Solutions Of Porous Medium Equation Research Articles

Existence of weak solutions for porous medium equation with a divergence type of drift term

We consider degenerate porous medium equations with a divergence type of drift terms. We establish existence of nonnegative $$L^{q}$$ -weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on q and m caused by nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider, if the drift term is divergence-free. Similar conditions of gradients of drift terms are also provided to ensure the existence of such weak solutions. Uniqueness results follow under an additional condition on the gradients of the drift terms with the aid of methods developed in Wasserstein spaces. One of our main tools is so called the splitting method to construct a sequence of approximated solutions, which implies, by passing to the limit, the existence of weak solutions satisfying not only an energy inequality but also moment and speed estimates. One of crucial points in the construction is uniform Hölder continuity up to initial time for homogeneous porous medium equations, which seems to be of independent interest. As an application, we improve a regularity result for solutions of a repulsive Keller-Segel system of porous medium type.

Homotopy Analysis Laplace Transform Method for Analytical Solution of Porous Medium Equation

Homotopy Analysis Laplace Transform Method for Analytical Solution of Porous Medium Equation

$L^1$-Convergence to Generalized Barenblatt Solution for Compressible Euler Equations with Time-Dependent Damping

The large time behavior of entropy solution to the compressible Euler equations for polytropic gas (the pressure $p(\rho)=\kappa\rho^{\gamma}, \gamma>1$) with time dependent damping like $-\frac{1}{(1+t)^\lambda}\rho u$ ($0<\lambda<1$) is investigated. By introducing an elaborate iterative method and using the intensive entropy analysis, it is proved that the $L^\infty$ entropy solution of compressible Euler equations with finite initial mass converges strongly in the natural $L^1$ topology to a fundamental solution of porous media equation (PME) with time-dependent diffusion, called by generalized Barenblatt solution. It is interesting that the $L^1$ decay rate is getting faster and faster as $\lambda$ increases in $(0, \frac{\gamma}{\gamma+2}]$, while is getting slower and slower in $[ \frac{\gamma}{\gamma+2}, 1)$.

Decay Estimates for Solutions of Porous Medium Equations with Advection

In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form \begin{equation} \notag u_t \,+\; \mbox{div}\,f(x,t,u) \;=\; \mbox{div}\,(\;\!|\,u\,|^{\alpha} \, \nabla u \;\!), \quad \;\; x \in \mathbb{R}^{n}\!\:\!, \; t > 0, \end{equation} where $ \alpha > 0 \, $ is constant, decrease to zero, under fairly broad conditions on the advection flux $f$. Besides that, we derive a time decay rate for these solutions.

Complicated asymptotic behavior of solutions for porous medium equation in unbounded space

In this paper, we find that the unbounded spaces Yσ(RN)(0<σ<2m−1) can provide the work spaces where complicated asymptotic behavior appears in the solutions of the Cauchy problem of the porous medium equation. To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimates, the growth estimates and the weighted L1–L∞ estimates for the solutions.

Proper spaces for the asymptotic convergence of solutions of porous medium equation

In this paper, we consider the problem that a weighted L∞ space Wϑ(RN) is proper or not for the asymptotic convergence of solutions of the porous medium equation. We find that there exists a critical exponent ϑ=σ of the proper spaces for the asymptotic problem proposed by Alikakos and Rostamian (1984).

Numerical Simulation that Provides Non-oscillatory Solutions for Porous Medium Equation and Error Approximation of Boussinesq's Equation

So far, so many works have been done in a different way to find the exact track that can grasp the true solution for the one-dimensional porous media equation (PME). For instance, Monika studied using relaxation [1], Q. Zhang and Z. Wu. have already done the similar work by local degenerate Galarkin (LDG) method [12] and so on. Still, this is a challenge to find an appropriate scheme that can track the true solution when adiabatic exponent increases monotonically. In this paper, we have studied numerical result for where we have used Explicit-Implicit Finite Difference Method (EIFDM). Since so far PME is a degenerate parabolic equation and analytically the existence and uniqueness occur weakly only in the Sobolev sense, it is very hard to track the true solution numerically. Our main objective is to study numerically the PME with mixed boundary conditions and shown the result is helpful to track the Barenblatt's self similar solution and its interface when adiabatic exponent larger than 3 that provides much less error. This paper will show a possibility that Finite Difference Method (FDM) is also helpful rather the Finite Elements Method to track the interface in the simulation with an appropriate initial guess. Also checked L1, L2 and L∞-error for Boussinesq's equation which is a fundamental equation of ground- water flow, hopefully, the simulated results can help when this equation is useful in the practical world. Finally, all studied results are given to show the advantage of the θ-scheme method in the simulation of the PME and its capability to capture accurately sharp interfaces without oscillation.

Gradient estimates for porous medium equations under the Ricci flow

A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compact Riemannian manifolds is also obtained.

Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension

In this paper we consider the following equation $$ u_t=(u^m)_{xx}+(u^n)_x, \ \ (x, t)\in \mathbb{R}\times(0, \infty) $$with a Dirac measure as initial data, i.e., $u(x,0)=\delta(x)$. The solution of the Cauchy problem is well-known assource-type solution. In the recent work [11] the authorstudied the existence and uniqueness of such kind of singularsolutions and proved that there exists a number $n_0=m+2$ suchthat there is a unique source-type solution to the equation when$0 \leq n < n_0$. Here our attention is focused on the nonexistenceand asymptotic behavior near the origin for a short time. We provethat $n_0$ is also a critical number such that there exits nosource-type solution when $n \geq n_0$ and describe the short timeasymptotic behavior of the source-type solution to the equationwhen $0 \leq n < n_0$. Our result shows that in the case ofexistence and for a short time, the source-type solution of suchequation behaves like the fundamental solution of the standardporous medium equation when $0 \leq n < m+1$, the uniqueself-similar source-type solution exists when $n = m+1$, and thesolution does like the nonnegative fundamental entropy solution inthe conservation law when $m+1 < n < n_0$, while in the case ofnonexistence the singularity gradually disappears when $n \geq n_0$that the mass cannot concentrate for a short time and no such a singular solutions exists. The results of previouswork [11] and this paper give a perfect answer to suchtopical researches.

The Solution of Porous Medium Equation by Aboodh Homotopy Perturbation Method

In this paper, the homotopy perturbation method (HPM) and Aboodh transform are employed to obtain analytical solution of the porous medium equation. The proposed method (ATHPM) is an elegant combination of the new integral transform “Aboodh Transform” and the homotopy perturbation method. The porous medium equations have importance in engineering and sciences and constitute a good model for many systems in various fields. The results tell us that the proposed method is more efficient and easier to handle when is compared with existing other methods in such partial differential equations.

A solution of porous media equation

A solution of porous media equation

Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian

We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=\Delta_\phi(u^p)$$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the $m$-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which generalize the results in [20] and [13]. Moreover, inspired by X. -D. Li's work in [19] we also study the entropy formulae introduced in [20] for porous medium equations and fast diffusion equations associated with the Witten Laplacian. We prove monotonicity theorems for such entropy formulae on compact Riemannian manifolds with non-negative $m$-dimensional Bakry-Emery Ricci curvature

Gradient estimates of porous medium equations under the Ricci flow

In this paper we consider an n-dimensional manifold Mn evolving under the Ricci flow and establish gradient estimates for positive solutions of porous medium equations on Mn. As applications, we derive Harnack type inequalities. In particular, our results generalize gradient estimates for positive solutions of the heat equations in Liu (Pacific J Math 243:165–180 [18]).

Instant blow-up solutions for porous medium equation with sources

Instant blow-up solutions for porous medium equation with sources

Global existence and blow-up properties of solutions for porous medium equation with nonlinear memory and weighted nonlocal boundary condition

In this paper, we investigate a initial boundary value problem of porous medium equation with nonlocal boundary condition. We find the effects of weight function in the boundary condition and competitive relationship between nonlinear memory term and inner absorption term on whether determining blow-up of solutions or not, and give the blow-up rate estimate under some suitable condition.

Hamilton’s gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds

Let M be a complete noncompact Riemannian manifold of dimension n. In this paper, we derive a local gradient estimate for positive solutions of porous medium equation ∂tu=Δuα,1<α<1+12n+1 on M×(−∞,0]. We also obtain a theorem of Liouville type for positive solutions of the porous medium equation.

Blow-Up Phenomena for Porous Medium Equation with Nonlinear Flux on the Boundary

We investigate the blow-up phenomena for nonnegative solutions of porous medium equation with Neumann boundary conditions. We find that the absorption and the nonlinear flux on the boundary have some competitions in the blow-up phenomena.

Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, V\'{a}zquez and Villani in [10]. Moreover, our results recover the ones of Davies in [4], Hamilton in [5] and Li and Xu in [7].

Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time. Recently several papers have considered the flow of convex hypersurfaces by speeds which are homogeneous functions of the principal curvatures of degree α > 1. Under suitable pinching conditions on the curvature of the initial hypersurface, the aim is to prove that solutions become spherical as they contract to points. This behaviour has been established for a wide range of flows where the speed is homogeneous of degree 1 in the principal curvatures, including the mean curvature flow (H1), the flow by nth root of Gauss curvature (Ch1), square root of scalar curvature (Ch2), and a large family of other speeds (A1,A3,A4). The first such result with degree of homogeneity higher than 1 was due to Ben Chow (Ch1), and concerned flow by powers of the Gauss curvature. He proved that flow by K β with β ≥ 1/n produces a spherical limiting shape provided the initial hypersurface is sufficiently pinched, i n the sense that hij ≥ C(β)Hgij. Later such results were proved by Schulze (S3) for powers of the mean curvature, by Alessandroni and Sinestrari for powers of the scalar curvature (A,AS), and by Cabezas-Rivas and Sinestrari (CRS) for normalized flows by powers of elementary symmetric functions. A feature of the results mentioned above is that the flows all have some divergence structure, a point which is used crucially in deriving sufficient regularity of solutions to deduce the existence of a smooth limiting hypersurface: In (Ch1) the divergence structure was used to deduce pinching of the principal curvatures using integral estimates in a manner similar to (H1). In (S3), (A), and (AS) the curvature pinching was proved using maximum principle arguments, but the divergence structure was still needed because higher regularity of solutions was established using results for divergence form porous-medium equations. To date we are not aware of any work which provides Holder continuity for solutions of porous medium equations in non-divergence form without assuming regularity of the coefficients. In this paper we provide a geometric estimate which circumvents this difficulty: We prove in Section 3 that if the ratios of principal curvatures at each point are bounded in terms of the maximum principal curvature by a function which approaches one at infinity, then the ratio of circumradius to inradius is as close to one as desired if the circumradius is sufficiently small. Using

Classification of solutions of porous medium equation with localized reaction in higher space dimensions

We consider the behavior of nonnegative solutions to the Cauchy problem of the porous medium equation with localized reaction term: \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = \Delta(u^m) + a(x)u^p, & (x,t) \in \mathbf{R}^n \times (0,T),\\ u(x,0) = u_0(x), & x \in \mathbf{R}^n, \end{array} \right. \end{eqnarray*} where $ m > 1 $, $ p > 0 $, $ a(x) \geq 0 $ is a compactly supported function, and $u_0(x)$ is continuous, nonnegative but not identical with zero, and has compact support as well. We show the relationship between the occurrence of blow-ups and the exponents $m$ and $p $: in two-dimensional space, all the solutions are globally defined if $0 < p \leq \frac{m+1}{2}$, and the solutions may blow up in finite time if $p \geq m$; in spaces higher than two-dimensional, all the solutions are global if $0 < p < m$, and there exist both global solutions and blow-up solutions if $p \geq m$. We also show that, for any solution, the intersection of its support and the support of $a(x)$ will be non-empty at some time.