Degenerate diffusion with a drift potential: a viscosity solutions approach

We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of the viscosity solutions theory, we show that the free boundary uniformly converges to the equilibrium as time grows. In the case of a convex potential, an exponential rate of free boundary convergence is obtained.


Introduction
Consider a C 2 function Φ(x) : R n → R, and consider a nonnegative, continuous function ρ 0 (x) : R n → R which has compact support Ω 0 . In this paper we study the porous medium equation with a drift ρ t = (ρ m ) + ∇ · (ρ∇Φ), (1.1) for m > 1, with initial data ρ 0 (x). Note that, at least formally (and proven in [4] for the weak solutions), the solution of (1.1) preserves its L 1 norm.
It will be convenient to change from the density variable to the pressure variable so that the equation becomes (PME-D) u t = (m − 1)u u + |∇u| 2 + ∇u · ∇Φ + (m − 1)u Φ (for more on the density to pressure transform see e.g., the discussions in [5]). We consider continuous and nonnegative solutions in the space-time domain Q = R n × (0, T ) for some T > 0, with prescribed initial conditions u(x, 0) = u 0 (x) ∈ C(R n ).
where u solves (PME). This suggests that the local behavior of (PME-D) is similar to that of (PME), with perturbations due to the inhomogeneity of Φ. We will illustrate this fact in the construction of various barriers in section 2 and 3.
The weak solution theory for (PME-D) in the case of bounded domains has been developed in [4] and [8]. Also, in [11], existence and uniqueness of solutions are established for the full space case under reasonable assumptions (either the initial data is compactly supported or the potential has less than quadratic growth at infinity).
Further, uniform convergence to equilibrium for (PME-D) have also been shown in [4] (see Theorem 3.1). In [12], the connection between the (PME) and the nonlinear Fokker-Planck equation is established, which facilitates the use of the entropy method to derive an explicit L 1 rate of convergence. In [13], an extensive study is made of a general form of the nonlinear Fokker-Planck equation, i.e., ρ t = ∇ · (∇ϕ(ρ) + ρ∇V ) with suitable assumptions on ϕ and exponential L 1 rate of convergence is obtained. (PME-D) falls under the framework of [13], and in fact it is the case that almost all of our results would also go through for a general equation of this form, but for ease of exposition we will restrict attention to (PME-D).
We introduce a notion of viscosity solution for the free boundary problem associated with this equation, which we will show to be equivalent to the usual notion of weak solutionssee [7] for the general theory of viscosity solutions. Note that, formally, the free boundary ∂{u > 0} moves with the outward normal velocity where the first equality is due to the fact that u = 0 on Γ(u). In this regard we closely follow the framework and arguments set out in [6] (see also [14] and [5]), where the viscosity concept is introduced and studied for the Porous Medium Equation.
We point out especially that [5] extends the result of [6] to the case where the diffusion term is multiplied by more general nonlinearities; our focus, however, is on the added drift term, which introduces spatial inhomogeneities. The key utility of the viscosity concept here is that we will be able to describe the pointwise behavior of the free boundary evolution by maximum principle arguments with local barriers. As an application, we are able to extend the results of [4] and [13] to a stronger notion of free boundary convergence.
Main Theorem. There exists a viscosity solution of u (PME-D) with ( m−1 (a) u is unique and coincide with weak solutions studied in [4] and [13].
(b) If |DΦ| > 0 except at x = x 0 where Φ achieves its minimum, then from [4] there exists a unique C 0 > 0 depending only on m 0 such that u uniformly converges to (c) For a strictly convex Φ with k 0 < ∆Φ, there exists K, α > 0 depending on u 0 , k 0 , the C 2 -norm of Φ and n such that Remark. 1. We point out that the free boundary convergence may not hold if |DΦ| vanishes at some points, even though the uniform convergence of the solution still holds.
2. In the case of Φ(x) = |x| 2 (that is for the renormalized (PME)) Lee and Vazquez [16] showed that the interface becomes convex in finite time. It is unknown whether such results hold for general convex potentials: we shall investigate this in an upcoming work.

Viscosity Solution
In this section we introduce the appropriate notion of viscosity solution for (PME-D) and show that it is equivalent to the usual notion of weak solution. Our definition descends from those in [6] and [14]. For more details we also refer the reader to the definitions, discussions and results in [5].

Definition and Basic Properties
Let Q := R n × (0, ∞). For a nonnegative function u(x, t) in Q, we denote the positive phase and the free boundary As in [6], to describe the free boundary behavior using comparison arguments we need an appropriate class of test functions to deal with the degeneracy of (PME-D).
We define a classical free boundary supersolution by replacing ≤ with ≥. Finally, u is a classical free boundary solution if it is both a sub and supersolution.
Before proceeding further it is convenient to introduce some auxiliary definitions.
Definition 2.2. Let ϕ be a continuous, nonnegative function. Now if ψ is another such function, then we say that ϕ touches ψ from above at ( We have a similar definition for ϕ touching ψ from below.
We note that e.g., due to the maximum principle, a classical free boundary subsolution that lies below a classical free-boundary supersolution at time t 1 ≥ 0 cannot cross the supersolution from below at a later time t 2 > t 1 . This observation leads to a notion of viscosity solution which takes into account the free boundary: Definition 2.4. Let u be a continuous, nonnegative function in Q.
• u is a viscosity subsolution of (PME-D) if, for any given smooth domain Σ given in (2.1), for every ϕ ∈ C 2,1 (Σ) that touches u from above at the point (x 0 , t 0 ), we have • u is a viscosity supersolution of (PME-D) if, for any given smooth domain Σ given in (2.1), (i) for every ϕ ∈ C 2,1 (Σ) that touches u from below at the point (x 0 , t 0 ) ∈ Ω(u) ∩ Σ, we have ϕ t ≥ αϕ ϕ + β|∇ϕ| 2 + ∇ · (ϕ∇Φ) (2.3) (ii) for every classical free-boundary subsolution ϕ in Σ, the following is true: If ϕ ≺ u on the parabolic boundary of Σ, then ϕ ≤ u in Σ. That is, every classical free-boundary subsolution that lies below u at a time t 1 ≥ 0 cannot cross u at a later time t 2 > t 1 .
• u is a viscosity solution of (PME-D) with initial data u 0 if u is both a super-and subsolution and u uniformly converges to u 0 as t → 0.
Remark 2.5. In general one can define viscosity sub-and supersolutions respectively as upper-and lower semicontinuous functions. Such a definition turns out to be useful when one cannot verify continuity of solutions obtained via various limits. This problem does not arise in our investigation here thanks to [4], and our definition assumes continuity of solutions.
It is fairly straightforward to verify that a classical free boundary sub-(super)solution is also a viscosity sub-(super)solution.
Lemma 2.6. If w is a classical free boundary sub-(super) solution to (PME-D), then w is also a viscosity sub-(super) solution.
Proof. We will be brief: The subsolution case presents no difficulty since if contact with some ϕ ∈ C 2,1 (Σ) occurs in Ω(w) then we use the fact that w is classical there, whereas no contact can occur on the free boundary unless |∇w| = 0, in which case the differential inequality is satisfied since then ϕ = |∇ϕ| = 0 and ϕ t ≤ 0.
If w is a classical free boundary supersolution, then (i) in Definition 2.4 follows as before.
To see (ii), let us note that if ϕ is a classical free boundary subsolution which crosses w, then since the free boundary is C 2 , Hopf's Lemma implies that at the touching point |∇ϕ| < |∇w| (see e.g., [17]). On the other hand, since ϕ started below w, at the touching point we must have v n (ϕ) ≥ v n (w), which leads to a contradiction since it is also the case that we have ∇w |∇w| = ∇ϕ |∇ϕ| .
Next we have the following stability result.
Lemma 2.7. Let u ε be a smooth solution of (PME-D) with initial data u 0 + ε and let u be its uniform limit. Then u is a viscosity solution of (PME-D) with initial data u 0 .
1. Let us first show that u is a subsolution. First suppose that ϕ touches u from above at the point (x 0 , t 0 ). We may assume that u − ϕ has a strict maximum at (x 0 , t 0 ) in if necessary. By uniform convergence there exists a sequence (x ε , t ε ) converging to (x 0 , t 0 ) such that u ε − ϕ has a local maximum at (x ε , t ε ). Now if we we let Then u ε −φ has a local maximum at (x ε , t ε ) with (u ε −φ)(x ε , t ε ) = 0. We can now conclude by taking the limit of the viscosity subsolution property of u ε .
2. Next we show that u is a supersolution. Let ϕ be a classical free-boundary subsolution such that ϕ(x, t 1 ) ≺ u(x, t 1 ). Since the u ε 's are strictly ordered, u < u ε and hence ϕ(x, t 1 ) ≺ u(x, t 1 ) < u ε (x, t 1 ). Now suppose ϕ touches u ε at some point (x 2 , t 2 ), then ϕ(x 2 , t 2 ) > 0 since u ε is positive, so by continuity, there is a parabolic neighborhood of (x 2 , t 2 ) in which both functions are classical and positive. By the Strong Maximum Principle, the touching cannot have occurred at (x 2 , t 2 ), a contradiction. We conclude that ϕ < u ε so that in the limit ϕ ≤ u ε .
An immediate consequence of above lemma is that weak solutions are viscosity solutions (see Corollary 2.12). We shall introduce the precise notion of weak solutions in the next subsection, and summarize some results from [4].

Weak Solutions
To be consistent with the setup in both [4] and [13], let us return to the density variable and consider the solution of (1.1) in a bounded domain Ω with Neumann boundary condition: We will see shortly that we need not worry about the fact that we are on a bounded domain, but for now we will let Q = Ω × R + and Q t = Ω × (0, t]. Following [4], We also define a weak subsolution (respectively supersolution) by (i) and (ii) with equality replaced by ≤ (respectively ≥).
¿From [4] we have existence, regularity, uniqueness and comparison principle for weak solutions: The problem (N) has a unique solution; (b) The solution is uniformly bounded in Q and is continuous in The existence of solutions is obtained as the uniform limit of solutions to uniformly problems (equicontinuity is obtained from [8]). For our purposes, a very simple approximation basically suffices and we summarize the relevant result in the following: Lemma 2.10 (From [4]). Let u ε be a solution of (N) with initial data u ε 0 = u 0 + ε, then u ε is equicontinuous and there exists a subsequence which uniformly converge to u which is the unique weak solution to (N) with initial data u 0 .
While a priori our viscosity solution is defined in all of R n , since (formally at least) solutions of (PME-D) should have finite speed of propagation, the boundary conditions should be inconsequential if we take Ω sufficiently large. (Later we will also establish finite propagation for viscosity solutions -see Corollary 2.17.) Control on the speed of expansion of the support can be done via comparison with any (weak) supersolution. In particular, when Φ is monotone (that is, when |DΦ| > 0 except at one point where Φ achieves its minimum), we can use the stationary profiles of the form Ψ(x) = (C − Φ) + with sufficiently large C as a supersolution (see Theorem 3.1).
Remark 2.11. Alternatively (and perhaps this is a cleaner line of reasoning), we can directly use the result of [11] on existence and uniqueness of solutions in all of R n , which implies in particular that the results of [13] also apply in that setting.
Combining Lemma 2.7 with Lemma 2.10 and the uniqueness statement in Theorem 2.9, we obtain: Corollary 2.12. Any weak solution is also a viscosity solution.
We will eventually establish uniqueness of viscosity solutions via maximum-principle type arguments, which culminates in the identification of the two notions of solution.

Construction of test functions
In this subsection we collect some test functions, i.e., (classical free boundary) sub-(super) solutions, to (PME-D) which will be useful for comparison purposes. In the first couple of lemmas (Lemmas 2.15 and 2.19), the idea is to control the Φ dependence via Taylor expansion in a small neighborhood of a point, so that we can appropriately perturb the test functions for (PME) constructed in [5] and [6] for our purposes.
The starting point is to observe that if we consider (PME-D) in some small cylinder Next proposition illustrates the necessary perturbation one needs to perform on solutions of (PME) to arrive at (2.5).
Proposition 2.13. Let u(x, t) be a viscosity subsolution of (PME) in B 1+α (0) × [−1, 1]. Then for 0 < α < 1, is a subsolution of (PME-sub) is a supersolution of (PME-super) Proof. We only show the supersolution part. Let u 2 be as given above. Suppose then that ϕ is classical and touches u 2 from below at some point (x 0 , t 0 ). We first note that there exists ( Next for any unit vectorb, let us consider Then we note that 1)φ(x 1 , t 1 ) = ϕ(x 0 , t 0 ) and so (u−φ)(x 1 , t 1 ) = 0 and 2) by the definition of u 2 as an infimum and by continuity of ϕ, in a small parabolic neighborhood of (x 0 , t 0 ), it is the case that u −φ ≥ u 2 −φ ≥ 0; we therefore conclude thatφ touches u from below at (x 1 , t 1 ) and so we have, Now the desired inequality is achieved by settingb = ∇ϕ |∇ϕ| (x 0 , t 0 ).
Indeed the above calculation shows that if u is a viscosity supersolution of (PME), then u 2 should be a viscosity supersolution of (PME-super): If a classical free boundary subsolution ϕ of (PME-super) crosses u 2 from below, then the correspondingφ is a subsolution of (PME) and crosses u, yielding a contradiction (there is no distinction between the interior and boundary cases).
[CV] Consider the function Then u is a classical free boundary supersolution of (PME) in the Proposition 2.13 and Lemma 2.14 yields the following: Corollary 2.15. Let us fix x 0 ∈ R n and let H be given as in Lemma 2.14. Then the inf convolution of H, given as is a classical (free boundary) supersolution of (2.5). Consequently, there exists C = C 0 which only depends on the is a classical (free boundary) supersolution of (PME-D) in Proof. By Lemma 2.6 and Proposition 2.13, H is a viscosity supersolution of (PME-super), so it is sufficient to show that it has the required regularity. For this, note that for simplicity we have only taken the supremum over space and the reader can readily check that in this case the infimum for H(x, t; α) is achieved at the point y which minimizes |y| subject to the constraint that |y − x| = α − αt, and thus an explicit expression is possible for H.
Remark 2.16. In fact due to the explicit form of H it follows that the free boundary velocity ofH is given by By comparison with these supersolutions, we immediately obtain with constants λ, K, C, τ > 0 such that Then B(x, t; τ, C) is a classical (free boundary) solution of (PME).
Using Proposition 2.13 (see also the proof of Corollary 2.15) once again, we obtain the following: Lemma 2.19. Let us fix x 0 ∈ R n and let B be a Barenblatt function. Then there exists C which only depends on the is a classical (free boundary) subsolution of (PME-D) in Remark 2.20. The reason for taking the hyperbolic scaling is because we will have occasion to require rather fine control on the boundary velocity (see Lemma 2.24) and this is the scaling which preserves the velocity -in contrast to the parabolic scaling, which dramatically reduces the effect of the drift Φ in the bulk (the positivity set), but unfortunately at the cost of severely disrupting the boundary velocity.
To establish the Comparison Principle, we will need the following weak analogue of (ii) in the definition of viscosity supersolutions for subsolutions, the proof of which utilizes an approximation lemma from [5].
Lemma 2.21. Let u be a viscosity subsolution of (P M E-D), and let ϕ be a classical free boundary supersolution from Lemma 2.15 which lies above u at some time t 0 . Then ϕ cannot cross u from above at a later time t > t 0 .
Proof. Let ϕ and u be as described in the statement, and suppose that ϕ touches u from above at some point (x 0 , t 0 ). From Lemma 2.15, we have that ϕ is given as the inf convolution of some spherical traveling waves from Lemma 2.14, which we denote ψ. Further, let us suppose the infimum is achieve at (x 1 , t 1 ) so that 1) ϕ(x 0 , t 0 ) = ψ(x 1 , t 1 ) and 2) by the definition of ϕ as an inf convolution, the translated functioñ also touches u from above at the point (x 0 , t 0 ). From Lemma 4.4 in [5], we know that ψ can be given as the monotone limit of classical positive supersolutions, and hence the same is true ofψ: I.e., there exsits ψ ε ψ , with ψ ε > 0 classical. But since u cannot touch ψ ε by the Strong Maximum Principle, we obtain in the limit that u ≤ψ, which is a contradiction.

Comparison Principle and Identification with Weak Solution
Here the outline of the proof closely follows that of the corresponding result for (PME) (Theorem 2.1 in [6]): We will give an abridged version of the proof, pointing out main steps and modifications for our problem. First note that W and Z preserve properties of v and u: • W is a supersolution and Z is a subsolution; • Z(·, r) ≺ W (·, r) for r sufficiently small. The proof of this can be done by comparison with the supersolutions (respectively subsolutions) constructed in Lemma 2.15 (respectively Lemma 2.19). We omit the details since with replacement of barriers it is no different from the proof of Proposition 6.2 in [5].
Thus if we can prove that W stays above Z for all choices of r and δ(sufficiently small), then we may take δ → 0 and then r → 0 to recover the conclusion for u and v. First let us note that due to the Strong Maximum Principle, W cannot touch Z from above, and therefore we are reduced to the analysis of a first contact point of W and Z at some P 0 = (A, t 0 ).
The key usefulness of Z and W lies in the fact that they enjoy interior/exterior ball properties: • The positivity set of Z has the interior ball property with radius r at every point of its boundary and at the points of the boundary of the support of u where these balls are centered we have an exterior ball; • The positivity set of W has the exterior ball property with radius less than r −δt (since in this case we really have an exterior ellipsoid in space-time) and at the points of the boundary of the support of v where these balls are centered we have an interior ball.
For detailed proofs of these statements we again refer the reader to [5]. 2) [The Contact Point] The first contact point P 0 = (x 0 , t 0 ) is located at the free boundary of both functions. Therefore by the definitions of Z and W , there is a point P 1 = (x 1 , t 1 ) on the free boundary of u located at distance r from P 0 and there is another point P 2 = (x 2 , t 2 ) on the free boundary of v at distance r 0 = r − δt 0 from P 0 . Let us also denote by H Z (respectively H W ) the tangent hyperplane to the free boundary of Z (respectively W ) at P 0 . (see Figure 1) Lemma 2.23. Neither H Z nor H W is horizontal. In particular, one can denote the spacetime normal vector to H Z , in the direction of P 1 − P 0 , as (e n , m) ∈ R n × R where |e n | = 1 and −∞ < m < ∞.
Proof. It is enough to show that t 1 > t 0 −r (i.e., Γ(Z) cannot propagate with infinite speed) and t 2 < t 0 + r (i.e., Γ(W ) cannot propagate with negative infinite speed). The desired conclusion then follows by the ordering of Z and W .
We first show that t 2 < t 0 + r. Otherwise H W is horizontal and after translation we have P 0 = (0, −r) and P 2 = (0, 0). Moreover Ω(v) has an interior ball at P 2 with horizontal tangency with radius 0 < r < r. Now in any parabolic cylinder On the lateral boundary of C η it may be the case that v = 0, so we will have to compare with a subsolution with support strictly contained in −η < |x| < η at time t = −η 2 and still contains 0 in its support at time t = 0, which rigorously implies that v cannot contract sufficiently fast for (0, 0) to be a free boundary point. The necessary subsolution can be constructed as the one in Lemma 2.19, adjusted for the parabolic scaling.
3) [Non-tangential Estimate] The next lemma states that the normal velocity V of Γ(Z) at (x 0 , t 0 ) satisfies, in the viscosity sense, Proof. The argument is parallel to the proof of Lemma 4.3. in [6]; the only difference for us is taking into account the change of reference frame introduced by the drift given by Φ. This is ensured by the local nature of the construction of our barrier in Lemma 2.15, which replaces the corresponding barriers used in [6].

4) [Conclusion]
Due to Lemma 2.24, we may place a small subsolution ϕ from Lemma 2.19 below Z at P 0 with speed close to m (again see Remark 2.20, which assures us that our subsolutions are constructed so that this is possible) such that it crosses anything with speed m < m. Since ϕ is also below W and hence v (after a small translation), v must expand by at least m , but then Γ(W ) has speed m + δ > m at P 0 , yielding a contradiction to the fact that Z touched W from below at P 0 .
We can now establish uniqueness of viscosity solutions: Proof. The existence of a continuous weak solution can be provided as the uniform limit of classical solutions with initial data u 0,ε = u 0 + ε, and by Lemma 2.7, such a limit, which we will denote by U , is also a continuous viscosity solution. Further, by comparison with u 0,ε and taking a limit, it is clear that such a limit U is also a maximal viscosity solution.
Uniqueness would follow if we can show that any other viscosity solution u also cannot be smaller than U . For this purpose, consider u n (x, t) with initial data u n (x, 0) := (u 0 − 1 n ) + . Now consider positive u εn n such that |u εn n − u n | < 1 n in R n × [0, T ]. It follows from Lemma 2.10 that u εn n uniformly converges to U 2 (x, t), which is then a continuous weak solution of (PME-D). Therefore, by uniqueness of weak solutions, U 2 is equal to U . On the other hand by Theorem 2.22 u n ≺ u and thus U = U 2 ≤ u. Hence we conclude.
Using Theorem 2.22 and Theorem 2.25, we can in fact prove a stronger comparison theorem for viscosity solutions (see [5], Theorem 10.2): Theorem 2.26. Let u 1 and u 2 be respectively a viscosity subsolution and a viscosity supersolution of (PME-D) in some parabolic cylinder Q with initial data u 0,1 and u 0,2 such that u 0,1 (x) ≤ u 0,2 . Then u 1 (x, t) ≤ u 2 (x, t) for all (x, t) ∈ Q.
This comparison theorem in particular allows us to restrict attention to only the (classical free boundary) supersolutions used to establish Theorem 2.22, and consequently, we can now strength Lemma 2.21 to enable comparison with any classical free boundary supersolution: Lemma 2.27. Let u be a viscosity subsolution of (PME-D), and let ϕ be a classical free boundary supersolution which lies above u at some time t 0 . Then ϕ cannot cross u from above at a later time t > t 0 .
Proof. Let us first replace ϕ by a viscosity solution of (PME-D) with the same initial data which we denote v. Since ϕ is a viscosity supersolution by Lemma 2.6, we have that ϕ ≥ v by Theorem 2.26. Finally, u ≤ v by Theorem 2.22.

Convergence to Equilibrium
We begin by discussing the set of equilibrium solutions to (PME-D) and reviewing some known results. Since by Theorem 2.25, the unique viscosity solution coincides with the continuous weak solution, we may carry out our discussion in the context of weak solutions.
The set of equilibrium solutions and uniform convergence of solutions to the equilibrium are established in [4]. Below we state the corresponding result in the pressure variable. or W + Φ = C for some constant C in a neighborhood of x} Further, given u 0 , there exists a unique W (x) ∈ S such that u(x, t) uniformly converges to W as t → ∞.
It is fairly immediate that S is contained in the set of equilibrium solutions; the converse containment and the convergence statement are established based on a L 1 contraction result in [4].
Under the assumption that Φ is convex. Note that the density function ρ(·, t) given in (1.2) preserves its L 1 norm over time. Therefore there is a unique equilibrium solution u ∞ = (C 0 − Φ) + to which u(·, t) uniformly converges as t → ∞, i.e., the one with An explicit exponential rate of convergence is then derived in [13] by the entropy method: Theorem 3.2. Suppose that Φ is strictly convex, i.e., there exists a constant k 0 > 0 such that x T · [(Hess Φ(x))x] ≥ k 0 |x| 2 for x ∈ R n . Let u ∞ = (C 0 − Φ) + be the equilibrium solution to which our solution u(x, t) converges as t → ∞. Then there exist constants K, α > 0 depending on m, k 0 and the L 1 norm of u 0 such that Remark 3.3. In fact the estimate in [13] is given in terms of the pressure variable ρ. Due to Corollary 2.17, for a convex (and in fact monotone) potential Φ u is uniformly bounded with its support contained in a compact set for all times. This allows us to derive the estimate for u from that of ρ for 1 < m < ∞. Further, due to the equivalence of all L p norms in our setting, we will take some liberties in passing between u and ρ in our estimates.

Convex potential
As an application of the viscosity solutions theory, we will convert the L 1 estimate in Theorem 3.2 into a pointwise estimate (see Lemmas 3.4 and 3.5); such an estimate in turn will yield a quantitative estimate on the rate of the free boundary convergence (see Theorem 3.6).
Rescaling u by cu(x, ct) if necessary, let us assume for the rest of this subsection that u ≤ 1 and max ∆Φ ≤ 1 on our domain of consideration, which is bounded (see Corollary 2.17) and we assume to be {|x| ≤ R} for some R > 0. Suppose, for (x 0 , t 0 ) ∈ R n × (0, ∞) and for 0 < a < 1, Then u(·, t 0 + a) ≥ a k in B a (x 0 ).
Thenũ is a supersolution of where C is a constant depending on the C 2 -norm of Φ (near x 0 ). Below we will construct a subsolution of (3.1) to compare withũ in order to establish the lemma.
2. Let us consider w(x, t) which satisfies in the weak sense (see e.g., [9]), with initial condition w(x, 0) : Then, say for t ≥ 1/2, w(·, t) is Hölder continuous due to [9]. Since u is bounded by 1, so is w(x, 0), by the Maximum Principle. Further, note that any solution of the (PME) is now automatically a supersolution of (3.2) and therefore, using an appropriate Barenblatt profile as a supersolution of (3.2), one can check that Moreover, integration by parts yields that and in particular we deduce that e.g., w(x, 1/2)dx ≥ a k /2. Let x * be the point where w(·, 1/2) assumes its maximum, then from (3.3) we see that w(x * , 1/2) ≥ C n a k for some dimensional constant C n . Due to the Hölder regularity of w, where k 2 = γ −1 k where 0 < γ < 1 depends only on m and n. is the Barenblatt profile given in Lemma 2.18, withm as the permeability constant and the conditions Cτ 2λ−1 = a k /4 (height), and C/Kτ λ ≤ a k 2 /2 (the size of initial support).
Next lemma establishes a uniform upper bound on u in terms of its L 1 norm. Here we would address our equation in pressure form, i.e., (1.1), to invoke regularity theory for divergence form operators studied in [15]. ρ(·, t)dx ≤ c 0 for t 1 ≤ t ≤ t 2 := t 1 + log(1/c 0 ).
then ρ(·, t 2 ) ≤ Cc Proof. 1. We proceed by induction. Suppose that u ≤ a = c where k > 0 is chosen such that 4Cc with C to be determined later. Our goal is to show that Then the desired result is obtained by iteration, beginning with a = 1 and continuing until a reaches the lower bound in (3.4). Note that the total number of iteration for this process, therefore the total time we need for the desired result, is of order log(1/c 0 ).
Due to (3.4) and the Hölder regularity ofρ 1 (see [15]) we havẽ if we choose C > 0 sufficiently large in (3.4) corresponding to the Hölder regularity for solutions of (P) for t ≥ 1.
As forρ 2 , arguments with test functions in the weak formulation of (P) (not very different from the case of the heat equation) in combination with the Hölder regularity estimates yield that, for sufficiently large C > 0, with a dimensional constant C. In particularρ 2 (x, 1) ≤ 1/4 in B C/a−C/4 (0).
Due to our assumption on ρ 0 , one can go through the above argument starting at any time 5. Repeating the above argument with a 2 := a/2 starting at t = 2, we get If we iterate up to of order log c 0 times, then a reaches the lower bound in (3.4), and we arrive at the desired result.
Next we use the uniform bounds obtained above to investigate the rate of free boundary convergence.
Proof. 0. We first show that Γ t (u) is close to the equilibrium profile from the inside, i.e., After t = T ≥ C ln a, L 1 -average of u in B a (x 0 ) is bigger than Ca k when the center x 0 is in Ω a := {Φ ≤ C 0 − a k }. Hence Lemma 3.4 yields that u(x 0 , T ) ≥ C 2 a k with T = a k/2 . As a result we conclude that Γ(u) lies outside of Ω a after t = C ln a.
It remains to show that Γ t (u) is also close to the equilibrium profile from the outside. This will be more involved.
We want to show that T * = ∞. Let us choose x 0 ∈ ∂D a , where a = Ke −α 2 T * . Since Φ is convex with |DΦ| > 0 in D a , there is an exterior ball B 1 (x 1 ) outside of D a such that x 0 ∈ ∂B 1 (x 1 ). Due to (3.6) and due to the fact T * ≥ T 0 , we have forã := 1/K, Let us first make a heuristic argument. Suppose that x 0 ∈ Γ T * (u) (by the definition of T * such an x 0 exists). Then at x 0 we have DΦ(x 0 , T * ) pointing in the direction of x 1 − x 0 , which is the inward normal of Γ(u) at (x 0 , T * ), which is parallel to − Du |Du| (x 0 , T * ). Hence formally the normal velocity of Γ(u) at (x 0 , T * ) should be (see Figure 3) The rest of the proof consists of a barrier argument to establish an appropriate version of (3.8). if K > 2C/A. Since u ≤H in Σ ∩ [T * , T * + ε), we conclude that u(·, T * + ε) = 0 in B Aε/2 (x 0 ).

Consider
Since x 0 ∈ ∂D a is chosen arbitrarily, we conclude that u(·, T * + ε) vanishes in Aε/2neighborhood of D a , which includes D a(T +ε) if K > 2/A for any ε > 0.
This contradicts the definition of T * .
We are not able to yield quantitative estimates on the rate of free boundary convergence, due to the lack of available L 1 -estimates. However we state the theorem below to illustrate that if we neglect the rate of convergence, then considerably simpler arguments already yield the free boundary convergence of u as t → ∞, Proof. 0. We first verify that for any compact subset K of Ω(u ∞ ) there exists T > 0 such that Γ t (u) lies outside of K for t ≥ T . This is immediate from the uniform convergence of u to u ∞ .