Fractional Porous Medium Equation Research Articles

On the global existence and analyticity of the mild solution for the fractional Porous medium equation

In this research article we focus on the study of existence of global solution for a three-dimensional fractional Porous medium equation. The main objectives of studying the fractional porous medium equation in the corresponding critical function spaces are to show the existence of unique global mild solution under the condition of small initial data. Applying Fourier transform methods gives an equivalent integral equation of the model equation. The linear and nonlinear terms are then estimated in the corresponding Lei and Lin spaces. Further, the analyticity of solution to the fractional Porous medium equation is also obtained.

Large time behavior of signed fractional porous media equations on bounded domains

Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.

The fractional porous medium equation on noncompact Riemannian manifolds

The fractional porous medium equation on noncompact Riemannian manifolds

Periodic measures for a class of SPDEs with regime-switching

We use the variational approach to investigate periodic measures for a class of stochastic partial differential equations (SPDEs) with regime-switching. The hybrid system is driven by degenerate Lévy noise. We use the Lyapunov function method to study the existence of periodic measures and show the uniqueness of periodic measures by establishing the strong Feller property and irreducibility of the associated time-inhomogeneous semigroup. The main results are applied to stochastic fractional porous medium equations with regime-switching.

Derivation of the fractional porous medium equation from a microscopic dynamics

In this article we derive the fractional porous medium equation for any power of the fractional Laplacian as the hydrodynamic limit of a microscopic dynamics of random particles with long range interactions, but the jump rate highly depends on the occupancy near the sites where the interactions take place.

An inverse problem for the fractional porous medium equation

We consider a time-independent variable coefficients fractional porous medium equation and formulate an associated inverse problem. We determine both the conductivity and the absorption coefficient from exterior partial measurements of the Dirichlet-to-Neumann map. Our approach relies on a time-integral transform technique as well as the unique continuation property of the fractional operator.

The fractional porous medium equation on manifolds with conical singularities II

AbstractThis is the second of a series of two papers that studies the fractional porous medium equation, with and , posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin–Sobolev spaces including the Rellich–Kondrachov theorem and Sobolev–Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for initial data and all . We further investigate a number of properties of the solutions, including comparison principle, contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.

Complexity of asymptotic behavior of solutions for the fractional porous medium equation

In this paper, we study the large time behavior of solutions to the fractional porous medium equation $ u_{t} = \nabla\cdot (u\nabla^{\alpha-1}u) $ in $ \mathbb{R}^{N} $ with $ 0<\alpha<2 $. More precisely, we reveal that for any given $ 0<\mu<\frac{2N}{N+\alpha} $ and $ \beta>\frac{2-\mu}{2\alpha} $, there exists an initial-value $ u_{0}(x) $ such that the complexity of asymptotic behavior for the rescaled solutions $ t^{\frac{\mu}{2}}u(t^{\beta}\cdot,t) $ occurs in $ C_{0}^{+}(\mathbb{R}^{N}) $. For this purpose, we apply the $ L^{1} $-$ L^{\infty} $ smoothing effect and establish the propagation estimates for the solutions.

Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger’s approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo–Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.

Group classifications, optimal systems, symmetry reductions and conservation law of the generalized fractional porous medium equation

This paper investigates the generalized fractional porous medium equations (GFPMEs) which are the generalizations of the dual porous medium equation with integer order derivative. The complete group classification of the equations in consideration are performed with respect to their arbitrary parameters. And all vector fields admitted by the GFPMEs are obtained with the help of Lie symmetry analysis. In addition, the corresponding optimal systems are provided and the group-invariant solutions to the GFPMEs are constructed by performing symmetry reductionsand the three-dimensional diagrams of the obtained group-invariant solutions are presented. Furthermore, the conservation law of one of the considered equations is established by means of new conservation theorem.

Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic p-Laplace equations

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional p-Laplace equations which includes the fractional parabolic p-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

The fractional porous medium equation on manifolds with conical singularities I

This is the first of a series of two papers which studies the fractional porous medium equation on a Riemannian manifold with isolated conical singularities. In this article, we show R-sectoriality for the fractional powers of possibly non-invertible R-sectorial operators. Applications concern existence, uniqueness and maximal \(L^{q}\)-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided, and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator.

Gevrey Regularity and Time Decay of Fractional Porous Medium Equation in Critical Besov Spaces

In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 α ≤ 2, we prove global well-posedness for initial data with 1 ≤ p q ≤ ∞, and analyticity of solutions with 1 p q ≤ ∞. In particular, we also proved that when α = 1, both u and belong to . We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is as t → ∞.

Group Classifications, Optimal Systems, Symmetry Reductions and Conservation Law of the Generalized Fractional Porous Medium Equation

Group Classifications, Optimal Systems, Symmetry Reductions and Conservation Law of the Generalized Fractional Porous Medium Equation

The Human Meniscus Behaves as a Functionally Graded Fractional Porous Medium under Confined Compression Conditions

In this study, we observe that the poromechanical parameters in human meniscus vary spatially throughout the tissue. The response is anisotropic and the porosity is functionally graded. To draw these conclusions, we measured the anisotropic permeability and the “aggregate modulus” of the tissue, i.e., the stiffness of the material at equilibrium, after the interstitial fluid has ceased flowing. We estimated those parameters within the central portion of the meniscus in three directions (i.e., vertical, radial and circumferential) by fitting an enhanced model on stress relation confined compression tests. We noticed that a classical biphasic model was not sufficient to reproduce the observed experimental behaviour. We propose a poroelastic model based on the assumption that the fluid flow inside the human meniscus is described by a fractional porous medium equation analogous to Darcy’s law, which involves fractional operators. The fluid flux is then time-dependent for a constant applied pressure gradient (in contrast with the classical Darcy’s law, which describes a time independent fluid flux relation). We show that a fractional poroelastic model is well-suited to describe the flow within the meniscus and to identify the associated parameters (i.e., the order of the time derivative and the permeability). The results indicate that mean values of λβ,β in the central body are λβ=5.5443×10−10m4Ns1−β, β=0.0434, while, in the posterior and anterior regions, are λβ=2.851×10−10m4Ns1−β, β=0.0326 and λβ=1.2636×10−10m4Ns1−β, β=0.0232, respectively. Furthermore, numerical simulations show that the fluid flux diffusion is facilitated in the central part of the meniscus and hindered in the posterior and anterior regions.

Relaxation to Fractional Porous Medium Equation from Euler–Riesz System

We perform asymptotic analysis for the Euler–Riesz system posed in either \({\mathbb {T}}^d\) or \({\mathbb {R}}^d\) in the high-force regime and establish a quantified relaxation limit result from the Euler–Riesz system to the fractional porous medium equation. We provide a unified approach for asymptotic analysis regardless of the presence of pressure in the case of repulsive Riesz interactions, based on the modulated energy estimates, the Wasserstein distance of order 2, and the bounded Lipschitz distance. For the attractive Riesz interaction case, we consider the periodic domain and estimate a lower bound on the modulated internal energy to handle the modulated interaction energy.

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>

Regularity of solutions of a fractional porous medium equation

This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely $$ \partial\_t u = { \nabla \cdot} \left(u \nabla(-\Delta)^{\frac{\alpha}{2}-1}u^{m-1} \right) $$ where $u:\mathbb{R}+\times \mathbb{R}^N \to \mathbb{R}+$, for $0 < \alpha < 2$ and $m\geq2$. We prove that the $L^1\cap L^\infty$ weak solutions constructed by Biler, Imbert and Karch (2015) are locally Hölder-continuous in time and space. In this article, the classical parabolic De Giorgi techniques for the regularity of PDEs are tailored to fit this particular variant of the PME equation. In the spirit of the work of Caffarelli, Chan and Vasseur (2011), the two main ingredients are the derivation of local energy estimates and a so-called "intermediate value lemma". For $\alpha\leq1$, we adapt the proof of Caffarelli, Soria and V\\'azquez\~(2013), who treated the case of a linear pressure law. We then use a non-linear drift to cancel out the singular terms that would otherwise appear in the energy estimates. \keywords{Parabolic regularity, De Giorgi method, porous medium equation (PME), H\\"older regularity, non local operators, fractional derivatives.

The fractional porous medium equation on the hyperbolic space

We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $$L^p$$ spaces or to larger (weighted) spaces determined either in terms of a ground state of $$\Delta _{\mathbb {H}^{N}}$$ , or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $$L^1-L^\infty $$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.

Fractional Porous Medium Equations to Non-Darcian Flow by Means of the Bernstein Polynomials

The Bernstein polynomials is used to develop the fractional differential Swartzendruber model in porous media. With the operator matrix of the Bernstein polynomials, the fractional differential Swartzendruber equation is transformed into the products of several dependent matrixs which can also be regarded as the system of linear equations after dispersing the variable. By solving the system of linear equations, the numerical solutions are acquired. Some experimental data are provided to show that the method is computationally efficient and accurate.