We consider the 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 $Δ_{\mathbb{H}^n}$, 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.
We consider the fractional porous medium flow introduced by Caffarelli-Vazquez and obtain local in time existence, uniqueness, and blow-up criterion for smooth solutions. The proof is based on establishing a commutator estimate involving fractional Laplacian operators.
In this paper, we developed the group analysis to the time Fractional Porous Medium Equation (FPME) and the time Fractional Dual Porous Medium Equation (FDPME). The symmetry groups and the corresponding optimal systems of these two equations are obtained. Based on the above results, similarity reductions are performed and some explicit group invariant solutions are constructed.
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.
We consider nonlinear nonlocal diffusive evolution equations, governed by fractional Laplace-type operators, fractional time derivative and involving porous medium type nonlinearities. Existence and uniqueness of weak solutions are established using approximating solutions and the theory of maximal monotone operators. Using the De Giorgi-Nash-Moser technique, we prove that the solutions are bounded and Hölder continuous for all positive time.
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.
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, based on the modulated energy estimates, the Wasserstein distance of order $2$, and the bounded Lipschitz distance.
Following the methodology of [Brasco and Volzone, Adv. Math. 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 prof…
In this work we prove that the time-fractional porous medium equation on the half-line with Dirichlet boundary condition has a unique compactly supported solution. The approach we make is based on a transformation of the fractional integro-differential equation into a nonlinear Volterra integral equation. Then, the shooting method is applied in order to facilitate the analysis of the free-boundary problem. We further show that there exists an exactly one choice of initial conditions for which the solution has a zero which guarantees the no-flux condition. Then, our previous considerations impl…
In this paper we prove existence of entropy solutions to the time-fractional porous medium type equation, $$\partial_t[k\ast(u-u_0)]-\operatorname{div} (A(t,x)\nablaφ(u))=f\text{ in }Q_T=(0,T)\timesΩ,$$ with Dirichlet boundary condition, initial condition $u(0,\cdot)=u_0$ in $Ω$, and $L^1$-data $f\in L^1((0,T)\timesΩ), u_0\in L^1(Ω)$. To this end we approximate the data by $L^\infty$-functions, use a known existence result of weak solutions for these more regular data, and additionally a known contraction principle for weak solutions, which can be adopted to the entropy solutions.
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 eq…
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 met…
Self-similar solutions to the time-fractional Porous-Medium Equation
Universidad Autónoma de Madrid（马德里自治大学）；Instituto de Ciencias Matemáticas, CSIC（西班牙国家研究委员会数学科学研究所）；Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）
We prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w.r.t. time belongs to the large class of ${\cal PC}$ kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the …
Symmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Type
利用浓度比较的概念，建立了线性分数阶扩散方程$\partial_t u +(-Δ)^{σ/2}u=f$及其椭圆方程$h v +(-Δ)^{σ/2}v=f$，$h>0$解的对称化结果.结果扩展到非线性版本，$\partial_t u+(-Δ)^{σ/2}A(u)=f$，但只有当$A:\re_+\to\re_+$是一个凹函数。在椭圆情形下，当$B(v)$是一个凸非负函数，$v>0$与$B(0)=0$时，证明了$\,B(v)+(-Δ)^{σ/2}v=f$ \的完全对称化结果，当$B$是凹函数时，证明了部分对称化结果。当$A$为凸时，分别构造了抛物型方程的显着反例.当$B$是凹的时，这样的反例在标准扩散情形$σ=2$中不存在。
In this paper we investigate existence of solutions for the system: \begin{equation*} \left\{ \begin{array}{l} D^α_tu=\textrm{div}(u \nabla p),\\ D^α_tp=-(-Δ)^{s}p+u^{2}, \end{array} \right. \end{equation*} in $\mathbb{T}^3$ for $0< s \leq 1$, and $0< α\le 1$. The term $D^α_t u$ denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian $(-Δ)^{s}$ represents the Lévy diffusion. We prove global existence of nonnegative weak solutions that satisfy a variational inequality. The proof uses several approximations steps, including an implicit Euler time discretizati…
This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely $\partial_t u = { \nabla \cdot} \left(u \nabla(-Δ)^{\fracα{2}-1}u^{m-1} \right)$ where $u:\mathbb{R}_+\times \mathbb{R}^N \to \mathbb{R}_+$, for $0<α<2$ and $m\geq2$. We prove that the $L^1\cap L^\infty$ weak solutions constructed by Biler, Imbert and Karch (2015) are locally H{ö}lder-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…
Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains
研究了分数阶多孔介质方程$u_t+(-Δ)^su^m=0$在$(0,\infty)\timesΩ$中，$m>1$和$s\in (0,1)$，以及Dirichlet边界数据$u=0$在$(0,\infty)\times({\mathbb R}^N\setminusΩ)$中，非负初始条件$u(0,\cdot)=u_0\geq0$的正解和正则解.我们的第一个结果是一个定量下限的解决方案，它适用于所有积极的时间$t>0$。因此，我们发现了一个全球性的Harnack原则，指出对于任何$t>0$解决方案是可比的$d^{s/m}$，其中$d$是距离$\partialΩ$。这与局部情况$s=1$形成鲜明对比，其中方程具有有限的传播速度。在此基础上，我们研究了解的正则性。我们证明了解决方案是经典的内部（$C^\infty$在$x$和$C^{1,α}$在$t$），并建立了一个尖锐的$C^{s/m}_x$正则性估计的边界。我们的方法是相当普遍的，并可以应用到更广泛的一类非局部抛物方程的形式$u_t-\mathcal L F(u)=0$在$Ω$，无论是在有界或无界域。
