\sigma(v):=1-v$ is the typical surface tension. Here $\rho \geq 0$ and c>0 are the inverses of the surface Peclet number and the modified capillary number. In addition to showing stability bounds for our approximation, we prove convergence in one space dimension when $\rho >0$ and either $a=\delta=0$ or $\delta > 0$. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.

A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations

Following the lead of [Carrillo, Arch. Ration. Mech. Anal. 147 (1999) 269–361], recently several authors have used Kružkov's device of “doubling the variables” to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order differential operator is not allowed to depend explicitly on the spatial variable, which certainly restricts the range of applications of entropy solution theory. The purpose of this paper is to extend a version of Carrillo's uniqueness result to a class of degenerate parabolic equations with spatially dependent second order differential operator. The class is large enough to encompass several interesting nonlinear partial differential equations coming from the theory of porous media flow and the phenomenological theory of sedimentation-consolidation processes.

Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

We analyze approximate solutions generated by an upwind difference scheme (of Engquist-Osher type) for nonlinear degenerate parabolic convection-diffusion equations where the nonlinear convective flux function has a discontinuous coefficient γ(x) and the diffusion function A(u) is allowed to be strongly degenerate (the pure hyperbolic case is included in our setup). The main problem is obtaining a uniform bound on the total variation of the difference approximation u Δ , which is a manifestation of resonance. To circumvent this analytical prob- lem, we construct a singular mapping Ψ(γ,·) such that the total variation of the transformed variable z Δ =Ψ ( γ Δ ,u Δ ) can be bounded uniformly in Δ. This establishes strong L 1 com- pactness of z Δ and, since Ψ(γ,·) is invertible, also u Δ . Our singular mapping is novel in that it incorporates a contribution from the diffusion function A(u). We then show that the limit of a converging sequence of difference approximations is a weak solution as well as satisfying a Kruzkov-type entropy inequality. We prove that the diffusion function A(u )i s Hcontin- uous, implying that the constructed weak solution u is continuous in those regions where the diffusion is nondegenerate. Finally, some numerical experiments are presented and discussed.