Degenerate Diffusion Equation Research Articles

Nonlocal approximation of nonlinear diffusion equations

We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of 2-Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation.

Local well-posedness of 1D degenerate drift diffusion equation

<abstract><p>This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.</p></abstract>

On the structure of weak solutions of the Riemann problem for a degenerate nonlinear diffusion equation

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coe cient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a nite number of variables. A similar result is true for Stefan’s problem. In the limit, when the number of phases tends to in nity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function.

Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct diffusion processes with infinite lifetime and explicit invariant measures. The processes provide weak solutions to infinite-dimensional Langevin dynamics. The second part deals with a general abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser. In order to treat stochastic (partial) differential equations, Grothaus and Stilgenbauer translated these concepts to the Kolmogorov backwards setting taking domain issues into account. To apply these concepts in the context of infinite-dimensional Langevin dynamics we use an essential m-dissipativity result for infinite-dimensional Ornstein–Uhlenbeck operators, perturbed by the gradient of a potential. We allow unbounded diffusion operators as coefficients and apply corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic is needed. Poincaré inequalities for measures with densities w.r.t. infinite-dimensional non-degenerate Gaussian measures are studied. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of a diffusion process enables us to show an L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-exponential ergodicity result for the weak solution. Finally, we apply our results to explicit infinite-dimensional degenerate diffusion equations.

Cauchy problems for the time-fractional degenerate diffusion equations

This paper is devoted to the Cauchy problems for the one-dimensional linear time-fractional diffusion equations with $\partial^{\alpha}_{t}$ the Caputo fractional derivative of order $\alpha\in(0,1)$ in the variable t and time-degenerate diffusive coefficients $t^{\beta}$ with $\beta >1-\alpha$. The solutions of  Cauchy problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative $\partial^{\alpha}_{t}$ of order $\alpha\in(0,1)$  in the variable $t$, are shown. In the "Problem statement and main results" section of the paper, the solution of the time-fractional degenerate diffusion equation in a variable coefficient with two different initial conditions are considered. In this work, a solution is found by using the Kilbas-Saigo function $E_{\alpha,m,l}(z)$ and applying the Fourier transform $F$ and inverse Fourier transform $\mathcal{F}^{-1}$. Convergence of solution of problem 1 and problem 2 are proven using Plancherel theorem. The existence and uniqueness of the solution of the problem are confirmed.&nbsp

Existence and stability of traveling waves for doubly degenerate diffusion equations

Existence and stability of traveling waves for doubly degenerate diffusion equations

Degenerate time-fractional diffusion equation with initial and initial-boundary conditions

Abstract In this paper, we study initial and initial-boundary problems for the time-fractional degenerate diffusion equations in bounded and unbounded domains. We obtain the existence and uniqueness of solutions using Fourier methods.

Monotone reducing mechanism in delayed population model with degenerate diffusion

In this paper, we study the monotone reducing mechanism of degenerate diffusion equations with time delay. We show the monotone dependence of critical wave speed on time delay. Due to the complex dynamics arising from degeneracy and time delay, we use a new phase transform approach to analyze the delicate local and global behaviors of critical sharp traveling waves and then derive the comparison of critical speeds.

Smooth traveling waves for doubly nonlinear degenerate diffusion equations with time delay

ABSTRACT This paper deals with the existence of traveling waves for doubly nonlinear degenerate diffusion equations with Nicholson's blowflies-type source term and time delay. We prove the existence of monotone and non-monotone traveling waves by the monotone iteration and Schauder's fixed point theorem respectively. Furthermore, the non-critical traveling waves are proven to be globally -stable. The adopted approach for the proof is the weighted energy method combining the compactness analysis.

Well-posedness of the initial-boundary value problems for the time-fractional degenerate diffusion equations

This paper deals with the solving of initial-boundary value problems for the one-dimensional linear timefractional diffusion equations with time-degenerate diffusive coefficients t^β with β > 1 − α. The solutions to initial-boundary value problems for the one-dimensional time-fractional degenerate diffusion equations with Riemann-Liouville fractional integral I^1−α_0+,t of order α ∈ (0, 1) and with Riemann-Liouville fractional derivative D^α_0+,t of order α ∈ (0, 1) in the variable, are shown. The solutions to these fractional diffusive equations are presented using the Kilbas-Saigo function Eα,m,l(z). The solution to the problems is discovered by the method of separation of variables, through finding two problems with one variable. Rather, through finding a solution to the fractional problem depending on the parameter t, with the Dirichlet or Neumann boundary conditions. The solution to the Sturm-Liouville problem depends on the variable x with the initial fractional-integral Riemann-Liouville condition. The existence and uniqueness of the solution to the problem are confirmed. The convergence of the solution was evidenced using the estimate for the KilbasSaigo function E_α,m,l(z) from and by Parseval’s identity.

Critical sharp front for doubly nonlinear degenerate diffusion equations with time delay

This paper is concerned with the critical sharp travelling wave for doubly nonlinear diffusion equation with time delay, where the doubly nonlinear degenerate diffusion is defined by with m > 0 and p > 1. The doubly nonlinear diffusion equation is proved to admit a unique sharp type travelling wave for the degenerate case m(p − 1) > 1, the so-called slow-diffusion case. This sharp travelling wave associated with the minimal wave speed c*(m, p, r) is monotonically increasing, where the minimal wave speed satisfies c*(m, p, r) < c*(m, p, 0) for any time delay r > 0. The sharp front is C 1-smooth for , and piecewise smooth for . Our results indicate that time delay slows down the minimal travelling wave speed for the doubly nonlinear degenerate diffusion equations. The approach adopted for proof is the phase transform method combining the variational method. The main technical issue for the proof is to overcome the obstacle caused by the doubly nonlinear degenerate diffusion.

Propagation Speed of Degenerate Diffusion Equations with Time Delay

We are concerned with a class of degenerate diffusion equations with time delay describing population dynamics with age structure. In our recent study [Nonlinearity, 33 (2020), 4013–4029], we established the existence and uniqueness of critical traveling wave for the time-delayed degenerate diffusion equations, and obtained the reducing mechanism of time delay on critical wave speed. In this paper, we now are able to show the asymptotic spreading speed and its coincidence with the critical wave speed $$c^*(m,r)$$ of sharp wave, and prove that the initial perturbation or the boundary of the compact support of the solution propagates at the critical wave speed $$c^*(m,r)$$ for the time-delayed degenerate diffusion equations. Remarkably, different from the existing studies related to spreading speeds, the time delay and the degenerate diffusion lead to some essential difficulties in the analysis of the spreading speed, because the time delay makes the critical speed of traveling waves slow down in a more complicated fashion such that the critical speed cannot be determined by the characteristic equation, and the degenerate diffusion causes the loss of regularity for the solutions. By a phase transform technique we construct upper and lower solutions with semi-compact supports and then we determine the asymptotic spreading speed. Furthermore, we propose a brand-new sharp-profile-based difference scheme to handle large variation of degenerate diffusion $$(u^m)_{xx}$$ near the sharp edge and carry out some numerical simulations which perfectly confirm our theoretical results.

On the time fractional heat equation with obstacle

We study a Caputo time fractional degenerate diffusion equation which we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any α∈(0,1) to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with α is the convergence speed. We also study the problem from the numerical point of view, comparing some finite different approaches, and showing the results of some tests. These results extend what recently proved in [1] for the case α=1.

Convergence Analysis of the Nonoverlapping Robin--Robin Method for Nonlinear Elliptic Equations

We prove convergence for the nonoverlapping Robin--Robin method applied to nonlinear elliptic equations with a $p$-structure, including degenerate diffusion equations governed by the $p$-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov--Poincaré operators based on the $p$-structure and the $L^p$-generalization of the Lions--Magenes spaces. This framework allows the reformulation of the Robin--Robin method into a Peaceman--Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions.

Initial-boundary value problem for the time-fractional degenerate diffusion equation

In this paper the initial-boundary value problems for the one-dimensional linear time-fractional diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t and time-degenerate diffusive coefficients t β with β ≥ 1 − α are studied. The solutions of initialboundary value problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t, are shown. The second section present Dirichlet and Neumann boundary value problems, and in the third section has shown the solutions of the Dirichlet and Neumann boundary value problem for the one-dimensional linear time-fractional diffusion equation. The solutions of these fractional diffusive equations are presented using the Kilbas-Saigo function Eα,m,l(z). The solution of the problems is discovered by the method of separation of variables, through finding two problems with one variable. The existence and uniqueness to the solution of the problem are confirmed. In addition, the convergence of the solution has been proven using the estimate for the Kilbas-Saigo function Eα,m,l(z) from [13] and Parseval’s identity

Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility

Abstract We consider density solutions for gradient flow equations of the form u t = ∇ · (γ(u)∇ N(u)), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ(u) = u α , and α &gt; 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u) = u α , with 0 &lt; α &lt; 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

Fundamental solution to 1D degenerate diffusion equation with locally bounded coefficients

In this work we study the degenerate diffusion equation ∂t=xαa(x)∂x2+b(x)∂x for (x,t)∈(0,∞)2, equipped with a Cauchy initial data and the Dirichlet boundary condition at 0. We assume that the order of degeneracy at 0 of the diffusion operator is α∈(0,2), and the coefficient functions (and their derivatives) are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution p(x,y,t) and prove several properties for p(x,y,t); by conducting a localization procedure, we obtain an approximation for p(x,y,t) for x,y in a neighborhood of 0 and t sufficiently small, where the error estimates only rely on the local bounds of a(x) and b(x) (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of α=1. Our work extends part of the existing results to the cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probabilistic view (e.g., wellposedness of stochastic differential equations).

Global stability of traveling waves for nonlocal time-delayed degenerate diffusion equation

This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Hölder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L1-stable, which is the first frame work on L1-wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L1-weighted energy estimates combining the compactness analysis, but with some new development.

A multi level linearized Crank–Nicolson scheme for Richards equation under variable flux boundary conditions

The Richards equation is a nonlinear degenerate advection diffusion equation that models flow in saturated/unsaturated porous media, it's crucially important for prediction of disasters when heavy rain attacks. Efficient and precise linearized numerical schemes are necessary, but there is few study related it, and the numerical theory is incomplete because of the degeneracy and strong nonlinearity. In this paper, we establish a linearized Crank–Nicolson finite difference scheme which is a three-level scheme with almost second-order accuracy. In stability analysis, we develop a creative technique to overcome the degeneracy by adding a small positive perturbation ϵ. We also propose the error estimates by applying Young's inequality and prove the convergence order is approximate to second-order. Numerical examples are also provided to verify our main results and show the relationship between the computational error and ϵ is linear.

Continuity results for degenerate diffusion equations with [formula omitted] drifts

In this paper, we study local uniform continuity of nonnegative weak solutions to degenerate diffusion-drift equations in the form of ut=Δum+∇⋅B(x,t)u,for m≥1assuming a vector field B∈LtpLxq. Regarding local Hölder continuity, we provide a sharp condition on p and q, which is referred to as the subcritical region. In the critical region, the divergence-free condition is essential to providing uniform continuity which depends on the modulus of continuity of the local norm of B.