Nonlinear Degenerate Parabolic Equations Research Articles

Asymptotic behavior for nonlinear degenerate parabolic equations with irregular data

ABSTRACT This paper focuses on the following degenerate parabolic equation where Ω is a smooth bounded domain in , is positive almost everywhere and satisfies proper degenerate conditions. The existence and uniqueness result is proved in the framework of entropy solutions. For the long-time behavior, we prove the existence of a global attractor in by using some delicate estimates on the solution, which are derived by taking advantage of both the leading operator and the zero-order nonlinear term. The aforementioned results improve some previous results in the literature in several aspects.

PRECONDITIONED ITERATIVE METHOD FOR REACTIVE TRANSPORT WITH SORPTION IN POROUS MEDIA

This work deals with the numerical solution of a nonlinear degenerate parabolic equation arising in a model of reactive solute transport in porous media, including equilibrium sorption. The model is a simplified, yet representative, version of multicomponents reactive transport models. The numerical scheme is based on an operator splitting method, the advection and diffusion operators are solved separately using the upwind finite volume method and the mixed finite element method (MFEM) respectively. The discrete nonlinear system is solved by the Newton–Krylov method, where the linear system at each Newton step is itself solved by a Krylov type method, avoiding the storage of the full Jacobian matrix. A critical aspect of the method is an efficient matrix-free preconditioner. Our aim is, on the one hand to analyze the convergence of fixed-point algorithms. On the other hand we introduce preconditioning techniques for this system, respecting its block structure then we propose an alternative formulation based on the elimination of one of the unknowns. In both cases, we prove that the eigenvalues of the preconditioned Jacobian matrices are bounded independently of the mesh size, so that the number of outer Newton iterations, as well as the number of inner GMRES iterations, are independent of the mesh size. These results are illustrated by some numerical experiments comparing the performance of the methods.

A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations

In this paper, we consider the asymptotic behavior of traveling wave solutions of the degenerate nonlinear parabolic equation: $u_{t}=u^{p}(u_{xx}+u)-\delta u$ ($\delta = 0$ or $1$) for $\xi \equiv x - ct \to - \infty$ with $c>0$. We give a refined one of them, which was not obtain in the preceding work [Ichida-Sakamoto, 2020], by an appropriate asymptotic study and properties of the Lambert $W$ function.

Nonlinear dynamics of dewetting thin films

Fluid films spreading on hydrophobic solid surfaces exhibit complicated dynamics that describe transitions leading the films to break up into droplets. For viscous fluids coating hydrophobic solids this process is called "dewetting". These dynamics can be represented by a lubrication model consisting of a fourth-order nonlinear degenerate parabolic partial differential equation (PDE) for the evolution of the film height. Analysis of the PDE model and its regimes of dynamics have yielded rich and interesting research bringing together a wide array of different mathematical approaches. The early stages of dewetting involve stability analysis and pattern formation from small perturbations and self-similar dynamics for finite-time rupture from larger amplitude perturbations. The intermediate dynamics describes further instabilities yielding topological transitions in the solutions producing sets of slowly-evolving near-equilibrium droplets. The long-time behavior can be reduced to a finite-dimensional dynamical system for the evolution of the droplets as interacting quasi-steady localized structures. This system yields <i>coarsening</i>, the successive re-arrangement and merging of smaller drops into fewer larger drops. To describe macro-scale applications, mean-field models can be constructed for the evolution of the number of droplets and the distribution of droplet sizes. We present an overview of the mathematical challenges and open questions that arise from the stages of dewetting and how they relate to issues in multi-scale modeling and singularity formation that could be applied to other problems in PDEs and materials science.

The behavior of solutions of nonlinear degenerate parabolic equations in nonregular domains and removability of singularity on boundary

The behavior of solutions of nonlinear degenerate parabolic equations in nonregular domains and removability of singularity on boundary

Null-controllability of perturbed porous medium gas flow

In this work, we investigate the null-controllability of a nonlinear degenerate parabolic equation, which is the equation satisfied by a perturbation around the self-similar solution of the porous medium equation in Lagrangian-like coordinates. We prove a local null-controllability result for a regularized version of the nonlinear problem, in which singular terms have been removed from the nonlinearity. We use spectral techniques and the source-term method to deal with the linearized problem and the conclusion follows by virtue of a Banach fixed-point argument. The spectral techniques are also used to prove a null-controllability result for the linearized thin-film equation, a degenerate fourth order analog of the problem under consideration.

Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states

In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn’t summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion problems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some energy balance models in climatology (see,e.g., the Budyko-Sellers model) and some models in population genetics (see,e.g., the Fleming-Viot model).

The Long Time Behavior of Periodic Entropy Solutions to Degenerate Nonlinear Parabolic Equations

We establish the asymptotic convergence of an entropy solution, periodic with respect to the spatial variable, of a degenerate nonlinear parabolic equation to a traveling wave. It is shown that, on a segment containing the essential values of the limit profile, the flow function is linear (with the angular coefficient equal to the traveling wave velocity) and the diffusion function is constant.

On the Weak Characteristic Function Method for a Degenerate Parabolic Equation

For a nonlinear degenerate parabolic equation, how to impose a suitable boundary value condition to ensure the well-posedness of weak solutions is a very important problem. It is well known that the classical Fichera-Oleinik theory has perfectly solved the problem for the linear case, and the optimal boundary value condition matching up with a linear degenerate parabolic equation can be depicted out by Fechira function. In this paper, a new method, which is called the weak characteristic function method, is introduced. By this new method, the partial boundary condition matching up with a nonlinear degenerate parabolic equation can be depicted out by an inequality from the diffusion function, the convection function, and the geometry of the boundary ∂Ω itself. Though, by choosing different weak characteristic function, one may obtain the differential partial boundary value conditions, an optimal partial boundary value condition can be prophetic. Moreover, the new method works well in any kind of the degenerate parabolic equations.

Application of the SBA Method to Solve the Nonlinear Biological Population Models

This paper presents numerical solution of some nonlinear degenerate parabolic equations arising in the spatial di¤usion of biological populations. The SBA method based on combination of Adomian decomposition method, principle of Picard and successive approximations is used for solving these equations. The analytical obtained solutions show that the SBA method leads to more accurate results.

Solving a two-dimensional problem of heat wave initiation by a boundary condition specified on a movable manifold

A boundary value problem for a nonlinear degenerate parabolic equation is considered for the case of two spatial variables. The boundary condition is specified on a movable manifold a time-varying closed line. A heat wave problem solution is constructed on the basis of the boundary element method. The numerical solution algorithm is based on the change of variables, when the roles of the required function and the radial coordinate of the polar system interchange. The BEM solution of the problem obtained in new symbols enables the zero front of the heat wave to be determined. The further solution of the problem with the known zero front yields a solution to the initial problem. The algorithm is implemented as a program. An exact (generalized self-similar) solution is constructed, whereby the program is tested.

Existence and uniqueness of a renormalized solution for nonlinear degenerate parabolic equations with diffuse measure data

In this paper, we prove the existence and uniqueness of a renormalized solution for some nonlinear parabolic equations with degenerate coercivity and diffuse measure data.

Partial boundary value condition for a nonlinear degenerate parabolic equation

In this paper, a nonlinear degenerate parabolic equation is considered. By proposing a new partial boundary value condition and constructing special test functions, we prove the stability of the solutions by means of the Kruzkov bi-variables method. Moreover, we show that if there is an interaction between the diffusion term and the convection term, the stability can be established without any boundary value condition.

Finite volume element approximation for nonlinear diffusion problems with degenerate diffusion coefficients

Some standard numerical methods, such as mimetic finite difference method, finite volume method and mixed finite element method, often fail in solving nonlinear diffusion problems with degenerate diffusion coefficients, since a harmonic average of diffusion coefficients is involved in these methods. To avoid such problem, we present some finite volume element schemes to solve a class of 1D degenerate nonlinear parabolic equations in this paper. Some fully discrete schemes are given by using linear and quadratic finite volume elements in space and a backward difference formulation in time. To deal with unphysical numerical oscillation, we apply two nonnegativity-preserving repair techniques based on a posteriori corrections to finite volume element solutions. One is a local approach, in which any negative energy corresponding to some mesh node is absorbed by the positive values near the current node. The other is a global strategy, in which the total negative energy is redistributed to each positive value in the numerical solution in proportion to its value. In addition, some monotonous finite volume element schemes with lumped-mass strategy are presented. Numerical examples are included to demonstrate the effectiveness and competitive behavior of the proposed methods.

Spherical Hellinger--Kantorovich Gradient Flows

We study nonlinear degenerate parabolic equations of Fokker--Planck type that can be viewed as gradient flows with respect to the recently introduced spherical Hellinger--Kantorovich distance. The driving entropy is not assumed to be geodesically convex. We prove solvability of the problem and the entropy-entropy production inequality, which implies exponential convergence to the equilibrium. As a corollary, we obtain some related results for the Wasserstein gradient flows. We also deduce transportation inequalities in the spirit of Talagrand, Otto, and Villani for the spherical and conic Hellinger--Kantorovich distances.

A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton–Jacobi equations

In this paper, a class of high order numerical schemes is proposed for solving Hamilton–Jacobi (H–J) equations. This work is regarded as an extension of our previous work for nonlinear degenerate parabolic equations, see Christlieb et al. [14], which relies on a special kernel-based formulation of the solutions and successive convolution. When applied to the H–J equations, the newly proposed scheme attains genuinely high order accuracy in both space and time, and more importantly, it is unconditionally stable, hence allowing for much larger time step evolution compared with other explicit schemes and saving computational cost. A high order weighted essentially non-oscillatory methodology and a novel nonlinear filter are further incorporated to capture the correct viscosity solution. Furthermore, by coupling the recently proposed inverse Lax–Wendroff boundary treatment technique, this method is very flexible in handing complex geometry as well as general boundary conditions. We perform numerical experiments on a collection of numerical examples, including H–J equations with linear, nonlinear, convex or non-convex Hamiltonians. The efficacy and efficiency of the proposed scheme in approximating the viscosity solution of general H–J equations is verified.

Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching

We study viscosity solutions to a system of nonlinear degenerate parabolic partial integro-differential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an n -dimensional Lévy process. We first establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependence estimate, we obtain the comparison principle and uniqueness of viscosity solutions as well as Lipschitz regularity in the spatial variables. Our main contribution is construction of suitable families of viscosity sub- and supersolutions which we use as “barrier functions” to prove Hölder continuity in the time variable, and, through Perron's method, existence of a unique viscosity solution. This paper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010) [5] and of Lundström, Nyström and Olofsson (2014) [21] , [22] to hold for more general systems of equations.

On a salt fingers model

We consider the model introduced in Paparella and von Hardenberg (2014), that consists in the homogeneous boundary value problem for a system of nonlinear degenerate parabolic equations. We prove the existence of global weak solutions and discuss their stability and asymptotic properties.

Global weak solutions to a higher order nonlinear degenerate parabolic equation

Our aim in this paper is to investigate the existence for a sixth-order parabolic equation, which arises in oil-water-surfactant mixtures. More precisely, under suitable assumptions, first we study the existence and regularity of weak solutions to an initial–boundary problem for a non-degenerate sixth-order parabolic equation. Second considering the limits of the solutions obtained in the first step, we prove the existence of weak solutions.

Existence of weak solutions for particle-laden flow with surface tension

We prove the existence of solutions for a coupled system modeling the flow of a suspension of fluid and negatively buoyant non-colloidal particles in the thin film limit. The equations take the form of a fourth-order non-linear degenerate parabolic equation for the film height $h$ coupled to a second-order degenerate parabolic equation for the particle density $ψ$ . We prove the existence of physically relevant solutions, which satisfy the uniform bounds $0 ≤ ψ/h ≤ 1$ and $h ≥ 0$ .