The notes begin with a discussion of the significance of the equation and some of its applications. Special attention is paid to the appearance of a free boundary, a consequence of the propagating properties of the nite. The purpose of these lectures is to provide an introduction to the mathematical theory of the so-called Porous Medium Equation (PME for short), i.e.
Perhaps the best known of them is the description of the flow of an isentropic gas through a porous medium M]. In a way, the property of nite propagation supports the physical soundness of the equation to model diusion or heat propagation. This free boundary or propagation front is an important and difficult subject of mathematical investigation.
The reader will find a version of many of the main results about the equation in the. According to them, a good approximation of the process is obtained with the PME for an exponent m close to 6. Such equations and systems therefore form an interesting possibility of generalization of the theory of the PME.
This solution is unique and serves as an absolute upper bound for all solutions of the Dirichlet problem.
Existence of a weak solution. Energy estimate
First of all, we note that the classical solution to the problem is necessarily a weak solution in our sense. According to the uniqueness result, no other weak solution can exist with the same data, so the classical solution does not exist. In this way, the problem is solved in the classical sense and the degeneracy of the equation is avoided.
To show that this u is the weak solution to the problem, we need some estimates. Since T is arbitrary, it follows that frumng is uniformly bounded in L2(Q), and therefore a subsequence of it should converge to a weak limit in L2(Q). To conclude this step, let's note that if we have two initial data au0u^0 such that u0 u^0, the above approximation process can produce ordered approximate sequences, u0n u^0n.
An alternative proof, where steps 2 and 3 are replaced by a single approximation step using a stability result, will be given at the end of x6. On the other hand, we see from the proof that the choice of Lm+1(&) as the initial data space essentially depends on the estimate (3.10). A priori estimates are one of the most powerful and widely used tools in the study of P.D.E.
This approach will be emphasized in our discussion of the existence, uniqueness, and qualitative properties of solutions to various problems. Before proceeding, we will make another important assessment, the boundedness of solutions for positive times. By absolute we mean that the limit does not depend on the data we are dealing with.
We will construct an explicit supersolution z(xt) with which we can compare the approximate solutions un. With those choices, the classical maximum principle implies that un(xt) z(xt) in Q, so in the limit u(xt) z(xt).
Existence of classical solutions
The basic L 1 -estimate
The above proof confirms the uniqueness of problem solutions by a technique (the L1 technique) that is completely different from that of Theorem 4. It is interesting to note that estimate (6.1) not only implies L1 dependence of solutions on data, but also a comparison statement. To conclude this section we give an alternative to steps 2 and 3 of the proof of Theorem 4, x3.
Now that we know that the solutions are dependent on the data in L1(&), we can approximate the general data u0 2 Lm+1(&) by u0n 2 Cc1(&), apply step 1 and go to the limit by Proposition 7 and energy assessment (3.10).
Solutions with data in L 1 (&)
Conversely, a weak solution in the new sense is also a weak solution by denition 1 if and only if fu0 2Lm+1(&). Given this definition, it is also obvious that if u(xt) is a solution with the data u0(x) and > 0, then v(xt) = u(xt;) is a solution corresponding to the data v0(x) = u( x) (this is commonly known as the semigroup property).
Further regularity
Unfortunately, the previous estimates do not allow direct control of the derivative ut appearing in Eq. This can only be done on the weak or distributive form of the inequality, which is obtained by multiplying by the test function '2 Cc1(Q), ' 0, and integrating by parts,. At least since u is the limit of a sequence of functions for which (un)t is locally bounded below uniformly in n, ut is a Radon measure.
To prove that ut is indeed an integrable function, we need to translate the estimate for (u(m+1)=2)t into the estimate for ut. Let K be a subset of Rd with finite measure, let I = t0t1] and assume that v is a function defined in KI that satisfies.
Strong solutions
A comment on continuity
Indeed, we show that there exists a function U which is the largest element in the class of functions that are weak solutions of the Dirichlet problem on Q. Then U is the maximal solution of the PME under Q with zero Dirichlet conditions and =fm is the unique positive solution of the nonlinear eigenvalue problem. i). As a monotonic limit of bounded solutions un in Q such that umn is bounded above by a function in L2(1 : H01(&)), it is clear to conclude that ~U is a strong one.
It is also clear that when applied to our latter sequence fung we get. check the start and limit values). iii). Let us now prove that ~U is larger than any solution of the Cauchy-Dirichlet problem in Q. A study of the properties of weak solutions to the Dirichlet problem was carried out by Aronson and Peletier in AP], who use a denition similar to our Denition 1.
Higher continuity of the solutions of this problem is proved by Gilding and Peletier in GP]. The first proof of the control of ut from below follows the proof of Caarelli and Friedman in CF], while an argument close to the second proof in CVW] was used in studying the regularity of the Cauchy problem. The existence of the special solution (11.1) is established in AP] by another method, which consists in studying the elliptic equation (11.3).
THE CAUCHY PROBLEM. L 1 THEORY
The fundamental estimate for the Cauchy Problem
Perhaps the most important novelty of the Cauchy problem is the existence of a lower bound for the pressure Laplacian. This bound will be used so often that we consider it a fundamental estimate for the Cauchy problem. Note also that (3.1) is optimal in the sense that equality is actually achieved by solutions of the original type or Barenblatt solutions, which are a kind of worst case with respect to this limit, a fact that has interesting implications.
- Boundedness of the solutions. Existence with general data
- Finite speed of propagation. The free boundary
- Local comparison
The Theorem can be derived as a consequence of the fundamental estimate (3.1), thanks to the following result. Therefore, for each x0 2 Rd the function. 2djx;x0j2 are subharmonics in Rd. d denotes the volume of the unit ball). Formula (4.1) not only asserts that solutions with L1data are bounded for positive times, but also gives a very precise quantitative estimate of the bound.
The same techniques can be used to prove a more general version of the smoothing eectProposition 12. We have observed this phenomenon on the source type solutions in the form of compact support of the solution at any time t >0. We are now in a position to establish the same result for a wide class of solutions of the PME that we have.
As a first result on the behavior of interfaces, we can combine estimates (5.1) and (5.4) to obtain the following asymptotic expression. It also gives an estimate of the rate of penetration of matter into the empty region with an exact exponent as a function of radius versus time. It is also possible to obtain the exact asymptotic value of the constant, i.e. we have jxj ct=d with c depending only on md and mass0k1.
The authors point out its optimality by checking it on the Barenblatt solutions and using the estimate to establish the existence of a strong solution of the Cauchy problem with L1 data. Sharp results are due to Vazquez Va2] in d = 1 and CVW] for d > 1, while the long-time behavior of the solution was described by Friedman and Kamin FK] cf. Peletier, Large-time behavior for solutions of the porous medium equation in bounded domains, Jour.
B"enilan, A Strong Regularity Lp for the Solution of the Porous Media Equation, Research Notes in Maths. Wolanski, Lipschitz Continuity of Solutions and Interfaces of the N-Dimensional Porous Media Equation, Indiana Univ. Kenig, Non-Negative Solutions of the Initial Dirichlet problem for generalized porous medium equations in cylinders, Jour.
Vazquez, Asymptotic behavior of the solutions of the porous medium equation with changing sign, SIAM Jour. Czhou, The Cauchy problem and boundary value problems for equations of the unstable ltration type, Izv.