Qifan Li
This paper is concerned with the self-improving property for obstacle problem related to the singular porous medium equation. We establish a local higher integrability result for the spatial gradient of the $m$-th power of nonnegative weak solutions, under some suitable regularity assumptions on the obstacle function.
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Local boundedness and higher integrability for the sub-critical singular porous medium system
该研究针对多孔介质型方程/组弱解梯度的可积性问题，解决了临界与次临界奇异情形下的局部有界性及更高可积性问题，完善了相关研究体系。
We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given by \begin{equation*} \partial_t u-Δ(|u|^{m-1}u)=\mathrm{div}\,F\,, \end{equation*} where $m>1$. More precisely, we prove that under suitable assumptions the spatial gradient $D(|u|^{m-1}u)$ of any weak solution is integrable to a larger power than the natural power $2$. Our analysis includes both the case of the lateral boundary and the initial boundary.
We show that signed weak solutions to parabolic obstacle problems with porous medium type structure are locally Hölder continuous, provided that the obstacle is Hölder continuous.
On two notions of solutions to the obstacle problem for the singular porous medium equation
本文证明对于足够正则且紧支撑的障碍，奇异多孔介质方程障碍问题的位势理论解与基于变分不等式的解这两种概念是一致的。
Continuity up to the boundary for obstacle problems to porous medium type equations
本文证明多孔介质型方程障碍问题在边界数据连续时，带符号弱解连续到抛物边界，该结果对奇异方程亦为新。
We deal with the obstacle problem for the porous medium equation in the slow diffusion regime $m>1$. Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a seq…
Higher integrability for the singular porous medium system
本文在快扩散范围内，通过内禀尺度、反向Hölder不等式和Vitali覆盖论证，建立了多孔介质系统弱解空间梯度的高阶可积性，并给出显式反向Hölder不等式，适用于一般快扩散系统及非负与带符号解。
We prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case $m<1$. The result holds for the natural range of exponents that is known from other regularity results for porous medium type equations. We also cover the case of signed solutions.
We prove continuity up to the boundary of the minimizer of an obstacle problem and higher integrability of its gradient under generalized Orlicz growth. The result recovers similar results obtained in the special cases of polynomial growth, variable exponent growth and produces new results for Orlicz and double phase growth.
We prove the existence and uniqueness of non-negative entropy solutions of the obstacle problem for stochastic porous media equations. The core of the method is to combine the entropy formulation with the penalization method.
We consider a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. The equation is posed in the whole space R^n. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density, which we call obstacle Barenblatt solutions. The theory for elliptic fractional problems with obstacles has been recently …
We prove global $W^{1,q}(Ω,\mathbb{R}^N)$-regularity for minimisers of $\mathscr{F}(u)=\int_ΩF(x,\mathrm{D}u)\mathrm{d} x$ satisfying $u\geq ψ$ for a given Sobolev obstacle $ψ$. $W^{1,q}(Ω,\mathbb{R}^m)$ regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on $F(x,z)$ are a uniform $α$-Hölder continuity assumption in $x$ and natural $(p,q)$-growth conditions in $z$ with $q<\frac{(n+α)p}{n}$. In the autonomous case $F\equiv F(z)$ we can improve the gap to $q<\frac{np}{n-1}$, a result new even in the unconstrained case.
This paper concerns a space-time homogenization limit of nonnegative weak solutions to porous medium equations. In particular, the so-called homogenized matrix will be characterized in terms of solutions to cell problems, which drastically vary in a scaling parameter $r > 0$. A similar problem has already been studied in [1], where the growth of the power nonlinearity is strictly restricted due to some substantial obstacles. In the present paper, such obstacles will be overcome by developing local uniform estimates for the gradients of nonnegative weak solutions.
Homogenization of a micropolar fluid past a porous media with non-zero spin boundary condition
本文研究了微极流体通过多孔介质的均质化问题，提出了非均匀微极旋转边界条件，并推导了类似经典微极达西定律的模型。
By using the unfolding operators for periodic homogenization, we give a general compactness result for a class of functions defined on bounded domains presenting perforations of two different size. Then we apply this result to the homogenization of the flow of a Bingham fluid in a porous medium with solid obstacles of different size. Next we give the interpretation of the limit problem in term of a non linear Darcy law.
We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse Hölder inequality in suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calderón-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution $u$.
For the logarithmically singular parabolic equation \[ u_t-Δ\ln u=0\qquad\text{weakly in}\ \ E\times(0,T], \] we establish a Harnack type estimate in the $L^1_{loc}$ topology, and we show that the solutions are locally analytic in the space variables and differentiable in time. The main assumption is that $\ln u$ possesses a sufficiently high degree of integrability, namely \begin{equation*} \ln u\in L^\infty_{loc}\big(0,T;L^p_{loc}(E)\big) \quad\text{for some} p\ge1. \end{equation*} These two properties are known for solutions of singular porous medium type equations ($0<m<1$), which formally…
We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.
We study the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions $u^m \in L^2_{\rm loc}(0,T;H^1_{\rm loc}(Ω))$ and $u^\frac{m+1}2 \in L^2_{\rm loc}(0,T;H^1_{\rm loc}(Ω))$ in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.
Multiscale Hyperbolic-Parabolic Models for Nonlinear Reactive Transport in Heterogeneously Fractured Porous Media
本文提出多尺度双曲-抛物模型，用于描述异质裂隙多孔介质中非线性反应输运的均质化过程，通过分析推导了耦合非线性界面条件的双曲系统，并给出了误差估计。
We prove a comparison principle for the porous medium equation in more general open sets in $\mathbb{R}^{n+1}$ than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.
We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. Th…
In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.
We investigate the behaviour of the solutions $u_m(x,t)$ of the fractional porous medium equation $$ u_t+(-Δ)^s (u^m)=0, \quad x\in {\mathbb{R}}^N, \ t>0. $$ with initial data $u(x,0)\ge 0$, $x\in {\mathbb{R}}^N$, in the limit as $m\to\infty$ with fixed $s\in (0,1)$. We first identify the limit of the Barenblatt solutions as the solution of a fractional obstacle problem, and we observe that, contrary to the case $s=1$, the limit is not compactly supported but exhibits a typical fractional tail with power-like decay. In other words, we do not get a plain mesa in the limit, but a mesa with tails…
Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: an entropy formulation
本文通过为随机变分不等式引入新熵公式，建立带保守梯度噪声的退化非线性扩散方程障碍问题的存在性、唯一性和稳定性结果，并得到带非线性反应项的确定性多孔介质方程障碍问题的新适定性结果。
In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type $$ \left\{ \begin{array}{llll} \sum\limits _{i=1}\limits^{n} D_i (a_i(x,Du(x)))=0, &x\in Ω,\\ u(x)=θ(x), & x\in \partial Ω. \end{array} \right. $$ Under some suitable conditions, they obtained an integrability result, which shows that, higher integrability of the boundary datum $θ$ forces solutions $u$ to have higher integrability as well. In the present paper, we consider ${\cal K}_{ψ,θ}^{(p_i)}$-obstacle problems of the nonhomogeneous anisotropic elliptic equ…
In this paper, we develop a viscosity method for Homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the Porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of th…
Dispersive estimates for wave and Schrödinger equations with a potential in non-trapping exterior domains
本文研究非捕获外部区域中带短程位势的波动方程与薛定谔方程，通过证明预解估计，得到局部能量可积性、光滑效应，并将自由方程的局部 Strichartz 估计推广为全局估计。
Well-posedness of the obstacle problem for generalized Dean-Kawasaki equation
本文研究由相关保守噪声驱动的广义Dean-Kawasaki方程障碍问题，通过动能解的存在性、唯一性和L^1稳定性建立，结合动能特征的Skorokhod条件和障碍项的反射测度描述，扩展了随机偏微分方程障碍问题理论至退化方程类。
We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.
Fractional porous medium equation on manifolds with nonnegative Ricci curvature: existence of solutions and smoothing effects via potential methods
本文在非负 Ricci 曲率流形上研究分数阶多孔介质方程，通过 Green 函数引入加权初值空间，证明解的存在性并建立最优的局部与全局光滑化估计，同时推广至一般渗流方程。
From the steady Stokes and Navier-Stokes models, a penalization method has been considered by several authors for approximating those fluid equations around obstacles. In this work, we present a justification for using fictitious domains to study obstacles immersed in incompressible viscous fluids through a simplified version of Brinkman's law for porous media. If the scalar function $ψ$ is considered as the inverse of permeability, it is possible to study the singularities of $ψ$ as approximations of obstacles (when $ψ$ tends to $\infty$) or of the domain corresponding to the fluid (when $ψ= …
We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $\varepsilon$ separated by distances $d_\varepsilon$ and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of $\frac{d_{\varepsilon}}\varepsilon$ when $\varepsilon$ goes to zero. If $\frac{d_{\varepsilon}}\varepsilon\to \infty$, then the limit motion is not perturbed by the porous medium, namely we recover the Euler solution in the whole space. On the contrary, i…
This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the {\bf time} regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latters are frequency dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well-controlled. This idea is originally due to Nguyen and…
Geometric Estimates for Solutions of Semilinear Equations with Singular Potentials
本文研究了半线性方程中奇异势的解的几何估计，扩展了尖锐正则性和非退化估计到无界、变号源的情况，分析了非线性与源的最小积分性假设的相互作用。
We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse Hölder inequality in suitable intrinsic cylinders. Relying on an intrinsic Calderón-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for $m\in\left(\frac{(n-2)_+}{n+2},1\right)$. Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for $m\geq 1$ (see [GS16] in the list of references) to the singular case…
Effective models for generalized Newtonian fluids through a thin porous medium following the Carreau law
本文通过Carreau定律研究薄多孔介质中广义牛顿流体的流动，推导并验证了不同有效达西定律，并分析了流变参数和障碍物形状的影响。
Homogenization of the compressible Navier-Stokes equations via two-scale convergence in perforated domains
通过双尺度收敛方法，研究周期穿孔区域中可压缩等熵Navier-Stokes方程的均匀化，导出Darcy定律和孔隙介质方程，并建立密度强双尺度收敛，将绝热常数扩展到γ>9/5。
For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes $d_{\varepsilon}$ is equal or much larger than the size of the holes $\varepsilon$. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when $(\varepsilon,d_{\varepsilon})\to (0,0)$. Smaller distance was avoided for mathematical reasons and for theses large distances, the geometry of the holes does not affect -- or f…
The Riemann problem for three-phase foam flow in porous media
本文研究了三相泡沫流在多孔介质中的非线性偏微分方程组，通过泡沫局部平衡假设简化了问题，分析了泡沫注入过程中的解分类及油池形成条件，为数值模拟校准提供理论支持。
We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton (1998) in their study of the porous medium equation or the degeneracy of the Heston operator (1993) in mathematical finance. Existence of a solution to the Dirichl…
We consider a simplified Boltzmann equation: spatially homogeneous, two-dimensional, radially symmetric, with Grad's angular cutoff, and linearized around its initial condition. We prove that for a sufficiently singular velocity cross section, the solution may become instantaneously a function, even if the initial condition is a singular measure. To our knowledge, this is the first regularization result in the case with cutoff: all the previous results were relying on the non-integrability of the angular cross section. Furthermore, our result is quite surprising: the regularization occurs for …
Global Existence and Asymptotic Equivalence to Barenblatt-type Solutions for the Physical Vacuum Free Boundary Problem of Damped Compressible Euler Equations in M-D
本文证明了2D和3D带阻尼可压缩Euler方程物理真空自由边界问题的整体存在性，通过提升扰动时间导数的衰减率确保时间可积性，并证明其渐近等价于Barenblatt自相似解，同时统一了时间依赖与时间无关阻尼情形的处理。
Resolving Sharp Gradients of Unstable Singularities to Machine Precision via Neural Networks
通过神经网络和多阶段架构，解决高梯度不稳定奇异性问题，实现高精度验证并发现新解。
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