Porous Medium Equation Research Articles

Interacting particle system approximating the porous medium equation and propagation of chaos

Interacting particle system approximating the porous medium equation and propagation of chaos

A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions

This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption∂tu=Δum−|x|σup, posed for (x,t)∈RN×(0,∞), N≥1, and in the range of exponents 1<m<p<∞, σ>0. We give a complete classification of (singular) self-similar solutions of the formu(x,t)=t−αf(|x|t−β),α=σ+2σ(m−1)+2(p−1),β=p−mσ(m−1)+2(p−1), showing that their form and behavior strongly depends on the critical exponentpF(σ)=m+σ+2N. For p≥pF(σ), we prove that all self-similar solutions have a tail as |x|→∞ of one of the formsu(x,t)∼C|x|−(σ+2)/(p−m)oru(x,t)∼(1p−1)1/(p−1)|x|−σ/(p−1), while for m<p<pF(σ) we add to the previous the existence and uniqueness of a compactly supported very singular solution. These solutions will be employed in describing the large time behavior of general solutions in a forthcoming paper.

Existence and uniqueness for p-adic counterpartof the porous medium equation

Existence and uniqueness for p-adic counterpartof the porous medium equation

Convergence of solutions of the porous medium equation with reactions

Consider the Cauchy problem of one-dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as t\to \infty to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behavior of the solutions for PME with monostable, bistable and combustion types of nonlinearities.

Stabilized equal lower-order finite element methods for simulating Brinkman equations in porous media

This paper demonstrates the mixed formulation of the Brinkman problem using linear equal-order finite element methods in porous media modelling. We introduce Galerkin least-squares (GLS) and least-squares (LS) finite element methods to address the incompatibility of finite element spaces, treating velocity, pressure, and vorticity as independent variables. Theoretical analysis examines coercivity and continuity, providing error estimates. Demonstrating resilience in theoretical findings, these methods achieve optimal convergence rates in the L 2 norm by incorporating stabilization terms with low-order basis functions. Numerical experiments validate theoretical predictions, showing the effectiveness of the GLS method and addressing finite element space incompatibility. Additionally, the GLS method exhibits promising capabilities in handling the Brinkman equation at low permeability compared to the LS method. The study reveals an increase in the average pressure difference in the Brinkman problem compared to the Stokes equations as the inlet velocity rises, providing insights into the behaviour of Brinkman equations.

Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions

AbstractWe prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ($$0&lt;a&lt;1$$ 0 &lt; a &lt; 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

Gradient flows of generalized relative entropy and functional inequalities on graphs

We derive the discrete forms of the porous medium equation and fast diffusion equation with drift terms on a finite graph as gradient flows of the m-relative entropy, with the global existence of solutions. We show the Łojasiewicz inequality to prove the convergence of solutions and some important functional inequalities. More specifically, the Łojasiewicz inequality holds near the stationary solution with exponent 12, which leads to the exactly exponential convergence rate with a finite trajectory length. It can also be applied to show the Talagrand-type inequality and the log-Sobolev-type inequality directly.

Machine-learning aided calibration and analysis of porous media CFD models used for rotating packed beds

The proposed research is an attempt to advance the state-of-the-art of the numerical modelling of RPB by combining Computational Fluid Dynamics and Machine Learning approaches. The latter creates an accurate framework that should help quantify the potential of Rotating Packed Beds (RPB) technology to intensify conventional CO2 capture processes. To this end, a direct sensitivity analysis is detailed to supplement a machine-learning (ML) algorithm built for calibrating resistance coefficients needed for porous media modelling. The algorithm is used to improve CFD predictions of dry pressure drop in rotating packed beds (RPBs) for a wide range of operating conditions. The sensitivity derivatives with respect to the packing resistance coefficients are demonstrated for the first time in RPBs, which is not available in the current CFD open source and commercial codes. In this regard, sensitivity differential equations are derived from three-dimensional Navier-Stokes equations for porous media in a rotating reference frame. These sensitivity equations are discretized using a finite volume scheme and solved to obtain the sensitivity pressure drop differences at the packing edges. The results are validated against the predictions of the analytical sensitivity analysis and the finite difference approximation. After, the Newton – Gauss method that employs the sensitivity pressure drop derivatives, is used to minimize the error (cost function) between the pressure drop obtained from CFD simulations and the available experimental data. This is achieved by tuning the packing resistance coefficients to the RBPs' operating conditions (gas flowrate and rotating speed) and correlate them using an artificial neural network (ANN). The results of the proposed approach show a significant improvement in porous media-based CFD predictions of RPBs' pressure drop across a wide range of operating conditions and this over conventional porous media-based CFD models. This is necessary for CFD models to be reliably used as a tool that can efficiently improve existing RPBs' designs and/or participate in RPBs' design innovation.

Buoyancy-driven heat transfer and entropy analysis of a hydromagnetic GO-Fe3O4/H2O hybrid nanofluid in an energy storage enclosure partially filled with non-Darcy porous medium under an oblique magnetic field

Purpose Thermal energy storage systems use thermal energy to elevate the temperature of a storage substance, enabling the release of energy during a discharge cycle. The storage or retrieval of energy occurs through the heating or cooling of either a liquid or a solid, without undergoing a phase change, within a sensible heat storage system. In a sensible packed bed thermal energy storage system, the structure comprises porous media that form the packed solid material, while fluid occupies the voids. Thus, a cavity, partially filled with a fluid layer and partially with a saturated porous layer, has become important in the investigation of natural convection heat transfer, carrying significant relevance within thermal energy storage systems. Motivated by these insights, the current investigation delves into the convection heat transfer driven by buoyancy and entropy generation within a partially porous cavity that is differentially heated, vertically layered and filled with a hybrid nanofluid. Design/methodology/approach The investigation encompasses two distinct scenarios. In the first instance, the porous layer is positioned next to the heated wall, while the opposite region consists of a fluid layer. In the second case, the layers switch places, with the fluid layer adjacent to the heated wall. The system of equations for fluid and porous media, along with appropriate initial and boundary conditions, is addressed using the finite difference method. The Tiwari–Das model is used in this investigation, and the viscosity and thermal conductivity are determined using correlations specific to spherical nanoparticles. Findings Comprehensive numerical simulations have been performed, considering controlling factors such as the Darcy number, nanoparticle volume fraction, Rayleigh number, bottom slit position and Hartmann number. The visual representation of the numerical findings includes streamlines, isotherms and entropy lines, as well as plots illustrating average entropy generation and the average Nusselt number. These representations aim to provide insight into the influence of these parameters across a spectrum of scenarios. Originality/value The computational outcomes indicate that with an increase in the Darcy number, the addition of 2.5% magnetite nanoparticles to the GO nanofluid results in an enhanced heat transfer rate, showing increases of 0.567% in Case 1 and 3.894% in Case 2. Compared with Case 2, Case 1 exhibits a 59.90% enhancement in heat transfer within the enclosure. Positioning the porous layer next to the partially cooled wall significantly boosts the average total entropy production, showing a substantial increase of 11.36% at an elevated Rayleigh number value. Positioning the hot slit near the bottom wall leads to a reduction in total entropy generation by 33.20% compared to its placement at the center and by 33.32% in comparison to its proximity to the top wall.

Two Finite Element Approaches for the Porous Medium Equation That Are Positivity Preserving and Energy Stable

In this work, we present the construction of two distinct finite element approaches to solve the porous medium equation (PME). In the first approach, we transform the PME to a log-density variable formulation and construct a continuous Galerkin method. In the second approach, we introduce additional potential and velocity variables to rewrite the PME into a system of equations, for which we construct a mixed finite element method. Both approaches are first-order accurate, mass conserving, and proved to be unconditionally energy stable for their respective energies. The mixed approach is shown to preserve positivity under a CFL condition, while a much stronger property of unconditional bound preservation is proved for the log-density approach. A novel feature of our schemes is that they can handle compactly supported initial data without the need for any perturbation techniques. Furthermore, the log-density method can handle unstructured grids in any number of dimensions, while the mixed method can handle unstructured grids in two dimensions. We present results from several numerical experiments to demonstrate these properties.

Porous medium type reaction-diffusion equation: Large time behaviors and regularity of free boundary

We consider the Cauchy problem of the porous medium type reaction-diffusion equation∂tρ=Δρm+ρg(ρ),(x,t)∈Rn×R+,n≥2,m>1, where g is the given monotonic decreasing function with the density critical threshold ρM>0 satisfying g(ρM)=0. We prove that the pressure P:=mm−1ρm−1 in Lloc∞(Rn) tends to the pressure critical threshold PM:=mm−1(ρM)m−1 at the time decay rate (1+t)−1. If the initial density ρ(x,0) is compactly supported, we justify that the support {x:ρ(x,t)>0} of the density ρ expands exponentially in time. Furthermore, we show that there exists a time T0>0 such that the pressure P is Lipschitz continuous for t>T0, which is the optimal (sharp) regularity of the pressure, and the free surface ∂{(x,t):ρ(x,t)>0}∩{t>T0} is locally Lipschitz continuous. In addition, under the same initial assumptions of compact support, we verify that the free boundary ∂{(x,t):ρ(x,t)>0}∩{t>T0} is a local C1,α surface.

Biological invasion with a porous medium type diffusion in a heterogeneous space

The knowledge of traveling wave solutions is the main tool in the study of wave propagation. However, in a spatially heterogeneous environment, traveling wave solutions do not exist, and a different approach is needed. In this paper, we study the generation and the propagation of hyperbolic scale singular limits of a KPP-type reaction–diffusion equation when the carrying capacity is spatially heterogeneous and the diffusion is of a porous medium equation type. We show that the interface propagation speed varies according to the carrying capacity.

On a Porous Medium Equation with Weighted Inner Source Terms and a Nonlinear Nonlocal Boundary Condition

On a Porous Medium Equation with Weighted Inner Source Terms and a Nonlinear Nonlocal Boundary Condition

Porous-DeepONet: Learning the Solution Operators of Parametric Reactive Transport Equations in Porous Media

Reactive transport equations in porous media are critical in various scientific and engineering disciplines, but solving these equations can be computationally expensive when exploring different scenarios, such as varying porous structures and initial or boundary conditions. The deep operator network (DeepONet) has emerged as a popular deep learning framework for solving parametric partial differential equations. However, applying the DeepONet to porous media presents significant challenges due to its limited capability to extract representative features from intricate structures. To address this issue, we propose the Porous-DeepONet, a simple yet highly effective extension of the DeepONet framework that leverages convolutional neural networks (CNNs) to learn the solution operators of parametric reactive transport equations in porous media. By incorporating CNNs, we can effectively capture the intricate features of porous media, enabling accurate and efficient learning of the solution operators. We demonstrate the effectiveness of the Porous-DeepONet in accurately and rapidly learning the solution operators of parametric reactive transport equations with various boundary conditions, multiple phases, and multi-physical fields through five examples. This approach offers significant computational savings, potentially reducing the computation time by 50–1000 times compared with the finite-element method. Our work may provide a robust alternative for solving parametric reactive transport equations in porous media, paving the way for exploring complex phenomena in porous media.

The Solution Comparison of Fractional Heat Transfer and Porous Media Equations Using Analytical Techniques

The Solution Comparison of Fractional Heat Transfer and Porous Media Equations Using Analytical Techniques

Fractional Newton Explicit Group Method for Time-Fractional Nonlinear Porous Medium Equations

Fractional Newton Explicit Group Method for Time-Fractional Nonlinear Porous Medium Equations

A New Numerical Method to Evaluate the Stability of Dike Slope Considering the Influence of Backward Erosion Piping

Backward erosion piping, a soil erosion phenomenon induced by seepage, compromises the stability of water-retaining structures such as dikes. During floods, the seepage in the dike body increases due to high water levels, which directly affects the progression of the piping channel. The formation of the piping channel then impacts the stability of the dike. In this paper, an improved piping model that considers the impact of seepage in the dike body is proposed based on Wewer’s model. Specifically, we added a seepage field of the dike body to the original model to account for the impact of dike-body seepage on the evolution of piping. The seepage field of the dike body is solved using Darcy’s law and the continuity equation for unsaturated porous media. In addition, this approach also incorporates the coupling effect of seepage stress. The accuracy of the model was verified through comparing the calculated results with the IJkdijk experiment and Wewer’s results. The effects of BEP on dike stability were investigated using the proposed improved piping model. The two major conclusions of the study are that (1) the incorporation of unsaturated seepage enhanced the performance of the piping model, allowing it to more accurately simulate the development of pipe length and the changing of pore pressure; and (2) the formation of the pipe impacted dike stability, leading to a substantial reduction in the safety factor of the dike slope.

Using metal foam and nanoparticle additives with different fin shapes for PCM-based thermal storage in flat plate solar collectors

This research focuses on employing heat transfer enhancement techniques as well as nanoparticles, porous metal foams, and extended surfaces, for the PCM-based thermal energy storage system of a flat plate solar collector. Numerical investigations are carried out for different combinations of the aforementioned techniques, with I-, Y-, and T-shaped fins as the extended surface. The mathematical model is developed with porous media equations, where nanoparticles are embedded in the PCM under the assumption of an equivalent single-phase medium. Results suggest that employing these techniques one by one augments heat transfer performances in terms of reduced melting time. The addition of both metal foam and nanoparticles in the straight wall (case A) reduces melting time by 85.16% when compared to the pure-PCM case, while a 84.38% reduction is reached if only foams are employed. The I-shaped fins (case B) are shown to have an impact too; they provide a 37.07% melting time reduction with only PCM, reaching 89.68% and 86.19% with nanoparticles with foams, respectively. This means that metal foams have a primary role in reducing melting time, where nanoparticles and I-shaped fins become helpful too. Furthermore, by considering all techniques together, the heat storage rate increases by an order of magnitude.

Local and global well-posedness for fractional porous medium equation in critical Fourier-Besov spaces

In this paper, we study the Cauchy problem for the Fractional Porous Medium Equation in Rn for n ≥ 2. By using the contraction mapping method, Littlewood-Paley theory and Fourier analysis, we get, when 1 β ≤ 2, the local solution v ∈ XT := LT ∞(FBp,r (2 − 2m −β + n/p' )(Rn))∩ LTρ1(FBp,r s1(Rn))∩ LTρ2(FBp,r s2 ( Rn)) with 1 ≤ p &lt; ∞, 1 ≤ r ≤ ∞, and the solution becomes global when the initial data is small in critical Fourier-Besov spaces FBp,r (2 − 2m −β + n/p' )(Rn) . In addition, We establish a blowup criterion for the solutions. Furthermore, the global existence of solutions with small initial data in FB∞,1 (2 − 2m −β + n )(Rn) is also established. In the limit case β = 1, we prove global well-posedness for small initial data in critical Fourier-Besov spaces FBp,1 (2 − 2m + n/p' )(Rn) with 1 ≤ p &lt; ∞ and FB∞,1 (2 − 2m + n )(Rn), respectively.

Hölder Regularity for the Linearized Porous Medium Equation in Bounded Domains

Hölder Regularity for the Linearized Porous Medium Equation in Bounded Domains