THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: REGULARITY AND LONG TIME BEHAVIOR OF WEAK SOLUTIONS A.L. BERTOZZI AND M. PUGH.

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1 THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: REGULARITY AND LONG TIME BEHAVIOR OF WEAK SOLUTIONS A.L. BERTOI AND M. PUGH April 1994 Abstract. We consider the fourth order degenerate diusion equation h t =?r (f(h)rh) in one space dimension. This equation, derived from a `lubrication approximation', models surface tension dominated motion of thin viscous lms and spreading droplets [14]. The equation with f(h) = jhj also models a thin neck of uid in the Hele-Shaw cell [9, 10, ]. In such problems h(x; t) is the local thickness of the the lm or neck. This paper will consider the properties of weak solutions which are more relevant to the droplet problem than to Hele-Shaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets. We consider two problems: The rst has initial data h 0 0 and f(h) = jhj n, 0 < n < 3. We show that there exists a weak nonnegative solution for all time and that this solution becomes a strong positive solution after some nite time T and asymptotically approaches its mean as t! 1. The weak solution is in a classical sense of distributions for 3 8 < n < 3 and in a weaker sense introduced in [1] for the remaining 0 < n 3 8. Furthermore, the solutions have suciently high regularity to just include the unique `source type' solutions [] with zero slope at the edge of the support. They do not include any of the less regular solutions with positive slope at the edge of the support. Secondly we consider strictly positive initial data h 0 m > 0 and f(h) = jhj n, 0 < n < 1. For this problem we show that even if a nite time singularity does occur of the form h! 0, there exists a weak nonnegative solution for all time t and that this weak solution becomes strong and positive again after some critical time T. As in the rst problem, we show that the solution approaches its mean as t! 1. The main technical idea is to introduce new classes of dissipative entropies to prove the existence and higher regularity. We show that these entropies are related to norms of the dierence between the solution and its mean to prove the relaxation result. 1

2 A.L. BERTOI AND M. PUGH (1) 1. Introduction We consider weak solutions of the fourth order degenerate diusion equation h t =?r (f(h)rh) in one space dimension. We consider the case where f(h) = jhj p, p > 0; the analysis can be directly extended to the case where f(h) is a sum of such terms. This equation, derived from a `lubrication approximation', models surface tension dominated motion of thin viscous lms and spreading droplets [14]. The equation with f(h) = jhj also models a thin neck of uid in the Hele-Shaw cell [9, 10, ]. In such problems h is the thickness of the the lm or neck. This paper considers weak solutions that are zero on a set of non-zero measure, hence are much more relevant to the droplet problem than to Hele-Shaw. We briey compare this fourth order problem to the well known second order degenerate diusion equation, () u t = (u m ); the `porous media' equation [0]. Some similarities between the fourth and second order cases are that both equations are parabolic and in divergence form with a `sub-diusive' nonlinear diusion coecient. Furthermore, both equations have `weak' solutions that are nonnegative. At any deeper level, the similarities between the second and fourth order problems cease to exist. One striking dierence is the lack of a maximum principle for the fourth order problem. In particular, analytical results for the lubrication approximation are not due to a maximum principle but due to the dissipation of nonlinear `entropies' present in these problems. Furthermore, the question of whether initially positive solutions can develop nite time singularities of the form h! 0 has been the subject of recent and ongoing study [6, 4]. The maximum principle prohibits such behavior in the second order case Elementary properties and exact solutions. For simplicity, we consider the 1-D problem with periodic boundary conditions. These boundary conditions have the physical interpretation of modeling a periodic array of spreading droplets. 1 The equation (3) h t + (f(h)h xxx ) x = 0 is derived from a conservation law. We now state some elementary properties that follow from integration by parts for strong solutions. We use these and other properties of strong solutions to prove results for weak solutions. The rst property is conservation of mass, h(x; t)dx = h 0 (x)dx: 1 One application of our theorems is a model for the spreading and eventual merging of the droplets to form a lm of uniform thickness. We explore this application in greater detail in a companion paper [5]

3 THE LUBRICATION APPROXIMATION FOR THIN FILMS 3 Second, we have dissipation of surface tension energy, 1 (4) jh x (x; T )j dx + f(h)h xxx dxdt = jh x (x; 0)j dx: [0;T ] In addition, we have the basic entropy dissipation: consider a function G(y) satisfying G 00 (y) = 1=f(y). The convexity of G and mass conservation allow us to choose G so that R S1 G(h(x; t)) dx 0 for all t. Integration by parts yields (5) G(h(x; T )) dx + h S QT xx dxdt = G(h(x; 0)) dx: 1 For n = 0, the linear problem, the entropy is merely the L norm. Bernis and Friedman rst introduced these entropies in [1]. The equation (6) h t + (jhj n h xxx ) x = 0 possesses a number of interesting exact solutions. Compactly supported nonnegative `source type' solutions exist for all 0 < n < 3 []. They have the scaling form (7) Where H() solves the ODE (8) h(x; t) = t? H(); = xt? = 1=(n + 4): H n H = H: For a given n and mass, there is more than one compactly supported symmetric solution to the ODE. However, if we impose the additional constraint that the solution have H = 0 at the edge of the support, we obtain a unique solution. This fact was proven in []. They also proved that these \most regular" solutions have the following behavior at the edge of their support: Let [?a; a] denote the support of H(). Then (9) (10) (11) for 0 < n < 3=; H() (a? ) as " a; for n = 3= H() (a? ) log(1=(a? )) =3 as " a; for 3= < n < 3 H() (a? ) 3=n as " a: The less regular solutions have H() (a? ). We note that the existence result we prove in Theorem.1 is for solutions in a regularity class that includes the source type solutions (9{11) and excludes the less regular ones. In a companion paper [5] we present numerical simulations of the weak solutions which suggests rapid convergence onto the above unique source type solutions. Starov [1] rst noted that there are no nite mass `source-type' solutions for n = 3. Brenner and Bertozzi [8] addressed the signicance of this fact for the physical problem of spreading droplets. The n = 3 case arises when there is a no-slip boundary condition at the liquid solid interface. The lack of such scaling solutions is consistent with the fact

4 4 A.L. BERTOI AND M. PUGH that a no-slip boundary condition leads to innite energy dissipation at the contact line for spreading drops with a nite contact angle [1, 17]. The non-existence of source type solutions for n 3 is due to the structure of the ODE (8) and is in sharp contrast to the source type solutions for the porous media equation () which exist for all m > 1. There are also traveling wave solutions of the form h(x; t) = H(x? ct) as described in [7]. Again, we see transitions in the allowable behavior at dierent values of n. We omit many details but it is noteworthy that there are no advancing front solutions for n 3 and that for n < there are compactly supported traveling wave solutions (we believe these are unphysical). Furthermore, for 3= < n < 3 there are advancing front solutions with the simple form 8 < A(x? ct) 3=n x > ct h(x; t) = c = ( 3 : 0 otherwise; n? )( 3 n? 1) 3 n An : Finally we remark that there are exact steady solutions for all n 8 < A? Bx ; jxj < A= p B (1) h(x; t) = : 0 otherwise: Bernis and Friedman [1] rst addressed the existence theory for `weak' nonnegative solutions. For simplicity they considered a bounded domain with boundary conditions h x = h xxx = 0 at both endpoints. In particular, they introduced the basic entropies (5) to prove positivity of solutions for n 4 and nonnegativity of weak solutions for all n 1. They considered two notions of `weak solution', the rst a very weak denition (see (0)) in which the integral of the ux term, f(h)h xxx, is only over the set where the solution h is positive. For this denition their methods directly prove existence of nonnegative solutions for n > 0. The only information needed for this very weak denition is the surface tension dissipation (4). They also proved existence of weak solutions in the sense of distributions (18) for nonnegative initial data for 1 < n < and for positive (or entropy bounded) data for n 1. They construct the weak solutions as the limit of smooth approximate solutions. The rst result of this paper extends the existence results in [1] to prove existence results in a sense of distributions for 3 < n < 3. We use several dierent formulations of distribution 8 solution ((18),(19) and modications of these). For these distribution solutions and weaker nonnegative solutions in the range 0 < n 3 we obtain greater regularity for the solution 8 than obtained in [1]. In particular the regularity class includes the unique most regular source type solutions (9{11) but not the less regular ones or the steady parabola solutions (1). The second result of this paper concerns the long time behavior of our weak solutions as t! 1. In particular, we show that they approach their mean in the limit as t! 1. Since the convergence is in the L 1 norm, there exists a time T after which the nonnegative weak solution becomes a positive strong solution. If we view the solution as describing a periodic

5 THE LUBRICATION APPROXIMATION FOR THIN FILMS 5 array of droplets, this implies that the drops will spread and merge to form a uniform layer in the limit as t! 1. The third result concerns the case of all n > 0 with positive (or entropy bounded) initial data. For this case we prove that even if the solution develops a singularity in nite time, it can be continued past this time as a nonnegative weak solution that approaches its mean in the innite time limit. In particular, there exists a critical time T after which the solution is guaranteed to be positive and strong again. The main technical tools are a new class of convex entropies which yield more rened existence results. Moreover, we show that for 0 < n < the basic entropies (5) are related to the L norm of h? c enabling us to prove a long time relaxation result. For the case n < 3 we relate the new entropies to L norms of nonlinear functions of h which again allow us to prove the relaxation result. In the case of positive initial data, we obtain this result for all n > 0. We use a regularization introduced in [1]. In a subsequent paper [5] we use the same regularization to numerically simulate the weak solutions. The simulations indicate that the support of the solution has nite speed of propagation and continuous ux, two properties desirable for a physically correct model. Moreover they show rapid convergence of the solution onto the similarity solutions (9{11) before the merging of support. The paper is organized as follows. Section presents the statements of the theorems for nonnegative initial data. Section 3 reviews the properties of the regularization scheme. Section 4 proves the existence results for nonnegative initial data for 0 < n < 3. Section 5 proves the long time results for nonnegative data for 0 < n < 3. Section 6 proves the existence and long time results for strictly positive initial data for all n > 0. Section 7 briey discusses unsolved problems and the ramications of our results. The appendix contains the proof of an interpolation lemma and the uniform convergence of the regularized diusion coecient and its derivatives used in Section 4. Before proceeding further, we discuss briey the physical problem of thin lms and contact lines. 1.. Thin Films, Contact Lines and the Lubrication Approximation. The lubrication approximation for a thin lm of liquid on a solid surface yields a fourth order degenerate diusion equation for the lm height [14]. In one space dimension, it is (13) h t + ((jhj 3 + b p jhj p )h xxx ) x = 0 where the parameter b p xes a `slip' length [15] and the specic slip model determines the power p. The derivation uses the Stokes equation for steady viscous ow combined with a depth averaging of the uid velocity in the direction perpendicular to the surface. A slip boundary condition on the liquid solid interface, = v; (h) hp? ;

6 6 A.L. BERTOI AND M. PUGH determines the power of p in the equation above [14]. For the purposes of this paper we consider p in the range 0 < p < 3. We remark that the paper [15] considers p = 0; 1; and. The case p = 1 models a thin lm spreading on a slightly porous surface [19]. 3 In this porous case, the parameter 0 < b 1 determines a microscopic length scale correlated with the porosity of the surface. The case p = is also a well known `slip model' (see e.g. [13]). Dussan [11] suggested that the choice of slip model is far less important that the slip length for the macroscopic dynamics of the drop. Nonetheless correct use of the various slip models requires a good understanding of the mathematics. The boundary of the support of the weak solution physically corresponds to the contact line 4 in the spreading droplet problem. In addition to a `slip' law (14) Greenspan [14] suggested the need for a `constitutive law' for the motion of the edge of the drop. Recent studies use such a relation [13, 15, 16]. The constitutive laws are motivated by the order of the equation (hence a need for an extra boundary condition) and are based on experimental observation. It is unclear that the problem remains well-posed or even solvable when constitutive laws are imposed at the edge of the support. We propose an alternative to constitutive laws in the case of complete wetting. We view this as a free boundary problem in which the motion of the boundary is determined by global properties of the solution, as in the porous media equation. Rather than enforcing a specic constitutive law, we advocate letting a regulatization scheme pick out a solution. This has the advantage of guaranteeing a true solution to the problem. An obvious concern is that, since we have no proof of uniqueness, the problem may not be well posed and other regularization schemes might converge to other solutions. We conjecture that within a higher regularity class well-posedness results exist for this problem. It is interesting to note that this regularity implies a zero local contact angle for the weak nonnegative solutions with 0 < n < 3. We address these issues in more detail in [5]. When h is small, the slip term in (13) dominates the behavior of the solution. Hence the results for the case f(h) = jhj n apply to the equation (13) with p = n.. Evolution From Nonnegative Initial Data We consider solutions of the equation (15) h t + (jhj n h xxx ) x = 0 However, the case p = 0 is the nondegenerate linear case which does not preserve positivity of the solution. Hence, we do not consider this a viable model. 3 We also remark that the 1-D equation with f(h) = jhj is models thin necks in the Hele-Shaw cell [9, 10, ] However, our the weak solutions do not make sense physically for that problem. One simple consequence of the entropy argument here which does apply to Hele-Shaw is that given positive initial data, if no singularity occurs by the critical time T, then the solution will stay strong for all time and decay exponentially fast to its mean. 4 the triple contact point of the air/liquid, air/solid, and solid/liquid interfaces, each of which has its own local interfacial energy

7 THE LUBRICATION APPROXIMATION FOR THIN FILMS 7 on the circle,, with periodic boundary conditions. This choice is made for simplicity, however it has the physical interpretation of modeling a periodic array of droplets. The case n = 0 is the fourth order linear heat equation. It is elementary to show that for this problem there are global strong solutions that decay to their average as t! 1. However, unlike the second order linear heat equation, the linear fourth order equation does not preserve positivity. For example, with periodic boundary conditions on [?1; 1] the initial condition (16) h 0 (x) = 0:8? cos(x) + 0:5 cos(x) yields a solution that is initially positive but which is negative at the origin for a nite interval of time: h(0; t) < 0 for t (t 1 ; t ). 5 Hence is it quite remarkable that for suciently strong nonlinearity, the equation preserves positivity. This section addresses the case of nonnegative intial data for 0 < n < 3. In section 6 we prove analogous results for the case of strictly positive intial data and 0 < n < 1. Let (17) = (0; T ); P (h) = f(x; t) j h(x; t) > 0g: The test functions, ', are in C 1 0 of weak solution: ((0; T ); C 1 ( )). We introduce the following denitions The strongest formulation uses two integration by parts (18) h' t? f(h)h xx ' xx? f 0 (h)h x h xx ' x = 0: (19) A second formulation uses a third integration by parts h' t + 3 f 0 (h)h x' xx + 1 f 00 (h)h 3 x' x + f(h)h x ' xxx = 0 and a nal, weakest, version has only one derivative on the test function, but integrates the ux over the set P, where h > 0 : (0) h' t + f(h)h xxx ' x = 0: P Versions (18) and (0) were introduced in [1]. We consider all three versions here. The main result for nonnegative initial data is 5 Numerical results indicate that for the nonlinear lubrication approximation for all n 0:6 nite time singularities occur at x = 0 with the initial condition (16) [4]. They have a similar blowup structure as the solutions described in [6] with dierent boundary conditions. In particular the fourth derivative blows up for all n > 0 and the third derivative blows up for all n > 1.

8 8 A.L. BERTOI AND M. PUGH Theorem.1. Given any nonnegative initial condition h 0 H 1 ( ), h 0 0 we have the following results Case 1: Given f(h) = h n, 1 < n <, s < min(? n; 1 ), and a time interval (0; T ) there exists h(x; t) 0 h L 1 0; T ; H 1 ( ) \ L 0; T ; H ( ) and h satises the equation in the following sense: (1) h' t? f(h)h xx ' xx? f 0 (h)h x h xx ' x = 0 Moreover, h(x; 0) = h 0 (x) 8x h x (; t)! h 0x strongly in L ( ) as t! 0. Furthermore, given 1? s=4, h has the additional regularity and h 1?s= L (0; T ; H ( )) (h ) x L 4 ( ): Moreover, there exist positive A and c such that for all t [0; T ], () kh(; t)? hk L 1 Ae?ct ; Where h is the mean value of h. A depends only on jh 0 j H 1, h, j j, n, and c, the rate of decay, depends only on n, and h. In particular, if h 0 is nonzero, there exists a time T after which the solution is a positive strong solution. Case 1A: If f(h) = h n, 3 8 a solution in the sense (3) < n 1 the above is true if we replace the equation ( 1) with h' t? f(h)h xx ' xx? QT nh n? ( h ) xh xx ' x = 0 and choose in the above so that n > 1? s. 4 Case : If f(h) = h n, < n < 3, given any 0 < r < 1 satisfying 0 < + r? n < 1 there exists h 0 such that on any time interval h L 1 0; T ; H 1 ( ) and h satises the equation in the following sense ( 19): h' t + 3 f 0 (h)h x' xx + 1 f 00 (h)h 3 x' x + f(h)h x ' xxx = 0: The initial data is achieved as above. Furthermore, h has the additional regularity h 1+r= L 0; T ; H ( ) and (h ) x L 4 ( ) 8 r=4 + 1=:

9 THE LUBRICATION APPROXIMATION FOR THIN FILMS 9 For all t [0; T ], (4) kh(x; t)? hk L 1 Ae?ct A and c have the same dependence as above. In particular, there exists a time T after which the solution is a positive strong solution. The statement for n = is as in Case with the minor change in the form of the equation. The details are presented in Section 4. We state and prove theorems for these cases in section 4. The basic existence result for 1 < n < is proved in Bernis and Friedman [1] using a dierent regularization. The other existence results, the additional regularity, and the asymptotic behavior () and (4) of the solutions is new. In addition, we prove the following for 0 < n 3=8. Theorem.. Given any nonnegative initial condition in h 0 H 1 ( ), h 0 0, f(h) = h n, 3=8 n > 0, and any time interval [0; T ], there exists h 0 h L 1 0; T ; H 1 ( ) \ L 0; T ; H ( ) ; satisfying the equation in the following sense ( 0): h' t + f(h)h xxx ' x = 0: P where h n h xxx L (P (h)): The initial data is achieved in the sense described in case 1 above. Furthermore, for any 0 < s < 1 and 1? s, there exists an h satisfying the above with the additional regularity 4 h 1?s= L (0; T ; H ( )) (h ) x L 4 ( ): Moreover, there exist positive A and c such that for all t [0; T ], (5) kh(; t)? hk L 1 Ae?ct : A and c have the same dependence as above. In particular, there exists a time T after which the solution is a positive, strong solution. Signicant Remark: Note that the additional regularity inherited by the weak solutions for 0 < n < 3 is in exact agreement with the regularity of the `zero contact angle' nonnegative source type solutions (9{11). That is, if we assume that the limiting solution h(x; t) has support compactly

10 10 A.L. BERTOI AND M. PUGH contained in and and h(x) x at the edge of the support for all t on some interval [0; T ], then the regularity constraints demand that (6) (7) 0 < n < 3= 3=n 3= < n < 3: Hence the similarity solutions (9{11) just t into our regularity class. Outline of Proof: First we regularize the initial data and the equation to obtain an approximate solution that is a strong, smooth solution for all time. The choice of regularization comes from [1]. In section 3 we discuss the regularization and some basic apriori bounds associated with the solution. The existence results require passing to limit in the regularization parameter. In section 4 we present the details, including new nonlinear estimates constructed specically for this problem. In section 5 we show that the various weak solutions constructed in the previous section all have the indicated long time behavior. The key tool is a lemma which shows that certain nonlinear entropies are equivalent to the L norm of the dierence between the solution and its average. The convexity of the entropy function allows us to prove such an estimate. In a companion paper [5] we show numerical simulations of the regularized equations for several dierent values of the regularization parameter. The numerics indicate the solutions have very nice properties including nite speed of propagation of the support and, for early times, rapid convergence onto the similarity solutions described in (9{11). 3. Regularized problem The regularization involves altering the equation and lifting the intial data. That is, we bound the initial data for the regularized problem away from zero by (8) h 0 (x) = h 0 (x) + (): In addition, we regularize the equation by considering h t + (f (h )h xxx ) x = 0 f (h ) = h 4 f(h ) f(h ) + h 4 : Note that f is still degenerate however for n < 4, f h 4 = as h! 0. Bernis and Friedman [1] proved that this approximate problem has global, positive, smooth solutions: Theorem 3.1. (Global existence of smooth positive solutions for the regularized problem [1]) Let h 0 H 1 ( ), h 0 0. Given an initial condition h 0 (x) = h 0 (x) + () there exists a unique positive solution to the regularized equation h t + (f (h )h xxx ) x = 0

11 THE LUBRICATION APPROXIMATION FOR THIN FILM1 f (h ) = h 4 f(h ) f(h ) + h 4 : We omit many details as they are presented in [1]. The main points are that Classical parabolic Schauder estimates guarantee existence of a smooth solution up to a time. In this short time of existence, the smooth solution satises (9) h x(x; t)dx + t 0 f (h )h xxxdx = h x(x; 0)dx guaranteeing an upper bound, independent of t, for R h x for any time t <. The Sobolev inequality and (9) provide an a priori bound for the Holder norm jh j C 1= ( ). (30) (31) One can use these to prove (3) jh (x 1 ; t)? h (x ; t)j Cjx 1? x j 1= 8t < ; jh (; t)j L 1 ( ) C 8t < : jh (x; t 1 )? h (x; t )j Cjt 1? t j 1=8 where C depends only on the H 1 norm of the initial data. Furthermore, on any interval of existence of a smooth solution, we have the basic entropy G (h ) = 6h + G 0(h ) + c i dx; where G 00 0(y) = 1=f(y) for y > 0: Integration by parts yields that t (33) G (h (x; t))dx + 0 h xx = G (h (x; 0))dx for any t <. The constant c is chosen to make R S G (h 1 ) 0 implying an apriori bound for R =h. This provides an a priori pointwise lower bound for the solution h (x; t). Indeed, let (t) be the minimum value of h (x; t), occurring at a point x 0 (t). Then Holder continuity implies h (x; t) (t) + Cjx? x 0 j 1= for all x. Hence C ((t) + Cjx? x 0 (t)j 1= ) h 1= 1 0 (1 + C(y) 1= ) dy C 1 j log j:

12 1 A.L. BERTOI AND M. PUGH Therefore (t) exp?(c =): This bound is quite crude. The entropy (45) described in the next section provides an algebraic in lower bound for the minimum height. Hence we have an a priori bound for the minimum in terms of the H 1 norm of the initial data and and for the maximum of the solution depending only on the H 1 norm of the initial data. I.e., the solution is uniformly parabolic on [0; ] and can be continued to any time T. Moreover, (30{3) imply that fh g is a uniformly bounded equicontinuous family of functions on. The Arzela-Ascoli theorem guarantees that a subsequence converges pointwise uniformly to a limit, h. We summarize the apriori bounds independent of : surface energy dissipation: (34) conservation of mass, (35) t h x (x; t) dx + 0 and the basic entropy dissipation (36) f (h )h xxx = h x h (x; T )dx = h (x; 0)dx; G (h (x; T )) dx + T 0 (x; 0) dx; h xx = G (h 0 (x)) dx: Additional constraints on the initial data guarantee an apriori bound for R G (h 0 ). In particular for 0 < n < we require that () c 1= and for n the initial data cannot be zero on a set of positive measure. This is the key reason for the upper bound of n < in theorem.1, Case 1. (37) (38) (39) 4. Existence of Weak Solutions Recall the regularized equation and initial data from the previous section: h 0 (x) = h 0 (x) + () h t + (f (h )h xxx ) x = 0 f (h ) = h 4 f(h ) f(h ) + h 4 We now show that with a good choice of () we can pass to the limit, proving the existence part of Theorems.1 and..

13 THE LUBRICATION APPROXIMATION FOR THIN FILM3 Recall that surface energy dissipation implies h x(x; t) dx h 0x dx C: The Sobolev embedding theorem implies there exists an M < 1 such that Moreover h > 0. jh (x; t)j M: 8x; t: Proposition 4.1. Given 1 < n <, h 0 0, and h 0 H 1 ( ), let h be the unique smooth solution to the regularized problem ( 37{39) with (40) () = ; < =5: Then on any time interval [0; T ], a subsequence h converges pointwise uniformly, weakly in L (0; T ; H ( )) \ L 1 (0; T ; H 1 ( )) to a solution h in the sense of distributions ( 18): h' t = f(h)h xx ' xx + f 0 (h)h x h xx ' x : Furthermore, given any 0 < s < min(? n; 1 ) and 1? s, there exists a solution with 4 the additional regularity: (41) (4) h 1?s= L (0; T ; H ( )); (h ) x L 4 ( ) : This existence result was proved in [1] using a dierent regularization. However we use the regularization (39) in the long time behavior results. For this reason, we only sketch the proof of existence. The additional regularity (41{4) is new. It follows from the existence of additional entropies, and its proof is presented in full. Proof. Since the regularized solution, h, is smooth it satises the following: For any test function ' C 1 0 ((0; T ); C 1 ( )) (43) h ' t = f (h )h xx ' xx + f(h 0 )h x h xx ' x : We introduce the entropy (44) G (h 0 ) = G 00 0 (y) = 1 f(y) 6h 0 + G 0 (h 0 ) for y > 0:

14 14 A.L. BERTOI AND M. PUGH The constraint (40) on () provides a uniform bound for C: 6h 0 As G 0 is only determined up to two constants of integration, Jensen's inequality allows us to choose them so that 0 R S G 0(h 1 ). Moreover, since n <, G 0 is a bounded function of h on the interval 0 h M, providing a uniform bound 0 G 0 (h 0 ) C: Combining these yields an a priori bound for the entropy of the initial data. As a result, h xx is bounded a priori since RR G (h (x; T )) + h xx = G (h 0 ) C: Weak compactness implies that a subsequence converges weakly in L (0; T ; H ( )) to the limit h. Surface tension dissipation (34) implies =@t is uniformly bounded in L (0; T ; H?1 ( )). The well-known Lions-Aubin lemma [18] then implies that a subsequence converges strongly in L 0; T ; H 1 ( )) to the limit h. We note that this is the only place in this existence section where we explicitly use this compactness argument. In the remaining proofs we prove strong convergence by using the regularity properties of the solution on the set where h > 0 combined with the fact that f(h) or a higher derivative vanishes on the set where h = 0. This idea was used in [1] to prove the very weak existence result in the sense of (0). The additional entropies (45) and (58) allow us to use this idea to prove a weak existence result in a distribution sense. We now argue that h is a weak solution in the sense (18). Passing to the limit requires f 0(h ) to converge uniformly on to f 0 (h). Since h converges uniformly to h on, it suces to show that f 0 (x) is a continuous function of x and f(x) 0 converges uniformly on [0; M] to f 0 (x). This is true for n > 1 and is proved in the appendix. Similarly, f (h ) converges uniformly on to f(h). Hence the limit h(x; t) solves the equation in the sense (18). In particular, the nonlinear terms converge to the desired limits. For example, f 0 (h )h x h xx ' x! f 0 (h)h x h xx ' x since h xx converges weakly in L to h xx, h x converges strongly in L to h x, and f 0 (h ) converges uniformly to f 0 (h). For the additional regularity, we introduce the following entropy: (45) G?s n (y) = 1 ( + s)(3 + s) y + 1 +s (? n? s)(1? n? s) y?n?s + c

15 THE LUBRICATION APPROXIMATION FOR THIN FILM5 chosen so that (G?s n ) 00 (y) = 1 y s f (y). Integration by parts yields (46) d G?s n (h ) =? h?s s(s + 1)?s? 4 h xx + h h dt 3 x: Note that for 0 s s s(s + 1) s?s? 4?s?1 h h 3 x = s h h x (h ) xx s h?s?s? 4 h xx h h x so that If s < 1, (47) (48) d G?s dt n (h ) =? h?s? h 4 x h?s h xx +?1 + 3s s (s + 1) s(s + 1) 3 h?s h xx:?s? 4 h h x h?s h xx =?C(s) h?s h xx < 0: The constant C(s) decreases to zero as s increases to 1. As before, Jensen's inequality implies that the constant c in (45) can be chosen so that 0 R G?s n (h ). Since 0 s < 1, the constraint (40) on (), implies +s C: h 0 Moreover, y?n?s is a bounded function of y on [0; M], implying?n?s h 0 C: Therefore we have an a priori bound for the entropy of the initial data, R G?s n (h 0). Hence (46) implies there is a C, independent of so that (49) Moreover, (48) implies (50) (51) 0 < h?s h xx? For smooth h(x) bounded away from zero, (h 1?s= ) xx = (1? s=) s(s + 1) 3 C(s) h?s h xx C: h?s? h 4 x C: S1 h?s h xx + [( s 4? s 3 (s + 1)] h 4 xh?s?! :

16 16 A.L. BERTOI AND M. PUGH Combining (49), (50), and (51), yields the a priori bound (h 1?s= ) xx C 1 (s): Thus there is a subsequence so that h 1? s converges weakly in L (0; T ; H ( )): Furthermore, the uniform bound in L 1 (0; T ; H 1 ( )) implies via standard parabolic arguments (see e.g. [1] sec. ) that fh g is a uniformly bounded, equicontinuous family of functions on, hence it has a subsequence 0 converging uniformly to a limit h. Hence 1 1?s= h, h?s=4 converge to h 1?s= and h 1?s=4. By the denition of distribution derivative, 1=?s=4 1?s= the weak limits of (h ) x and (h ) xx are (h 1=?s=4 ) x and (h 1?s= ) xx respectively. The weak solution, h, inherits the a priori bounds (h 1? s ) xx C Moreover, jjh(; t)jj 1 M, implies QT (h ) 4x C for all 1? s 4. (h 1? s 4 ) 4 x C (h 1? s 4 ) 4 x C h xx C: For 3 8 < n 1, we do not have uniform convergence of f 0 (x) to f 0 (x) on (0; M]. However, we prove in the appendix for 0 < n < 1 that given any < n, h 1? f 0 (h) converges uniformly to nh n?. This, combined with the additional regularity above, yields the following theorem: Proposition 4.. Given 3 8 < n 1, h 0 0, and h 0 H 1 ( ), let h be the unique smooth solution to the regularized problem ( 37{39) with (5) () = ; < =5: Then on any time interval [0; T ], given 0 < s < 1, and n > 1? s, there exists a 4 subsequence h converging pointwise uniformly, and weakly in L (0; T ; H ( )) \ L 1 (0; T ; H 1 ( )) to a solution h which satises the additional regularity conditions, (53) (54) h 1?s= L (0; T ; H ( )); (h ) x L 4 ( ): and satises the equation in the following sense h' t = f(h)h xx ' xx + h QT n? h! h xx ' x : x

17 THE LUBRICATION APPROXIMATION FOR THIN FILM7 Proof. Recall the entropy introduced in the previous proof: G?s 1 (y) = 1 ( + s)(3 + s) y? 1 +s s(1? s) y1?s : Exactly as before, (5) implies that the entropy of the initial data is uniformly bounded, yielding a priori bounds for the following: h?s? 4 1? h x h s xx Similarly the entropy G (y) (44) provides an a priori bound of h xx: Exactly as before, and h * h weakly in L (0; T ; H (S )) h 1? s * h 1? s weakly in L (0; T ; H (S )): To prove that h is a weak solution, we need the following lemma to prove convergence of the nonlinear tems: Lemma 4.3. Let 0 < s < 1, 3 < n 1, n > 1? s. The limit h satises 8 4 h 1?s= L (0; T ; H ( )); (h ) x L 4 ( ): Let be compactly contained in. Then f(h 0 )h x! h n? h! strongly in L (): Proof. Fix > 0.! f(h 0 )h x? h n? ( h ) x = \fh>g + x! f(h 0 )h x? h n? ( h ) x \fhg f 0 (h )h x? h n? ( h ) x By the regularity theory of uniformly parabolic equations, h is smooth in \ fh g, and h and its derivatives converge uniformly to h and its derivatives on this set. Hence, by taking to zero,! f(h 0 )h x? h n? ( h \fh>g ) x! 0!

18 18 A.L. BERTOI AND M. PUGH For the second integral, we expand out the square and bound each term. One such term is (f 0 (h )h x ) C (h 1? f 0 (h )) (h ) x \fhg h " # C sup (h 1? f 0 (h )) fhg " C sup (h 1? f 0 ))# (h : fhg (h ) x Here we use the fact that since 1=? s=4, (h ) x is uniformly bounded in L 4 ( ). We also use the fact (proved in the appendix) that y 1? f 0 (y) converges uniformly on [0; M] to ny n?. As before, this and the uniform convergence of h to h imply h 1? f 0(h ) converges uniformly on to h n?. Therefore, by taking small, " sup (h 1? f(h 0 ))# C (n?) : fhg The other two terms from the integral over \ fh g are bounded in the exact same manner. By taking! 0, we have the result. This lemma implies f 0 (h )h x h xx ' x! 1 1? s= QT h n? ( h ) xh xx ' x since any test function ' has support compactly contained in. Convergence of the remaining terms follows in the same way. The additional regularity results (53), (54) follow as in the proof for 1 < n <. We now consider the case n < 3. For n xed, we take r satisfying (55) (56) 0 < r < 1; 0 < + r? n < 1: Proposition 4.4. Let < n < 3 and r be as dened above, h 0 0, and h 0 H 1 ( ). Take h to be the unique smooth solution to the regularized problem ( 37{39) with (57) () = ; < 1 : Then there exists a subsequence h that converges pointwise uniformly and weakly in L 1 (0; T ; H 1 ( )) to a limit h. The limit h is a weak solution in the following sense: h' t =? 1 f 00 (h)h 3 x' x? 3 f 0 (h)h x' xx? f(h)h x ' xxx :

19 THE LUBRICATION APPROXIMATION FOR THIN FILM9 Furthermore the solution has the additional regularity Proof. We introduce the entropy G r n (58) G r n(y) = where c is chosen so that R Gr n(h ) 0. As before, we can integrate by parts: d G r dt n(h ) =? (59) Recall that for a smooth function h > 0, (h 1+r= ) xx = (1 + r=) h 1+ r L (0; T ; H ( )) (h ) x L 4 ( ) for all r : yr (y) chosen so that Gr00 n (y) = f (y) y r?n+ (? n + r)(1? n + r) + c + y r? (r? )(r? 3) h r h xx r(r? 1) h r? 4 h x: S1 h r h xx + f r 4? r 3 (r? 1)g for y > 0. h 4 xh r? Combining these, we nd for 0 r 1, d G r r 1+ dt n(h )?C r (h ) xx where C r = 16(1?r) hence (60) (4?r)(r+) G r n (h r 1+ (x; T )) dx + C r (h ) xx dx dt G r n (h 0(x)) dx: As before, the constraint (57) on (), and r? n + > 0 provide an a priori bound of the entropy of the initial data R Gr n(h 0 ) C. This in turn, implies that the following are bounded uniformly in : 1+ (h ) xx ; r r? 1 h (h ) 4 x = ( r )4! h r=4+1= 4 x : These bounds imply that for xed r, there exists a subsequence so that (61) (h 1+ r )xx * (h 1+ r )xx in L (0; T ; L ( )); (6) (h ) x * (h ) x in L 4 (0; T ; L 4 ( )) 8 r ; (63) hx * hx in L 1 (0; T ; L ( )): We need to show convergence of all the nonlinear terms. We present the argument for RR f 00 3 (h )h x' x. The others are simpler to show and follow analogously. :

20 0 A.L. BERTOI AND M. PUGH Lemma 4.5. Let be compactly contained in. Then for < n < 3, f 00 (h )h x! f 00 (h)h x strongly in L (): Proof. We use the fact that f 00 (y)! f 00 (y) = n(n? 1)y n? uniformly on [0; M] as! 0 for n >. The details are otherwise identical to proof of Lemma 4.3. In the appendix, we prove the uniform convergence of f 00 (y). As before, this lemma implies f 00 (h )h 3 x ' x! f 00 (h)h 3 x' x : since ' has support compact in and h x converges weakly in L ( ). Proposition 4.6. For n =, 0 < r < 1, h 0 0, and h 0 H 1 ( ), let h be the unique smooth solution to the regularized problem ( 37{39) with (64) () = ; < 1= Then on any time interval [0; T ], for any 1 > > r + 1, a subsequence h 4 converges pointwise uniformly, and weakly in L 1 (0; T ; H 1 ( )) to a limit h, and the limit h is a weak solution in the following sense: h' t =? 1 h QT 1? ( h ) xh x ' x? 3 f 0 (h)h x ' xx? f(h)h x ' xxx : Furthermore, the solution has the additional regularity h 1+ r L (0; T ; H ( )): The proof is idential to that for 3=8 < n 1 and is left to the reader. The following lemma is needed: Lemma 4.7. Let 1 > > r 4 + 1, n=. Let. Then f 00 (h )h x! (h1? ( h ) x) strongly in L (): Outline of proof: The proof is identical to the proof of Lemma 4.3, and is left to the reader. It uses the fact that for n =, and > 0, y f 00 (y) converges uniformly on [0; M] to y s and the additional a priori bound (h ) 4 x C: for r

21 THE LUBRICATION APPROXIMATION FOR THIN FILM Proposition 4.8. Given 0 < n, h 0 0; h 0 H 1 ( ) let h be the unique smooth solution to the regularized problem ( 37{39) with (65) () = ; < =5: Then a subsequence h converges pointwise uniformly, weakly in L (0; T ; H ( )) and L 1 (0; T ; H 1 ( )) to h where, h L (0; T ; H ( )) \ L 1 (0; T ; H 1 ( )): Moreover, h is a weak solution in the following sense: h' t =? f(h)h xxx ' x where f(h)h xxx L (P (h)): The constraint on is not used here but is necessary for the long time behavior proved in the next section. The proof of this proposition is identical to that used in [1] for n > 1 and is left to the reader. It uses the following lemma: Lemma 4.9. Let. Then f (h )h xxx converges strongly in L () to the following function 8 < f(h)h xxx h > 0; fl(x) = : 0 h = 0: Proof. Recall that the surface energy dissipation for the regularized problem (37{39) yields the following a priori bound: f (h )h xxx C: Let > 0. In the appendix, we prove that for 0 < n, f (x) converges uniformly on [0; M] to f(y) = y n. To show the strong convergence in L () we note that (66) (f(h )h xxx? fl) = (f(h )h xxx? fl) + (f(h )h xxx? fl)) : \fh>g P \fhg Again, on \ fh g f (h ) 1= h xxx converges uniformly to f(h) 1= h xxx by the uniform parabolicity of the equation. Therefore the rst integral goes to zero. This also yields the regularity result f(h) 1= h xxx L (P ).

22 A.L. BERTOI AND M. PUGH To compute the second integral note that (f(h ) h xxx C sup Moreover, \fhg \fhg fl = \f0<hg Taking! 0, we have the desired result. h f(h) h xxx n jf (h )j C n : 5. Long Time Behavior of Solutions P f(h)h xxx Cn : We now show that the solutions from the previous section satisfy the long time bounds (), (4), (5). Recall the regularized problem (67) (68) (69) h t =?(f (h )h xxx ) x ; f (h ) = h 4 n h h n + h ; 4 h 0 (x) = h 0 (x) + (); = ; < =5: In this section we prove Proposition 5.1. Given h 0 H 1 ( ), h 0 0, let h be a uniform limit of the regularization scheme ( 67{69) on [0; T ] with 0 < n < 3. h is the mean of the initial data, h R = 1 j j S h 1 0 then there exist positive A and c such that (70) kh(; t)? hk L 1 ( ) Ae?ct : A is determined by jh 0 j H 1, h, n, and j j, and c is determined by n and h. In particular, if h 0 is not identically zero there is a time T after which h is a positive strong solution. To prove this proposition, we rst prove the exponential convergence in the L norm and then use an interpolation lemma to get L 1 convergence. For technical reasons, we prove it in two steps, rst for 0 < n < and then for 1 < n < 3. For 0 < n <, we show that the basic entropy R G 0(h) is equivalent to R (h? h). A standard Poincare inequality and Gronwall argument then proves the result. Lemma 5.. Given 0 < n < and any 0 < y 0 < M, there exists positive constants C n;y0 and C n;m;y0 (71) so that the basic entropy, 8 < 1 (1?n)(?n) G 0;y0 (y) = y?n? y1?n 0 y + y?n 0 n 6= 1 (1?n)?n : y log y? (1 + log y 0 )y + y 0 n = 1

23 THE LUBRICATION APPROXIMATION FOR THIN FILMS 3 satises (7) (73) G 0;y0 (y) C n;y0 (y? y 0 ) 8 0 < y; C n;m;y0 (y? y 0 ) G 0;y0 (y) 8 0 < y < M: In the above, the integration constants were chosen so that both G 0;y0 and its rst derivative vanish at y 0. The geometric interpretation of this lemma is that for any xed 0 < y 0 < M, the graph of G 0;y0 (y) can be \sandwiched" between two parabolas with the vertex (y 0 ; 0). The proof follows from the convexity of G 0 (y) and the boundedness of G 0 (y) as y decreases to zero. Since jjhjj 1 M, the lemma proves that R G 0; h(h) is equivalent to R (h? h). To apply this lemma to the entropy dissipation, we recall Poincare's inequality. Let h C ( ) and consider x 0. Then (h(x)? h(x 0 )) dx j j 4 h xx dx: The regularized entropy G ; h+(y) = G 0; h+(y) + 6y satises d G dt ; h+(h (x; t)) dx =? h (x; t) xx dx: Applying Poincare's inequality and (7), G ; h+(h (; T ))? G ; h+(h (; 0))?c (h? ( h + )) Q T?c G 0; h+(h ): Dene B = sup t[0;t ] 6h (; t) : Gronwall's lemma, (73), and the a priori bound on the entropy of the initial data imply (74) C n;m;y0 jjh (; t)? ( h + )jj L G ; h+(h (; t)) B + Ae?ct : Furthermore, B! 0 as! 0. Indeed, taking 0 < s < min( 1 ;? n), the entropy (45) guarantees an a priori bound independent of for h (x; t) +sdx C:

24 4 A.L. BERTOI AND M. PUGH Hence, by Holder's inequality Taking! 0 in (74), h (x; t) dx C! =(+s) dx s=(+s) C s=(+s) : h (x; t) +s (75) We prove the following in the appendix jjh(; t)? hjj L Ae?ct : Lemma 5.3. (interpolation inequality) Let w L 1 ( ) \ C ( ), 0 < < 1. Let j j denote the Holder? seminorm. Then (76) jwj L 1 ( 1 + )=(1+) jwj 1=(1+) jwj =(1+) L + ( )jwj L 1=jS1 j: The limit h is uniformly bounded in C 1=8;1= ( ). Applying the interpolation inequality with = 1, (77) jwj L 1 Cjwj 1=3 L 1 C 1 jwj 1=3 L : This nishes the proof for 0 < n <, since (75) and (77) imply jjh(; t)? hjj L 1 Ae?ct : The rate of decay, c, has bad dependence on n as n increases to since C n;s0 from (7) blows up, forcing c to zero. This is articial, as a better rate of decay follows from the 1 < n < 3 result, which we now prove. The argument is similar to the 0 < n < case so we leave some of the details to the reader. Recall from the previous argument that on [0; M], the graph of the basic entropy, G 0;y0, was bounded above and below by the graphs of parabolas (73{73). In the following, we use a similar idea, bounding the entropy G r 0;y 0 (y) above and below with the graphs of \nonlinear parabolas" C(y 1+ r? y 1+ r 0 ). Lemma 5.4. Given 1 < n < 3, 0 < r < 1, 0 <? n + r < 1, 0 < y 0 < M 1+r= there exists constants C n;y0 and C n;m;y0 so that the entropy, (78) satises G r 0;y 0 (y) = y?n+r (1? n + r)(? n + r)? y1?n+r 0 (1? n + r) y + y?n+r 0? n + r (79) (80) G r 0;y 0 (y) C n;y0 (y 1+r=? y 1+r= 0 ) 8 0 < y C n;m;y0 (y 1+r=? y 1+r= 0 ) G r 0;y 0 (y) 8 0 < y < M:

25 THE LUBRICATION APPROXIMATION FOR THIN FILMS 5 Again, the integration constants were chosen so that both G r 0;y 0 (y) and its rst derivative vanish at y 0. As in the case of lemma 5., the lemma has the geometric interpretation of sandwiching the graph of the convex function G r 0;y 0 (y) between two `nonlinear' parabolas. We apply Lemma 5.4 and the Poincare inequality to the regularized entropy (81) G r ;y 0 (y) = G r 0;y 0 (y) + 6y :?r From section 4 we have (60) d G r 1+r= (8) ;y dt 0 (h (x; t)) dx?c r (h (x; t)) xx dx; where C r tends to zero as r tends to 1. 1+r= 1+r= We apply Poincare's inequality to the function h at a point x 0 where h equals ( h + ()) 1+r=. This and (79) imply T G r ; h+ ( h(x; T )) dx? G r ; h+ (h (x; 0)) dx?c G r (h 0 0;( h+) 1+r= (x; t)) dxdt: Again, dene B = sup t[0;t ] 6h (x; t)?r dx Gronwall's lemma, (80), and the a priori boundedness of the entropy of the initial data imply C n;m;( h+) 1+r=jjh(; t)1+r=? ( h + ) 1+r= jj L G r 0; h+ (h (x; t)) dx B + Ae?ct : Furthermore, B! 0 as! 0. Indeed, for any ~r satisfying the conditions of the lemma, 0 < ~r < 1, 0 <? n? ~r < 1, we have the uniform bound dx < C h (x; t)?~r from the G ~r ; h+ entropy dissipation. In particular, we choose ~r < r satisfying these conditions. Holder's inequality then implies that Taking! 0, h?r C :!?r?~r r?~r r?~r?~r?r C?r : h jjh(; t) 1+r=? h 1+r= jj L Ae?ct : The interpolation inequality then gives an L 1 bound for h(; t) 1+r=? h 1+r= which implies an L 1 bound for h(; t)? h. The rate of decay, c, decays to zero as r increases to 1. We note that as n increases to 3, the condition 0 <? n + r forces r to 1, resulting in a slower rate of decay. This is

26 6 A.L. BERTOI AND M. PUGH consistent with the fact that for the n = 3 case, there are no nonnegative source-type or advancing front exact solutions. 6. Evolution from Positive Initial Data The preceding parts of this paper consider general nonnegative initial data. For such data, we prove the existence of a weak solution for 0 < n < 3 that becomes strong in nite time and approaches its mean in the L 1 norm as t! 1. In this section we show that for all n > 0, if the initial data strictly positive there exists a nonnegative (possibly weak) solution to the equation that approaches its mean in the innite time limit. With strictly positive initial data, there are two possible scenarios. In the rst, h remains positive for all time and we have a global strong solution. Bertozzi et al showed in [6] that this is precisely what happens for n 3:5. 6 The second possibility is that the solution is initially positive but \touches down" somewhere in nite time. Indeed, for suciently small n (e.g. 0 < n < 1) numerical evidence shows that there is initial data that yields such behavior [4]. For such solutions, the following theorem proves the existence of a nonnegative weak continuation of the solution past the singularity time t c, moreover, after a second critical time, T, the solution becomes positive again and converges to its mean. We have the following result Theorem 6.1. Let 0 < m h 0 M, h 0 H 1 ( ), n > 0 and T > 0. Given 0 s < 1 and n > 1? s then there exists a weak nonnegative solution to the lubrication approximation 4 in the following sense of distributions for n > 1; h' t dxdt? nh n?1 h x h xx ' x dxdt? h n h xx ' xx dxdt = 0; for 3 8 Q < n 1; h' t dxdt? h n h xx ' xx dxdt? h T QT n? h! h xx ' x dxdt = 0 x for 0 < n < 3 8 Q, h' t dxdt + h n h xxx ' x dxdt = 0 T for all ' C 1 0 (83) (84) P ( ). In all cases the solution has the additional regularity h 1?s= L (0; T ; H ( )); (h ) x L 4 ( ): Moreover, in all cases there exist positive A and c so that (85) kh(; t)? hk L 1 Ae?ct : A depends on M, m, h, and n, and c depends on n and h. In particular, there exists a critical time T after which the solution is strong and positive. 6 the proof in [6] is an extension of the basic entropy argument used in [1]

27 THE LUBRICATION APPROXIMATION FOR THIN FILMS 7 Remarks: Since the initial data is strictly positive the solution will at least exist as a positive smooth solution on a nite time interval [0; t c ). For the case n = 0, there exists a unique strong solution for all time that approaches its mean in the innite time limit. However, for this linear equation, as the example in Section shows, the solution does not necessarily preserve positivity. We remind the reader that for n > 0 uniqueness is not known, even for nonnegative solutions. For this reason, we cannot exclude the possibility that at the singularity time t c, there might be more than one way to continue the solution. We note that the lower bound m on h 0 implies that for any n, the initial entropy R G 0(h 0 ) is nite. As was proved in [1], if R G 0(h 0 ) < 1 and n, then at any time t, h(; t) can only vanish on a set of measure zero. This result can be rened to n 3= by using the entropy (45). Also, as in [1] the theorem is true with the slightly milder condition of entropy bounded initial data. To prove theorem 6.1 we consider the regularization (86) h t =?(f (h)h xxx ) x ; (87) f (h) = hn+4 h n + h 4 : Since 0 < m h 0, we do not lift the initial data. The bound R S G 0(h 1 0 ) < 1 allows us to apply the arguments from section 4 to prove the existence result. Indeed, for the case n 1, we apply the argument used to prove Proposition 4.1 for nonnegative data with (1 < n < ). Furthermore, the solutions inherit the higher regularity proved for the solutions in section 4, and the long time behavior has already been proven for 1 n <. For the case 0 < n < 1 both the existence and long time behavior follow as in the proof of Theorem.. We need to prove the longtime behavior for n < 3:5. The proof of the longtime behavior (4) in Theorem.1 would suce for n < 3, but the rate of decay decreases to zero as n approaches 3. We show in this section that if there is an a priori bound for R S G 0(h 1 0 ), this is an articial eect. Recall that the basic entropy (88) G ;y0 (y) = 6y + G 0;y 0 (y) satises d G dt ; h(h (x; t)) dx =? h xx(x; t) dx: We now prove a variant of Lemma 5.. Lemma 6.. Given n and 0 < y 0 < M, there exists positive constants C n;y0 and C n;m;y0 (89) so that the basic entropy 8 < 1 (1?n)(?n) G 0;y0 (y) = y?n? y1?n 0 y + y?n 0 n 6= 1 (1?n)?n : y log y? (1 + log y 0 )y + y 0 n = 1