Porous Medium Equation Research Articles

Lower Bound Estimate of Blow Up Time for the Porous Medium Equations under Dirichlet and Neumann Boundary Conditions

Lower Bound Estimate of Blow Up Time for the Porous Medium Equations under Dirichlet and Neumann Boundary Conditions

Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation

<p style='text-indent:20px;'>The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincaré, Nash, Super Poincaré and logarithmic Sobolev type inequalities, on complete Riemannian manifolds satisfying some mild geometric assumptions. Second, based on the derived nonlocal functional inequalities, we analyze the asymptotic behavior of the solution to the fractional porous medium equation, <inline-formula><tex-math id="M1">\begin{document}$ \partial_t u +(-\Delta)^\sigma (|u|^{m-1}u ) = 0 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ m>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \sigma\in (0, 1) $\end{document}</tex-math></inline-formula>. In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.</p>

Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.

Mixing solutions for the Muskat problem

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type $$H^5$$ initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed.

Correction to: Finite Element Solvers for Biot’s Poroelasticity Equations in Porous Media

Correction to: Finite Element Solvers for Biot’s Poroelasticity Equations in Porous Media

Thermal transport equations in porous media from product-like fractal measure

In this study, we have used the concept of product-like fractal measure to analyze the fractal heat transfer in anisotropic media. This concept was introduced by Li and Ostoja-Starzewski in order to study anisotropic fractal elastic and continuum media. The theory is characterized by extended fluid, mass and heat transfer fractal equations besides the emergence of a damping heat term in the fractal advection-diffusion equation. Several applications are discussed and a number of features are revealed.

Extinction in finite time of solutions to fractional parabolic porous medium equations with strong absorption

In this article we study the solutions of a general fractional parabolic porous medium equation with a non-Lipschitz absorption term. We obtain the existence of weak solutions, \(L^p\)-estimates, and decay estimates. Also, we show that weak solutions must vanish after a finite time, even for large initial data. For more information see https://ejde.math.txstate.edu/Volumes/2021/29/abstr.html

Quenching for Porous Medium Equations

This paper studies the following two porous medium equations with singular boundary conditions. First, we obtain that finite time quenching on the boundary, as well as kt blows up at the same finite time and lower bound estimates of the quenching time of the equation kt = (kn)xx + (1 − k)−α, (x,t) ∈ (0,L) × (0,T) with (kn)x (0,t) = 0, (kn)x (L,t) = (1 − k(L,t))−β, t ∈ (0,T) and initial function k(x,0) = k0 (x), x ∈ [0, L] where n > 1, α and β and positive constants. Second, we obtain that finite time queching on the boundary, as well as kt blows up at the same finite time and a local existence resultbythehelpofsteadystateoftheequationkt =(kn)xx,(x,t)∈(0,L)×(0,T)with (kn)x (0,t) = (1 − k(0,t))−α, (kn)x (L,t) = (1 − k(L,t))−β, t ∈ (0,T) and initial function k (x, 0) = k0 (x), x ∈ [0, L] where n > 1, α and β and positive constants.

Global existence of solutions and smoothing effects for classes of reaction\u2013diffusion equations on manifolds

We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on p and m in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequalities hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in {{mathbb {R}}}^n.

Finite time extinction for the 1D stochastic porous medium equation with transport noise

We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate smash {O(t^frac{1}{2})} whereas the support of solutions to the deterministic PME grows only with rate smash {O(t^{frac{1}{m{+}1}})}. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.

Sobolev-Type Inequalities on Cartan-Hadamard Manifolds and Applications to some Nonlinear Diffusion Equations

We investigate the validity, as well as the failure, of Sobolev-type inequalities on Cartan-Hadamard manifolds under suitable bounds on the sectional and the Ricci curvatures. We prove that if the sectional curvatures are bounded from above by a negative power of the distance from a fixed pole (times a negative constant), then all the Lp inequalities that interpolate between Poincaré and Sobolev hold for radial functions provided the power lies in the interval (− 2,0). The Poincaré inequality was established by H.P. McKean under a constant negative bound from above on the sectional curvatures. If the power is equal to the critical value − 2 we show that p must necessarily be bounded away from 2. Upon assuming that the Ricci curvature vanishes at infinity, the nonradial version of such inequalities turns out to fail, except in the Sobolev case. Finally, we discuss applications of the here established Sobolev-type inequalities to optimal smoothing effects for radial porous medium equations.

Wave Equations of Porous Media Saturated With Two Immiscible Fluids Based on the Volume Averaging Method

Simulating and predicting wave propagation in porous media saturated with two fluids is an important issue in geophysical exploration studies. In this work, wave propagation in porous media with specified structures saturated with two immiscible fluids was studied, and the main objective was to establish a wave equation system with a relatively simple structure. The wave equations derived by Tuncay and Corapcioglu were analyzed first. It was found that the coefficient matrix of the equations tends to be singular due to the inclusion of a small parameter that characterizes the effect of capillary stiffening. Therefore, the previously established model consisting of three governing equations may be unstable under natural conditions. An improved model based on Tuncay and Corapcioglu’s work was proposed to ensure the nonsingularity of the coefficient matrix. By introducing an assumption in which one fluid was completely wrapped by the other, the governing equation of the wrapped fluid was degenerated. In this way, the coefficient matrix of wave equations became nonsingular. The dispersion and attenuation prediction resulting from the new model was compared with that of the original model. Numerical examples show that although the improved model consists of only two governing equations, it can obtain a result similar to that of the original model for the case of a porous medium containing gas and water, which simplifies the complexity of the calculations. However, in a porous medium with oil and water, the predictions of dispersion and attenuation produced by the original model obviously deviate from the normal trend. In contrast, the results of the improved model exhibit the correct trend with a smooth curve. This phenomenon shows the stability of the improved model and it could be used to describe wave propagation dispersions and attenuations of porous media containing two immiscible fluids in practical cases.

Solving One-Dimensional Porous Medium Equation Using Unconditionally Stable Half-Sweep Finite Difference and SOR Method

A porous medium equation is a nonlinear parabolic partial differential equation that presents many physical occurrences. The solutions of the porous medium equation are important to facilitate the investigation on nonlinear processes involving fluid flow, heat transfer, diffusion of gas-particles or population dynamics. As part of the development of a family of efficient iterative methods to solve the porous medium equation, the Half-Sweep technique has been adopted. Prior works in the existing literature on the application of Half-Sweep to successfully approximate the solutions of several types of mathematical problems are the underlying motivation of this research. This work aims to solve the one-dimensional porous medium equation efficiently by incorporating the Half-Sweep technique in the formulation of an unconditionally-stable implicit finite difference scheme. The noticeable unique property of Half-Sweep is its ability to secure a low computational complexity in computing numerical solutions. This work involves the application of the Half-Sweep finite difference scheme on the general porous medium equation, until the formulation of a nonlinear approximation function. The Newton method is used to linearize the formulated Half-Sweep finite difference approximation, so that the linear system in the form of a matrix can be constructed. Next, the Successive Over Relaxation method with a single parameter was applied to efficiently solve the generated linear system per time step. Next, to evaluate the efficiency of the developed method, deemed as the Half-Sweep Newton Successive Over Relaxation (HSNSOR) method, the criteria such as the number of iterations, the program execution time and the magnitude of absolute errors were investigated. According to the numerical results, the numerical solutions obtained by the HSNSOR are as accurate as those of the Half-Sweep Newton Gauss-Seidel (HSNGS), which is under the same family of Half-Sweep iterations, and the benchmark, Newton-Gauss-Seidel (NGS) method. The improvement in the numerical results produced by the HSNSOR is significant, and requires a lesser number of iterations and a shorter program execution time, as compared to the HSNGS and NGS methods.

Blow up for Porous Medium Equations

In various branches of applied sciences, porous medium equations exist where this basic model occurs in a natural fashion. It has been used to model fluid flow, chemical reactions, diffusion or heat transfer, population dynamics, etc.. Nonlinear diffusion equations involving the porous medium equations have also been extensively studied. However, there has not been much research effort in the parabolic problem for porous medium equations with two nonlinear boundary sources in the literature. This paper adresses the following porous medium equations with nonlinear boundary conditions. Firstly, we obtain finite time blow up on the boundary by using the maximum principle and blow up criteria and existence criteria by using steady state of the equation $k_{t}=k_{xx}^{n},(x,t)\in (0,L)\times (0,T)\ $with $ k_{x}^{n}(0,t)=k^{\alpha }(0,t)$, $k_{x}^{n}(L,t)=k^{\beta }(L,t)$,$\ t\in (0,T)\ $and initial function $k\left( x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $n>1$, $\alpha \ $and $\beta \ $and positive constants.

On an exact solution to the nonlinear heat equation with a source

The paper is devoted to the study of a singular nonlinear second-order parabolic equation, which is called the porous medium equation or the nonlinear heat equation. One of the important classes of its solutions is heat waves propagating over a zero background with a finite velocity. This property is not typical for parabolic equations and is a consequence of singularity. The main object of study is exact solutions of mentioned type. A new way of separating variables is used to represent them. We obtain conditions when it is possible to make a reduction to the Cauchy problem for an ordinary second-order differential equation with a singularity. It is shown that the Cauchy problem describes a heat wave whose front moves exponentially. We construct a solution to the Cauchy problem as a power series and determine the cases when the series breaks off, i.e. the solution has the form of a polynomial, and the corresponding heat wave can be written explicitly. If the Cauchy problem cannot be explicitly integrated, the solution is constructed numerically. An algorithm based on the boundary element method is proposed. We perform a computational experiment and conclude the properties of the found solutions. Besides, the accuracy of the calculation results is analyzed.

Global Well-Posedness and Analyticity of Generalized Porous Medium Equation in Fourier-Besov-Morrey Spaces with Variable Exponent

In this paper, we consider the generalized porous medium equation. For small initial data u0 belonging to the Fourier-Besov-Morrey spaces with variable exponent, we obtain the global well-posedness results of generalized porous medium equation by using the Fourier localization principle and the Littlewood-Paley decomposition technique. Furthermore, we also show Gevrey class regularity of the solution.

A novel approach for measuring plane anisotropic permeability through steady-state flow between two concentric cylinders

Anisotropy is a common feature for most of fibrous and porous media. This article reports a new method for measuring the 2D anisotropic permeability of a porous rock utilizing the measurements obtained from the steady-state flow between two concentric cylinders. The mathematical analysis of the experimental data is based on the general solution of the pressure equation derived for the steady-state flow equation for an anisotropic porous medium between two concentric cylinders. The general solution, incorporating the degree of anisotropy and the ratio between inner and outer cylinder radii, is presented in a dimensionless form. The solution shows that pressure distribution between the two cylinders is composed of an isotropic part plus a series representing the contribution of the anisotropy. The solution is validated through the comparison against numerically derived results and the analytical solution of two extreme cases; when the clearance between the two cylinders vanishes, and when the material is isotropic. The validation shows a perfect performance of the proposed solution even in the vicinity of the inner cylinder not like the approximate solutions existing in the literature. The validation also shows that the truncation of the series part has an impact on the results specially in the case of extreme anisotropy. Two experimental procedures 1) basic approach and 2) modified approach, utilizing the “pear” and “tent” charts derived from the general solution, demonstrate how to obtain the principle permeabilities and their directions. The applicability of the proposed experimental techniques is attained through the application of the modified approach over real experimental data from a holed core experiment to estimate anisotropic permeability parameters. The application of the proposed technique and the required modifications for measuring the anisotropy in tight cores are discussed. Also, the relation between the three-dimension anisotropy and the presented plane anisotropy is presented.

Ergodicity for Singular-Degenerate Stochastic Porous Media Equations

The long time behaviour of solutions to generalized stochastic porous media equations on bounded intervals with zero Dirichlet boundary conditions is studied. We focus on a degenerate form of nonlinearity arising in self-organized criticality. Based on the so-called lower bound technique, the existence and uniqueness of an invariant measure is proved.

Numerical Simulation of Fluid Induced Fracturing: Micro-seismicity and Fluid Type Dependency

To monitor the integrity of the storage formation in relation to CO2 injection, several methods and technologies will be employed, including microseismic monitoring. Injection of CO2 leads to increased pore-pressure which in turn can activate existing weaknesses, e.g. faults, at the risk of leakage. Presented is a hydro-mechanical model that describes single-phase fluid flow in a deforming porous media with an existing weakness that can fracture. In the intact porous media, the fluid flow is governed by Darcy's law and the geomechanical behavior is described by Biot's theory of linear poroelasticity. The weakness zone is defined as a zero-thickness domain and the governing equations are obtained by integrating the governing equations of the intact porous media over the virtual thickness of the weakness. This makes possible the seamless transition of an intact porous media formulation to an open fracture formulation as the weakness zone activates, deteriorates and eventually ruptures. The deterioration of the weakness zone is described using the Cohesive Zone Model (CZM). The hydro-mechanical model is used to study the impact of different fluid types on activation and fracturing of a weakness, in particular mode-I, tensile failure, and potential implications for micro-seismicity. Firstly, it was confirmed that fluid type matters, e.g. under the same pressurization, different fluids (different mobility and compressibility) propagates a fracture at different velocities. Additionally, three hypotheses from observations in the literature was tested. One hypothesis was confirmed, as simulation results show that for a high-viscosity fluid, a strong pore pressure gradient occurs ahead of the fracture tip, in the propagation direction, thus driving the fluid flow and the propagation of the fracture, favoring longer fractures. However, for the other two hypotheses, the hydro-mechanical model could neither confirm that low-viscosity fluids will tend to produce smaller but more abundant fractures (as reported in the literature), nor reproduce observations of lower breakdown- and initiation pressure of rocks for low-viscosity fluids. The reason for this discrepancy could be differences in the premises of the observation and the case studied here, or due to other factors not considered in this study, e.g. chemical effect, differences in wetting properties of the rock, or that super critical CO2 is not representative for typical low-viscosity fluids.

Higher regularity estimates for the porous medium equation near the heat equation

In this paper we investigate regularity aspects for solutions of the nonlinear parabolic equation u_t= \Delta u^m, \quad m > 1, usually called the porous medium equation. More precisely, we provide sharp regularity estimates for bounded nonnegative weak solutions along the free boundary \partial\{u > 0\} , when the equation is universally close to the heat equation. As a consequence, local Lipschitz estimates are also established for this scenario.