Bernis, F. & Friedman, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Di#. Eq. 83, 179--206.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the construction of globally nonnegativity preserving solutions. Basically, this is a consequence of the so called entropy estimate u(T,x) dr ds dx u 0 (x) dr ds dx. 4) Here, it is formulated for W 0 in its most simple version as it has been found by Bernis and Friedman in [5] (for the multi dimensional equivalent, see [17] Note that for a generic mobility like u and for values of n 2 the entropy G(u) dr ds behaves like u 2 n to leading order. Hence, solutions are positive almost everywhere provided initial data are strictly positive. For ....

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83, (1990), 179--206.

....Yin Jingxue [23] The existence result we present is for arbitrary space dimensions and uses a weak formulation which is different to the formulation of Yin Jingxue. Furthermore we allow the bulk energy Psi to have singularities when B degenerates. We refer also to the work of Bernis and Friedman [3] for results on fourth order degenerate parabolic equations in one space dimension. In section 4 we prove a similar existence result for a viscous Cahn Hilliard type equation of the form u t = Gammar Delta J ; J = GammaB(u)rw ; w = Gammafl Deltau Psi 0 (u) ffu t ; ff 2 IR where ....

....2 and Psi(u) 1 2 (1 Gamma u 2 ) We note that we have not proved the convergence of the whole sequence. This is due to the fact that so far there is no uniqueness result for the Cahn Hilliard equation with a degenerate mobility. 4. 3 Other applications In a paper by Bernis and Friedman [3] the equation u t = Gamma(f (u)u xxx ) x (4.7) where f(u) juj m f 0 (u) f 0 2 C 1 ff (IR) f 0 0 and m 1 (4.8) was studied. They proved the existence of a nonnegative continuous solution and properties of the support of the solution. For example they proved that the support increases ....

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Bernis F. and Friedman A., Higher order nonlinear degenerate parabolic equations, Journal of Differrential Equations 83 (1990), 179--206.

.... of dislocation densities in the theory of plasticity (Norton Hoff type models, cf. 14] For an overview on degenerate parabolic equations of higher order and their applications we refer to Bernis [2] The mathematical investigation of problem (P T ) started with a paper by Bernis and Friedman [5]. In one space dimension they were able to show the existence of a nonnegative Holder continuous solution for all values n 1. The Holder continuity of the solution is important for their analysis because it ensures the smoothness of the solution where it is positive and it implies its ....

....u ff 1 = Gamma Z Omega u n ff Gamma1 u 2 xx Gamma (n ff Gamma1) 2 Gammaff Gamman) 3 Z Omega u n ff Gamma3 u 4 x : E) A careful analysis shows that identity (E) gives a priori estimates for real numbers ff satisfying 1 2 ff n 2. In the paper by Bernis and Friedman [5] identity (E) was applied only for ff = 1 Gamma n. Using the new estimates Beretta, Bertsch, Dal Passo [1] and Bertozzi, Pugh [7] were able to prove regularity results that are optimal in the sense that they are sharp for the source type similarity solutions. Integral estimates derived from a ....

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Equ. 83 (1990), pp. 179--206.

....and in a plasticity model (cf. 13] and the references therein) u stands for the density of dislocations. Crucial for these applications is the fact, that it is possible to construct solutions of (1) which preserve nonnegativity as has been proved for space dimension N = 1 by Bernis and Friedman [6] and for higher space dimensions in the papers by Grun [13] and by Elliott, Garcke [10] This behaviour is in strong contrast to that of solutions to nondegenerate parabolic equations of fourth order which in general become negative even in the case of strictly positive initial values. Moreover, ....

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Equ. 83(1990), pp. 179--206.

....1 Consider initial data satisfying u 0 0, u 0 # H 1(# , and 1 2 ## # x u 0 2 dx ## Q(u 0 ) dx #, 2.1) and assume that 0 n m, m # 3 in (1.4) Then a unique positive smooth solution of (1.1 1.4) exists for all t 0. Proof: Following arguments from previous papers [4, 5] it su#ces to derive a priori pointwise upper and lower bounds for the solution. We derive an a priori bound on the H 1 norm and show this implies pointwise bounds. Then uniform parabolicity implies the solution is completely smooth. Similar arguments are presented in [4] for the case P (u) ....

....from previous papers [4, 5] it su#ces to derive a priori pointwise upper and lower bounds for the solution. We derive an a priori bound on the H 1 norm and show this implies pointwise bounds. Then uniform parabolicity implies the solution is completely smooth. Similar arguments are presented in [4] for the case P (u) 0 and in [5] for a destabilizing non singular P (u) First we note that the Liapunov functional implies that for any time T 0, 1 2 ## # x u(T ) 2 dx # 1 2 ## # x u 0 2 dx ## Q(u 0 ) dx ## Q(u(T ) dx. The initial data is in H 1 and positive. ....

F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J. Di#. Eqns., 83:179--206, 1990.

....are modeled by fourth order degenerate parabolic equations (see Bernis [B1] for an overview and Elliott and Garcke [EG] and Grun [G] for applications in materials science and plasticity) In applications, especially growth exponents n 2 (0; 3] appear. In a fundamental paper, Bernis and Friedman [BF] studied an initial boundary value problem to equation (1) in the case of space dimension one, and they showed that there exist nonnegative solutions provided the initial data were chosen nonnegative. This fact is remarkable not only because in general there is no comparison 1 2 or maximum ....

....We suppose 0 equals a function u 0 2 H 1 (R N ) and assume appropriate conditions for jxj large. A formal computation using equation (1) and integrating from 0 to t; gives 1 2 Z R N jruj 2 (t) Z t 0 Z R N u n jr Deltauj 2 = 1 2 Z R N jru 0 j 2 : 3) Bernis and Friedman [BF] used a variant of this energy identity for appropriate approximate problems to show existence of a Holder continuous solution of the initial boundary value problem (IBP ) 8 : u t div (juj n r Deltau) 0 in Omega Theta (0; 1) u = u n Deltau = 0 on Omega Theta (0; ....

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F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Equ. 83 (1990), 179-206.

....equations such as the porous medium equation u t Gamma r Delta (juj m ru) 0 (3.4) and its fourth order analogue u t r Delta (juj m r Deltau) 0 : 3.5) The equation (3.5) was introduced to model the motion of viscous droplets spreading over a solid surface. Bernis and Friedman [2] studied this equation in one space dimension and proved existence of a positive weak solution and results on the behaviour of the support of the solution. 2. Difficulty: Because the equation is a fourth order parabolic equation there is no maximum principle valid. Therefore many techniques which ....

Bernis F. and Friedman A., Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations 83 (1990), 179--206.

....The second aim of this paper is to use this numerical scheme to verify the above conjectures numerically. 2 In recent years, the question of positivity preservation for the dynamics of nonlinear fourth order equations was thouroughly investigated in the context of lubrication type equations [2, 3, 4, 5, 7, 21]. They arise in the study of thin liquid lms and spreading droplets (for an overview see [6] and the references therein) Numerically, several discretization methods have been presented. In [8] a positivity preserving numerical scheme with nite di erences has been developed. Finite element ....

F. Bernis, A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Di. Eqs. 83, 179-206 (1990).

....[JP00] deduced under much weaker assumptions the existence of a non negative global solution n in one space dimension. In the last years the question of positivity preservation for the dynamics of fourth order equations was thoroughly investigated in the context of lubrication type equations [BF90, BP98, PGG98], which read h t div (f(h) r h) 0: 1.3) They arise in the study of thin liquid lms and spreading droplets (for an overview see [Ber98] and the references therein) Numerically, there are two ways of dealing with Equation (1.3) Bertozzi et al. BZ00] designed a space discretization using ....

F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J. Di. Eqns., 83:179-206, 1990.

.... common for fourth order nonlinear degenerate parabolic problems; see e.g. 10] for the logarithmic Cahn Hilliard equation with a degenerate mobility ( P) with v 0 0) The restriction of existence to one space dimension in [9] is because they adopt the very weak solution concept introduced in [4] for fourth order nonlinear degenerate parabolic equations. It should be noted that the degenerate system (P) is far more complicated than the corresponding degenerate Cahn Hilliard equation. This is because for (P) the mobility b, 1.3a,b) vanishes only at the vertices of Q; whereas the ....

....00 Therefore combining the above results and repeating (3.68a,b) for all 0 , we obtain the desired results (3.51a,b) 3.52a,b) and (3.53a d) ut Remark. The weak formulation (3. 53a d) is slightly di erent to the one studied in [9] However, both are based on the solution concept introduced in [4] for fourth (and higher) order nonlinear degenerate parabolic equations. Hence they are both restricted to one space dimension. 4 Numerical Experiments We consider four numerical experiments, highlighting (i) the di erences between the mobility b being degenerate, 1.3a,b) and b being constant, ....

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations , J. Di. Eqns. 83 (1990), 179-206.

....of Theorem 3.1 in [2] ut We note that b U 0 h u 0 and b V 0 h v 0 satisfy the assumptions of Theorem 4.2. Finally, we remark that the weak formulation (4. 6a d) is slightly di erent to the one studied in [8] However, both are based on the solution concept introduced in [4] for fourth (and higher) order nonlinear degenerate parabolic equations. Hence they are both restricted to one space dimension. 5 Numerical Experiments We consider four numerical experiments, highlighting (a) the di erences between the mobility b being degenerate, b b 0 (1.4) with = 0, ....

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Di. Eqns. 83 (1990), 179-206.

....and in a plasticity model (cf. 13] and the references therein) u stands for the density of dislocations. Crucial for these applications is the fact that it is possible to construct solutions of (1. 1) which preserve nonnegativity as has been proved for space dimension N = 1 by Bernis and Friedman [6] and for higher space dimensions in the papers by Grun [13] and by Elliott and Garcke [10] This behavior is in strong contrast to that of classical solutions to linear parabolic equations of fourth order which in general become # Received by the editors June 12, 1996; accepted for publication ....

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Di#erential Equations, 83 (1990), pp. 179--206.

....literature in recent years. A key feature of this problem is that there is no uniqueness result. In addition to establishing well posedness of our finite element approximation for all d 3, we proved convergence in one space dimension to solutions using the very weak solution concept introduced by Bernis and Friedman (1990) for this problem. This basically states that u is a solution if Z T 0 h u t ; ji dt Gamma Z fjuj 0g b(u)r Deltaurj dx dt = 0 8 j 2 L 2 (0; T ; H 1( Omega Gamma4 : The restriction of convergence to one space dimension is due to the fact our a priori bounds on the finite element ....

....is due to the fact our a priori bounds on the finite element approximation only guarantee in the case of d = 1 uniform boundedness and equicontinuity of the approximate solutions, which is necessary to be able to pass to the limit in the discrete problem. For similar reasons, the results of Bernis and Friedman (1990) were restricted to one space dimension. In this paper we extend the techniques in Barrett, Blowey and Garcke (1997) to the Cahn Hilliard equation with degenerate mobility, 1.3a c) This paper is organised as follows. In Section 2 we formulate a fully practical finite element approximation, P ....

F. Bernis and A. Friedman (1990). Higher order nonlinear degenerate parabolic equations.

....which leads to the fullest balance will in each case be considered and in the current section we consider that in which Phi(w) w m Gamman as w 0 with n 3; m 3; 6.2) a representative form being Phi(w) w m w m w n ; 6. 3) which with m = 4 has been adopted in the literature ([4], 1] this form of regularisation ensures positivity of solutions, so that, in particular, the earlier loss of regularisation that occurs when a dead core is present ceases to be a difficulty. The solution we are seeking to select via the limit 0, ffi 0 is accordingly a compactly supported ....

....are selected as 0 , satisfying (5.4) and (3.1) If q = m Gamma 3) m Gamma n) we obtain (6.9) with quasi steady large time behaviour of the form u 0 t Gamma1=7 f(x=t 1=7 ) as t 1 (cf. 4. 10) Finally, for q (m Gamma 3) m Gamma n) the regime into which the analyses of [4] and [1] fall) we recover (2.43) and (3.4) The condition (6.9) requires that the interface move outward (correspondingly, 6.5) has a solution only for s 0 0) so dead cores (dry patches) are unable to develop even for n 3=2. Indeed, the comment above concerning the limit 0 ignores a ....

F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J. Diff. Eq., 83:179--206, 1990.

.... homogeneous boundary condition s 0 = 0 on fs 0g : 9) Existence and qualitative properties of weak solutions for the evolution problem given by (6) and (9) have been established by Bernis Friedman, by Grun, by Beretta, Bertsch Dal Passo, by Elliot Garcke, by Bertozzi Pugh, and others [7, 20, 4, 14, 8]. The reader will find a survey in [5] To us, it seems that the non zero contact angle case is qualitatively different and analytically harder to handle. Let us explain why. The relevant energy for the zero contact angle case is E(s) Z js 0 j 2 : 10) The functional (10) is a convex, ....

....will also give us (19) Proposition 1.3) Let us point out that the non negativity but not the strict positivity is build into our scheme. In this respect our approximation by a time discrete scheme consisting of variational problems differs considerably from the approximation used by [7, 20, 14, 4, 8] to prove existence for the zero contact angle situation. 7, 4, 8] regularize the equation by modifying the mobility and lift the initial data in such a way that the solutions are strictly positive. 20, 14] regularize the equation by modifying the mobility such that it is no longer ....

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F. Bernis, A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Equations, 83 (1990), 179 -- 206.

....The di erential equation in (1. 10) is a particular case of the so called thin lm equation t h x (h n 3 x h) 0; n 2 R ; subject to a di erent boundary condition, namely x h = 0, the thin lm equation has been studied recently by many authors; we only quote the pioneering paper [3] and [4] where further references can be found. 6 Lorenzo Giacomelli and Felix Otto The metric tensor g h is then de ned by g h (v 1 ; v 2 ) Z h 0 1 0 2 ; where v i are related to i through (1.12) With this choice we obtain 0 = g h(t) t h(t) v) hdE(h(t) vi = Z v Z ....

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Di. Equations 83 (1990), 179-206. Lubrication approximation of Hele-Shaw 25

....eld of intensive research. Even for strictly positive analytical solutions, the solution of a naive discretization scheme may become negative, causing unwanted numerical instabilities [Ber98] In the last years this question was thoroughly investigated in the context of lubrication type equations [BF90, BP98, dPGG98] which read h t div (f(h) r h) 0: 1.3) They arise in the study of thin liquid lms and spreading droplets (for an overview see [Ber98] and the references therein) Here, the main ingredient for the proof of the non negativity or positivity property is to exploit the special ....

F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J. Di. Eqns., 83:179-206, 1990.

....for ( which leads to the fullest balance will in each case be considered and in the current section we consider that in which (w) w m n as w 0 with n 3; m 3; 6.2) a representative form being (w) w m w m w n ; 6. 3) which with m = 4 has been adopted in the literature ([BF90], BBP95] this form of regularisation ensures positivity of solutions, so that, in particular, the earlier loss of regularisation that occurs when a dead core is present ceases to be a diculty. The solution we are seeking to select via the limit 0, 0 is accordingly a compactly supported ....

.... (m 3) m n) xed front solutions are selected as 0 , satisfying (5.4) and (3.1) If q = m 3) m n) we obtain (6.9) with quasi steady large time behaviour of the form u 0 t 1=7 f(x=t 1=7 ) as t 1 (cf. 4. 10) Finally, for q (m 3) m n) a regime into which the analyses of [BF90] and [BBP95] fall) we recover (2.43) and (3.4) The condition (6.9) requires that the interface move outward (correspondingly, 6.5) has a solution only for s 0 0) so dead cores (dry patches) are unable to develop even for n 32 3=2. Indeed, the comment above concerning the limit 0 ....

F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J. Di. Eq., 83:179-206, 1990.

.... upon these here; see, Bernis (1996) for example) involving, in particular, droplets of viscous fluid spreading under surface tension (the second order equation (2) applies instead when the driving force is gravity) Equation (3) has been the subject of much recent analysis (see, for example, Bernis Friedman (1990), Bernis et al. 1992) and Bertozzi et al. 1994) but not, to our knowledge, for boundary conditions of the form (4) a more detailed discussion of such boundary conditions in the semi infinite domain context is given in Bernis et al. 1999) Introducing a pressure p, equation (3) can also be ....

Bernis, F. & Friedman, A., 1990. Higher order nonlinear degenerate parabolic equations. J. Diff. Eq.

....is in contrast to linear parabolic equations of fourth order, where solutions which are initially positive may become negative in certain regions. We review what is known for problem (P) in one space dimension. Existence of Holder continuous nonnegative solutions to problem (P) was first shown by Bernis and Friedman (1990). They used a very weak solution concept, which basically says that a function u solves (1) if Z T 0 h u t ; ji dt Gamma Z fjuj 0g b(u)r Deltaurj dx dt = 0 8 j 2 L 2 (0; T ; H 1( Omega Gamma3 ; 3) 1 where h Delta; Deltai is the standard H Gamma1 Theta H 1 duality pairing. ....

....standard H Gamma1 Theta H 1 duality pairing. Although there exists a nonnegative solution to nonnegative initial data, it seems to be possible that positive initial data may lead to solutions which can become zero in finite time. Here the exponent p in the diffusional mobility is crucial. Bernis and Friedman (1990) showed that for p 4 positivity is preserved. For small values of p, numerical simulations by Bertozzi (1995) suggest that solutions which are initially positive can become zero in subsets with positive measure, i.e. the film breaks (see also Beretta et al. 1995) for a result in this direction ....

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F. Bernis and A. Friedman (1990). Higher order nonlinear degenerate parabolic equations. J. Diff. Eqns. 83, 179--206.

....sign of the initial data: 5] 27] This is in sharp contrast to the properties of the heat equation and other second order parabolic equations. During the last ten years it has been established that the thin lm equation for n 0, unlike (1. 2) preserves the nonnegativity of the initial data [9], 4] 14] Then two natural questions arise: 1) Is the limit as n 0 of the Cauchy problem for (1.1) a parabolic variational inequality And (2) How does the obstacle free boundary problem for (1.1) compare with the Cauchy problem when n 0 In order to address these questions we set up a ....

....datum u 0 satis es (2.1) the Cauchy problem for Equation (1.1) has a strong solution with nite speed of propagation, which is also a solution of Problem (OFB) see [7] 29] 20] and references therein. Strong solutions of (1. 1) are functions that satisfy the de nition of weak solution of [9] and, in addition, the entropy estimates of [4] and [14] Some technical details of the de nition of Problem (OFB) are adapted to the current stage of the theory for n 0. New developments (for n 0 or for n 0) may lead to simpli cations of this de nition. Weak solutions in the sense of [9] ....

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F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Dierential Equations, 83 (1990), pp. 179-206.

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Bernis, F. & Friedman, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Di#. Eq. 83, 179--206.

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Bernis, F. & Friedman, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Di#. Eq. 83, 179--206.

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F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J. Di. Eqns., 83:179-206, 1990.

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F. Bernis and A. Friedman. Higher order nonlinear degenerate parabolic equations. J Di Eq, 83:179{ 206, 1990.

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