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Z. Angew. Math. Phys. (2018) 69:79 Zeitschrift fur ¨ angewandte c 2018 The Author(s) Mathematik und Physik ZAMP https://doi.org/10.1007/s00033-018-0969-y Marco A. Fontelos, Georgy Kitavtsev and Roman M. Taranets Abstract. For a nonlinear system of coupled PDEs, that describes evolution of a viscous thin liquid sheet and takes account 1 2 of surface tension at the free surface, we show exponential (H ,L ) asymptotic decay to the ﬂat proﬁle of its solutions considered with general initial data. Additionally, by transforming the system to Lagrangian coordinates we show that the minimal thickness of the sheet stays positive for all times. This result proves the conjecture formally accepted in the physical literature (cf. Eggers and Fontelos in Singularities: formation, structure, and propagation. Cambridge Texts in Applied Mathematics, Cambridge, 2015), that a viscous sheet cannot rupture in ﬁnite time in the absence of external forcing. Moreover, in the absence of surface tension we ﬁnd a special class of initial data for which the Lagrangian solution exhibits L -exponential decay to the ﬂat proﬁle. Mathematics Subject Classiﬁcation. 35B40, 35G31, 76D45, 76D27, 35D30, 35D35. Keywords. Free viscous sheets, Non-rupture result, Exponential asymptotic decay, Nonlinear PDEs, Porous medium equations. 1. Introduction The last decades showed a considerable progress in the mathematical understanding of singularity forma- tion during dewetting of thin viscous liquid ﬁlms, see, e.g. reviews [2–4]. In particular, topics of ﬁnite time rupture of thin ﬁlms and pinch-oﬀ of a liquid thread driven by attractive intermolecular van der Waals forces and curvature, respectively, were intensively studied both analytically and numerically starting from the pioneering articles [5–7]. Fundamental forces in these processes include van der Waals, surface tension, viscous and inertia ones [8–14]. Let us also mention recent articles [15, 16] showing the inﬁ- nite time rupture of viscous sheets driven by Marangoni forces in presence of temperature or surfactant gradients at the sheet free surface. In the modelling of dewetting processes in micro- and nanoscopic liquid ﬁlms, the lubrication ap- proximation resulting in high-order degenerate parabolic equations occurs to be especially eﬀective, see, e.g. [2, 4, 17–21] and references therein. In this article, we study asymptotic behaviour of solutions to the system describing evolution of a thin viscous sheet [1]: v + vv = σh + [hv ] , (1a) t x xxx x x h = − (hv) . (1b) Here, v(x, t)and h(x, t) denote the average velocity in the lateral direction and the height proﬁle for the free surface, while σ, ν are dimensionless surface tension and viscosity, respectively. The high order of system (1) is a result of the contribution from surface tension at the free boundary, reﬂected by the linearised curvature term σh . The terms v + vv and ν(hu ) /h in (1a) represent inertial and Trouton xx t x x x viscosity terms, respectively. 0123456789().: V,-vol 79 Page 2 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP Existence of weak solutions to (1) considered in bounded domains Q =(0,T ) × Ω, where Ω = (0, 1), with zero velocity and homogeneous Neumann boundary conditions for h: h (0,t)= h (1,t)= v(0,t)= v(1,t) = 0 (2) x x and initial data with positive height: h(x, 0) = h (x) > 0,v(x, 0) = v (x)for x ∈ [0, 1], 0 0 was shown by Kitavtsev et al. [22] via an additional introduction of the regularising Lennard-Jones potential in (1a). Observe, that the boundary conditions (2)for v guarantee the mass conservation h(x, t)dx = ||h || := M> 0 for all t 0. (3) 0 1 To summarise what is known or widely accepted about pinch-oﬀ singularities of system (1). Finite time rupture of solutions of (1) under additional presence of van der Waals forces in (1a) was investigated recently both numerically and analytically in [14, 23]. They were concerned with existence and stability of the self-similar solutions dynamically describing the neighbourhood of the pinch-oﬀ. In [14], it was shown that the rupture occurs due to combined competition between inertia, viscosity and van der Waals terms, while surface tension staying negligible. The self-similar solutions in this case exhibit the rupture in ﬁnite time with the minimum height evolving as ∗ 1/3 min h(x, t) ∼ (T − t) , x∈Ω where T is the rupture time. In turn, inﬁnite time rupture of solutions to (1) considered without van der Waals forces but rather coupled through a Marangoni term to a diﬀusion equation for temperature or surfactant distribution at the sheet free surface was described recently in [15, 16]. In [16]itwas shown that the minimum height in this case follows the self-similar law min h(x, t) ∼ exp{−aνt}, x∈Ω where a> 0 is the thinning rate of the sheet depending both on the initial data and physical parameters of the system. Finally, for system (1) in the absence of any additional forces and in the inviscid case, i.e. ν =0, the ﬁnite time rupture can occur for suitable initial conditions [24–26]. The neighbourhood of the pinch point is described then by the similarity solution of [26] with ∗ β ∗ −γ min h(x, t) ∼ (T − t) , max v(x, t) ∼ (T − t) with β =0.7477,γ =0.3131, (4) x∈Ω x∈Ω and the rupture is driven by the competition of inertia and curvature terms in (1). In contrast to the mentioned studies, the main focus of this article is to consider system (1) with ν> 0 and to prove that ﬁnite time rupture is not possible in this case, but rather the exponential asymptotic decay to the ﬂat proﬁle (h, v)=(M, 0), where M is from (3), occurs. The fact that viscous sheets cannot rupture in the absence of van der Waals or external forcing is formally well accepted in physical literature (cf. [15, 16] and references therein). In fact, in book [1] it was pointed out that system (1) with σ> 0and ν> 0 does not admit consistent self-similar scalings. In particular, if one considers the self-similar scaling (4) for the case ν> 0, one ﬁnds out that the viscous Trouton term enters the leading order balance and becomes dominant, by that breaking the self-similar ansatz (4), when the minimum height decreases to the critical range: h (t) ν /σ. (5) min β−2 Indeed, as it shown in Example 7.7 of [1] the surface tension and viscous terms in (1a) scale like στ −β−1 β ∗ β and ντ / σ, respectively, where τ =(T − t) is the self-similar spatial scaling for the inviscid ZAMP Asymptotic decay and non-rupture of viscous sheets Page 3 of 21 79 Fig. 1. Left: Values of global minimum height divided by ν attained dynamically by numerical solutions to system (1)–(2) with σ =1 and ν =0.1, 0.05, 0.01, 0.005, 0.001, 0.0001. Initial data h(x, 0) = 1 − 0.2cos(πx/2),u(x, 0) = π sin(πx/2) were chosen as in the example of the inviscid pinch-oﬀ considered in [26], cf. formula (11) there. Right: numerical evolution of the minimal sheet height in time for the same initial data and ν =0.1. For obtaining numerical solutions to (1), the numerical code developed in [16, 27] was used 4β−2 pinch-oﬀ analysed in [26]. Balancing these two terms and using the fact h ≈ τ result together in min (5). On the other hand, the threshold range (5) suggests that for small viscosity ν 1 solutions to (1) may exhibit signiﬁcant decrease of height at initial and intermediate times. Indeed, numerical solutions to problem (1), (2) indicate (see Fig. 1) that for the initial data considered in [26], that lead to the ﬁnite time rupture in the inviscid case, the minimum of the height tends to zero initially until it enters the range given by (5) and only thereafter starts to converge to M . Nevertheless, up to our knowledge, mathematical results showing non-rupture or asymptotic decay of solutions to (1) are still lacked in the literature. The main goal of this article is to ﬁll this gap and to show that (5) provides the sharp uniform in space and time lower bound for the solutions to (1) considered with ν> 0and σ> 0 as well as the exponential asymptotic decay of them to the ﬂat proﬁle (h, v)=(M, 0). 1 2 In Sect. 2, we start with showing the (H ,L ) exponential asymptotic decay for the non-negative solutions to (1)to(h, v)=(M, 0). More precisely, by using the dissipation of entropy found in [22], in the case ν> 0and σ> 0 we show that any suﬃciently regular solution with initially non-negative height converges to (M, 0) at an exponential rate (Theorem 2.1). Nevertheless, this result still does not exclude a possibility of rupture in a ﬁnite time that could lead to loss of regularity properties of the solutions and by that terminating the asymptotic decay at the time when the height touches the zero. Therefore, in Sects. 3–5 we additionally show that the solutions to (1) cannot touch zero in a ﬁnite time. For this purpose, transformation of (1) to the Lagrangian coordinates following ideas suggested in [28–30] for the systems describing viscous jets turns out to be very useful. In Sect. 4, we show that system (1) considered without curvature term (the case σ = 0) transforms to a porous medium type Eq. [31]bythe Lagrangian formalism. In this case, existence, uniqueness and asymptotic behaviour of the solutions to it are understood and completely rigorous (Theorem 4.1). In particular, the solutions cannot either blow-up or rupture in a ﬁnite time. Moreover, for a special class of initial data we are able to show exponential asymptotic decay to the ﬂat proﬁle in the absence of surface tension [see estimate (28)]. In Sect. 5, we consider the Lagrangian formulation of the full system (1) and provide an analytical argument that general solutions to it cannot rupture in a ﬁnite time, but rather obey the lower bound (5). Finally, in “Appendix A” we extend results of Sect. 1 to show the exponential asymptotic decay of the radially symmetric solutions (Theorem A.2) to the two-dimensional generalisation of (1), which appear in applications to the axisymmetric sheet point rupture [14]. For this purpose, we derive for the 79 Page 4 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP ﬁrst time, up to our knowledge, the entropy and energy estimates for the radially symmetric solutions (Lemma A.1). Finally, we comment on the regularity classes of solutions considered in this article. Results of Sect. 2 1 2 and Appendix A are rigorous and apply to suitably deﬁned (H ,L ) nonnegative weak solutions to systems (1) and (A.2), respectively. Results of Sect. 4 are rigorous, and the strong solution obtained in Theorem 4.1 is positive and unique. In Sect. 5, we present a formal analytical argument for positive classical solutions to Eq. (16) considered with general initial data. 2. Asymptotic decay for non-negative solutions In this section, we show that solutions to (1) having non-negative height for all times asymptotically 1 2 decay in (H ,L )-norm to the ﬂat proﬁle (h, u)=(M, 0). First, we deﬁne energy and entropy functionals as 1 2 2 E(v, h):= hv + σh dx, 1 2 S(v, h):= h (v +(G(h)) ) + σh dx, where G(h)= ν log(h). Recall the following entropy and energy equalities [22] which hold for the solutions of (1): S(v, h)+ νσ h dx =0, (6) xx dt d 2 E(v, h)+ ν hv dx =0. (7) dt 1 2 Theorem 2.1. (asymptotic exponential decay) Assume that initial data (h ,v ) ∈ H (0, 1) × L (0, 1), 0 0 h 0,and σ> 0 and ν> 0. Then, there exist positive constants A, B depending on initial data and parameters σ, ν such that the height h(x, t) asymptotically exponentially decays to zero in H -norm: 2 −Bt h dx Ae for all t 0, (8) hence 1 0,1/2 h → M in H (0, 1) ∩ C [0, 1] as t → +∞, (9) where M is deﬁned in (3). Moreover, for the velocity ﬁeld we have v → 0 in L (0, 1) as t →∞. (10) Proof of Theorem 2.1. Using the Poincar´ e inequality 1 1 2 1 2 h dx h dx, h (0) = h (1) = 0, (11) 2 x x x xx 0 0 ZAMP Asymptotic decay and non-rupture of viscous sheets Page 5 of 21 79 from the entropy equality (6) we ﬁnd that 1 t 1 σ 2 2 2 h dx + νσπ h dxdt S(v ,h ). (12) 0 0 2 x x 0 0 0 By (12)the is dominated by the solution of 2 2 y (t)= −By(t),y(0) = A with A = S(v ,h ),B =2νπ . 0 0 Solving for y(t), we deduce that 2 −Bt ||h || y(t)= Ae for all t 0, whence (8) follows. Using the Poincar´ e inequality and (8), we deduce 1 1 2 1 2 A −Bt (h − M ) dx h dx e . 2 2 π x π 0 0 From here we have 2 1 −Bt ||h − M || 1 A 1+ e , H (Ω) 1 ∞ whence (9) follows. To show (10), ﬁrst (7) and the compact embedding H (0, 1) → L (0, 1) imply h(x, t) C for all (x, t) ∈ Q , where here and below C denotes a generic constant depending only on σ, ν and h ,v , and one estimates 0 0 3/2 3/2 ||(h v) || 1 ||h v || 2 +2||h || 2 || hv|| 2 x L x L x L L C || hv || 2 + ||h || 2 , x L xx L where the last inequality follows by (6) and (11). Next, using L Poincar´ e inequality and continuous 1,1 p embedding W (0, 1) → L (0, 1) one gets 3/2 p 2 2 ||h v|| C || hv || + ||h || for p ∈ [1, ∞]. (13) L p x L xx L On the other hand, we have 1 1 2 3 2 C hv dx h v dx. 0 0 Using this estimate, summing up (6) and (7), and taking into account (13) one arrives at 1 T 1 3 2 3 2 h v dx + B h v dxdt A , 1 1 0 0 0 with A and B depending on ν, σ, S(v ,h )and E(v ,h ). The comparison argument then implies 1 1 0 0 0 0 3/2 2 −B t ||h v|| A e . (14) 2 1 L (0, 1) ∗ ∗ The last estimate together with the fact that h> 0 for all t>T for some T > 0 because of (9) implies (10). 79 Page 6 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP 3. Transformation to Lagrangian coordinates Using the mass conservation (3), we make the change of coordinates as in [23, section 4.1] K ds x (s, t)= ,x(s, 0) = K ,x (s, t)= v(x(s, t),t), (15) s t h(x(s, t),t) h (s ) where ds K =1 > 0. h (s) Moreover, due to the mass conservation (3) and the second relation in (15) one has necessarily K = M. Note, that (15) deﬁnes a monotonic mapping of s ∈ [0, 1] into x(·,t) ∈ [0, 1] for each t 0 provided h(x, t) > 0. Then, Eq. (1b) is trivially satisﬁed and Eq. (1a) transforms into σM 1 1 1 x ts x = + ν tt x x x x x s s s s s s s s s Next, by diﬀerentiating the last equation on s and introducing the new function u := x , one arrives at the equation 1 1 1 1 1 u = σM − ν . (16) tt u u u u u s s tss Note, that due to (15) the relation between new function u and the solutions of (1) is given by u(s, t)= . (17) h(x(s, t),t) The boundary conditions x(0,t)=0 and x(1,t) = 1, which follow from the last relation in (15), imply the conservation of Lagrange function: u(s, t)ds =1. (18) Moreover, the boundary conditions for height proﬁle in (2), h (0,t)= h (1,t)=0, x x due to (15) and (17), imply the Neumann boundary conditions for u(s, t): u (0,t)= u (1,t)=0. (19) s s Finally, using (18) and (19) one can rewrite the entropy and energy estimates (6)-(7) in the Lagrange setting for (16). Namely, the following relations hold for solutions to (16) for all t> 0: ⎡ ⎤ ⎛ ⎞ 1 s 1 2 2 1 u 3u u ⎢ ⎥ ss 1 d 2 s 2 s ⎝ ⎠ u ds + ν + σM ds = −νσ M − u ds, (20) ⎣ t ⎦ 2 dt 5 5 4 u u u u 0 0 0 ⎡ ⎤ ⎛ ⎞ 1 s 1 u u ⎢ ⎥ t 1 d 2 ⎝ ⎠ u ds + σM ds = −ν ds. (21) ⎣ t ⎦ 2 dt u u 0 0 0 Although, a priori estimates (20)–(21) can be derived for problem (16), (19) independently, the equivalence of Euler and Lagrange formulations given by transformation (15) for classical solutions with positive for ZAMP Asymptotic decay and non-rupture of viscous sheets Page 7 of 21 79 all times height implies them directly. We note also that this equivalence valid also for local solutions with positive h(x, t) for all t ∈ (0,t ), where t > 0 is the ﬁrst time at which h(x, t) touches zero. In r r particular, this observation is important for application of (20)–(21) in the vicinity of potential rupture in Sect. 5. 4. Boundedness and asymptotic decay in the case σ =0 In the absence of surface tension, Eq. (16) after integration in time reduces to −2 u + ν = f (s)or u = ν u u + f (s), (22) t t s ss where the integration factor is determined by the initial conditions and (15)as M h (x(s, 0)) 0,x f (s)= v (x(s, 0)) + ν . (23) h (x(s, 0)) h (x(s, 0)) 0 0 For compatibility with boundary conditions (19) and conservation of Lagrange function (18) one has to impose the following integral condition onto initial data: f (s)ds =0, (24) which is immediately true due to (23) and (15) if the initial data are compatible with (2), i.e. h = v =0 at x =0, 1. 0,x 0 We note that Eq. (22) coincides with the porous medium equation in [32, (1.27)] with exponent m = −1, which arises from the backward parabolic super fast diﬀusion equation u =Δu after reversing the time in the latter. Therefore, for investigation of regularity and qualitative behaviour of solutions to (22) in the next theorem we apply the methods developed for porous medium equations in [31, 32]. Theorem 4.1. Suppose ν> 0, u ,f ∈ L (0, 1) and u > 0, (25) 0 0 then a unique strong solution u(s, t) to problem (22), (19) with initial data u(s, 0) = u (s)= M/h (x(s, 0)) 0 0 exists for all t> 0. For it one has a linear in time upper and uniform lower bounds: 0 <C u(s, t) ||u || + ||f || t for (x, t) ∈ (0, 1) × (0, ∞), (26) 0 ∞ ∞ where constant C depends on u and ν.Hence, u(s, t) cannot blow up in a ﬁnite time or touch zero. Moreover, if f =0 then there exist positive constants ln ||u || 0 ∞ A := (u − 1) ds, B := 2ν (27) ||u || − 1 0 ∞ such that ||u(·,t) − 1|| A exp{−Bt} for all t> 0 (28) L (0, 1) holds, i.e. L -exponential asymptotic decay of u to the ﬂat proﬁle. 79 Page 8 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP Proof of Theorem 4.1. Existence part follows by the quasilinear parabolic theory [33] and the fact that the main term in (22), namely ss is uniformly coercive unless u becomes inﬁnite at some point. In particular, the upper bound (26) follows by an application of the maximum principle [31, Lemma 3.3, p. 34]. Note, that this upper bound does not depend on viscosity ν. Indeed, this can be checked by rescaling time t = νt, f = νf and applying the maximum principle. For the global well-posedness of the solution for all t> 0, it remains to show the uniform lower bound for u(s, t). First, we note that the entropy and energy equalities reduce for (22)to ⎛ ⎞ 1 s ⎝ ⎠ u ds + ν ds =const, (29) 0 0 ⎛ ⎞ 1 s 1 1 d ⎝ ⎠ u ds ds = −ν ds. (30) 2 dt 0 0 0 2 2 2 Combining them together and using the inequality (y + z) y /2 − z one estimates ⎛ ⎞ ⎛ ⎞ 2 2 1 s 1 1 s 1 2 1 ⎝ ⎠ ⎝ ⎠ u ds + ν ds ds − u ds ds, t t u u s s 0 0 0 0 0 and therefore, one concludes that sup ds const. (31) t∈(0, ∞) Next inequality can be considered as an energy type one for (22)(see[31, 32]). Testing (22) with −1/u and subsequently integrating by parts provides the equality ⎛ ⎞ 1 1 1 1 1 u 1 ds ⎝ ⎠ ds +2ν ds = − f (s) − dsdt, (32) dt 5 2 2 u u u u 0 0 0 0 wherewehaveusedalso(24). Let us integrate the last equality in time and estimate the right hand side of it using L Poincare inequality as follows. ⎛ ⎞ T 1 1 T 1 T 1 1 ds 1 u ⎝ ⎠ f (s) − dsdt ||f || dsdt = ||f || dsdt ∞ ∞ 2 2 2 3 u u u u 0 0 0 0 0 0 0 ⎡ ⎤ T 1 1 1 T 1 u 1 ||f || 1 T u s s ⎣ ⎦ ||f || ds dsdt ε sup ds + dsdt . 5 5 u u 2 u ε u t∈(0,T ] 0 0 0 0 0 0 2 2 Combining this with (32) implies for T = νε = ν/||f || that 1 1 1 1 1 sup ds + C(ν, ||f || ) dsdt ds. u u u t∈(0,T ] Q T s 0 0 ZAMP Asymptotic decay and non-rupture of viscous sheets Page 9 of 21 79 But the latter inequality can be extended to 1 1 1 1 sup ds ds (33) u u t∈(0, ∞) 0 0 0 via repeating the last argument iteratively on small time intervals (t, t + ν/||f || ). We conclude that estimates (31) and (33) imply together that const. ∞ 1,1 u L (0,∞;W (0,1)) 1,1 ∞ In turn, the compact embedding W (0, 1) → L (0, 1) implies C, u L ((0, 1)×(0,∞)) i.e. the uniform lower bound in (26). It remains to show (28) when f = 0. For that we test (22) with u and integrate by parts to get: 1 1 1 d 2 2 u ds + ν (ln u) ds =0. (34) 2 dt 0 0 Using Poincare inequality and (18) one gets 1 1 |ln(u))| ds (ln(u)) ds. 0 0 Next, using (26) one estimates ln ||u || 2 0 ∞ |ln(u(x, t))| (u − 1) for all (x, t) ∈ Q . ||u || − 1 0 ∞ Combining the last two estimates one gets from (34): 1 t 1 2 2 (u − 1) ds + B (u − 1) dsdt A 0 0 0 with A and B deﬁned as in (27). Here, we have used also the equality 1 1 2 2 u ds = (u − 1) ds +1. 0 0 Whence by comparing y(t):= (u − 1) ds to the solution y ¯(t) of the ODE y ¯ (t)+ By ¯(t)=0, y ¯(0) = A, one obtains 0 y(t) A exp{−Bt} for all t> 0. The last estimate gives (28). 79 Page 10 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP Remark 4.2. Estimates (26) together with (17) show for the original system (1) considered with general initial data and σ = 0 that the height proﬁle stays bounded for all times from above and below, respec- tively, but do not preclude it from approaching zero in inﬁnite time with the rate independent of ν. Note that the upper bound in (26) holds also in the limiting case of (22) considered with ν = 0, which can be shown in this case via an explicit integration. Remark 4.3. We note that estimate (28) implies the exponential asymptotic decay to constant proﬁle (h, v)=(M, 0) for a special class of solutions to original viscous sheet system (1). Indeed, using La- grangian transformation (15) and formula (23) one obtains that any solution to (1) having initial data satisfying h (x) 0x v (x)= −ν and h > 0 (35) 0 0 h (x) should decay to (h, v)=(M, 0) in a ﬁnite time. Relation (35) written in Lagrangian coordinates takes the form: v (x(s, 0)) = − h (x(s, 0)). (36) 0 0s Figures 2 and 3 present numerical conﬁrmation of the convergence to the ﬂat proﬁle for ν =1. In the ﬁrst row, we have taken special initial data h (x(s, 0)) = cos(πs)+ π, v (x(s, 0)) = √ sin(πs), 0 0 π − 1 tan( s)(π − 1) ds 2 x(s, 0) = M = arctan √ (37) h (x(s , 0)) π 0 π − 1 which satisfy (35) and (36). The solution to (1) considered with σ = 0 and initial data deﬁned by (37) converges to the ﬂat proﬁle (h, v)=( π − 1, 0). Next, we slightly modiﬁed the initial data (37)to h (x(s, 0)) = cos(πs)+ π, v (x(s, 0)) = π sin(πs), 0 0 tan( s)(π − 1) ds 2 x(s, 0) = M = arctan √ , (38) h (x(s , 0)) π π − 1 so that in this case f (s)= π cos(πs) 1 − √ =0. (39) π − 1 The second row in Fig. 2 shows the dynamical snapshots of the solution to (1) with initial data given by (38). The solution converges then in a ﬁnite time to the stationary solution of (22) satisfying the ODE: ν = f (s). (40) ss The last ODE with the right hand side (39), ν = 1 and boundary conditions (19) can be integrated to give explicitly u and the corresponding limiting height proﬁle as M 1 h (x (s)) = = −M cos(πs) 1 − √ + C , ∞ ∞ 0 u (s) π − 1 s s ds ds x (s)= M = − , (41) h (x (s )) 1 ∞ ∞ cos(πs ) 1 − + C 0 0 π −1 where C ≈−1.2 is ﬁxed by the conservation of mass laws (3) or equivalently (18). 0 ZAMP Asymptotic decay and non-rupture of viscous sheets Page 11 of 21 79 Fig. 2. Convergence to stationary proﬁles of solutions to (1)with ν =1 and σ = 0 illustrated by several snapshots of the height (left column) and velocity proﬁles (right column). First row: convergence to the ﬂat proﬁle (M, 0) for the initial data (37). Second row: convergence to nonstationary height proﬁle h with v = 0 for the initial data (38). The initial data in ∞ ∞ both cases are depicted by blue solid lines. The red dotted curve corresponds to the analytical formula (41)for h (x)and is superimposed onto the ﬁnal numerical height proﬁle in the second simulation. For obtaining numerical solutions to (1) the numerical code developed in [16, 27] was used Finally, in Fig. 3 we present the numerically calculated decay of ||u(s, t) − 1|| , with u(s, t)= h(x(s, t),t) according to (17), for the solution to (1) with initial data (37). The decay is profoundly exponential with the numerical saturation eﬀect when ||u(s, t) − 1|| becomes small. Comparison with the derived analytical bound (28) shows that the latter is the upper bound but is naturally not sharp. 79 Page 12 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP Fig. 3. Semilog plot (blue markers) of the decay of ||u(s, t) − 1|| corresponding to the solution to (1) with initial data (37). Solid line indicates the upper bound (28)with ||u || = π − 1/(π − 1) and B =1.4252,A =0.0549. For obtaining numerical solutions to (1), the numerical code developed in [16, 27] was used 5. Non-rupture in the case σ> 0. Now, we consider full Eq. (16). We assume without loss of generality σM = 1, by an appropriate rescaling of time variable and ν parameter, and rewrite it in terms of the variable h(s, t):= h(x(s, t),t) using (17) as − + νh = h (h (hh ) ) . tss s s s 2 s The last equation is equivalent to h h 2 3 − + νh = h + h h . (42) tss ss h 2 t ss Note that from the energy and entropy inequalities (20)–(21), using again (17), we have a priori bounds: C C ||h || ∞ 2 and ||h || 2 . (43) s t L (0,T ;L (0,1)) L (Q ) ν ν For showing the second estimate in (43) we have used additionally equivalence of the Euler (1)and Lagrange (42) systems provided by transformation (15) and the uniform upper bound h(s, t) C following from (7). In (43) and below in this section C denotes a constant that may depend on the initial data only, but not on the parameters ν and T of the problem. Using the ﬁrst estimate in (43) one naturally gets the uniform H¨ older continuity in space: 1/2 |h(s ,t) − h(s ,t)| |s − s | . (44) 1 2 1 2 Next, applying [31, Lemma 7.19], in particular (7.32) there, one obtains from (43) 1/4 1/4 |h(s, t ) − h(s, t )| C||h || ∞ 2 ||h || 2 |t − t | |t − t | . (45) 1 2 s L (0,T ;L (0,1)) t L (Q ) 1 2 1 2 3/2 1/2, 1/4 From the last two estimates, we conclude h ∈ C (Q ). Note that a priori estimates (43)–(45)do s,t not depend on smallness of h and, therefore, should be also valid in the vicinity of a rupture point if it happens. ZAMP Asymptotic decay and non-rupture of viscous sheets Page 13 of 21 79 Below we provide a formal analytical argument for the fact that solutions to (42) considered with general initial data cannot rupture in a ﬁnite time. The proof proceeds by contradiction. We will assume that there is a solution to (42) that ruptures in a ﬁnite time. Let us integrate (42) once in time and write it as 2 2 2 3 2 h = νh h − h h + h h dτ − h f (s), (46) t ss ss ss where f (s):= − h (s, 0)/h (s, 0) + νh (s, 0). t ss Close to the expected rupture time t = t we will zoom into the neighbourhood of the pinch-oﬀ point and assume that there the rupture solution to (42) is well approximated by the solution to ⎛ ⎞ 2 2 2 3 ⎝ ⎠ h = νε h − ε h + h h dτ − f (s) , (47) t ss ss ss considered now in the whole real line s ∈ R via a suitable extension of h to a small constant proﬁle outside of the pinch-oﬀ region. Additionally, we will assume that the following analogues of a priori bounds (43)–(45) ||h || ∞ 2 , (48) L (0,T ;L (R )) C C 1/2 1/4 |h(s ,t) − h(s ,t)| |s − s | , |h(s, t ) − h(s, t )| |t − t | , 1 2 1 2 1 2 1 2 3/2 ν ν for all s ,s ∈ R and t ,t ∈ [0,t (ε)), (49) 1 2 1 2 r still hold for the approximating solution to (47), where constant C does not depend on ε.Next, by applying an appropriate time shift we assume that the initial data for (47) satisﬁes the uniform spatial bounds: 0 <ε h(s, 0) = h (s) Cε with C> 1. (50) Note, that the above time shift implies also that the ﬁnite rupture time of the solution to (47) t = t (ε) →0as ε → 0. (51) r r Naturally, for the small initial data (50) and times close to t = t (ε) one would expect the solution of approximate equation (47) stay close to the rupture solution to the original equation (46). Finally, we will make a technical assumption that the approximate solution to (47) satisfying (50) stays of the same order smallness up to the rupture time, i.e. h(s, t) Cε for t ∈ [0,t (ε)). (52) Let us now write the exact integral representation of solutions to (47): h(s, t)= K (s − ξ, t)h (ξ)dξ ν,ε 0 −∞ ⎛ ⎞ t ∞ τ 2 2 3 ⎝ ⎠ − ε K (s − ξ, t − τ ) h + h h dτ ˜ + f (s) dξdτ, (53) ν,ε ξξ ξξ −∞ 0 0 where K (s, t) denotes the heat kernel: ν,ε 1 s K (s, t):= √ exp − ν,ε 4νε t 4πνε t 79 Page 14 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP having the well-known property ∂K ∂ K ν,ε ν,ε = νε . ∂t ∂ξ Using assumption (50) one can show that the following estimate from below for the ﬁrst integral in (53): ∞ ∞ K (s − ξ, t)h (ξ)dξ exp[−r ]dr = ε. (54) ν,ε 0 −∞ −∞ In the rest of the proof we will estimate the second integral in (53) from above and show that contributions from the nonlinear curvature term become smaller then the right hand side of (54) when ε is suﬃciently small and satisﬁes the relation (62) below. First, one estimates t ∞ ∞ 2 2 2 2 ε K (s − ξ, t − τ )f (s)dξdτ ε ||f (s)|| √ exp[−r ]dr = ε t||f (s)|| . (55) ν,ε ∞ ∞ 0 −∞ −∞ Next, one shows using several times integration by parts that t ∞ τ 2 2 3 ε K (s − ξ, t − τ ) h + h h dτ ˜dξdτ ν,ε ξξ ξξ −∞ 0 0 t ∞ τ 1 ∂K (s − ξ, t − τ ) h ν,ε 2 3 = − h + h h dτ ˜dξdτ ξ ξξ ν ∂τ 2 0 −∞ 0 t ∞ 1 h 2 3 = K (s − ξ, t − τ ) h + h h dξdτ ν,ε ξξ ν 2 0 −∞ t ∞ t ∞ 5 1 ∂K (s − ξ, t − τ ) ν,ε 2 2 4 = − K (s − ξ, t − τ )h h dξdτ − h dξdτ ν,ε 2 2 2ν 4ν ε ∂τ 0 −∞ 0 −∞ t ∞ t ∞ 5 1 h 2 2 = − K (s − ξ, t − τ )h h dξdτ − K (s − ξ, t − τ ) dξdτ ν,ε ν,ε 2 2 2ν 8ν ε (t − τ ) 0 −∞ 0 −∞ t ∞ 4 2 1 h (s − ξ) + K (s − ξ, t − τ ) dξdτ ν,ε 2 2 2 4ν ε (t − τ ) 4νε (t − τ ) −∞ =: I + I + I . (56) 1 2 3 Let us estimate consequently the integrals I ,i =1, 2,3in(56) at the special point s = s , where the i r rupture occurs at time t = t . At this point, using the H¨ older estimates (49) one obtains C C C 1/2 1/4 1/2 h(ξ, τ ) h(s ,τ)+ |ξ − s | |t − τ | + |ξ − s | . (57) r r r r 3/2 ν ν ν First, using (48) and (52) one estimates at t = t (ε) t ∞ t ∞ 5 Cε dτ Cε t 2 2 2 |I | = K (s − ξ, t − τ )h h dξdτ √ √ h dξ √ . (58) 1 ν,ε r ξ ξ 3 3 2ν t − τ ν ν 0 −∞ 0 −∞ ZAMP Asymptotic decay and non-rupture of viscous sheets Page 15 of 21 79 Next, using (57) and (52) one obtains t ∞ 1 h (ξ, τ ) |I | = K (s − ξ, t − τ ) dξdτ 2 ν,ε r 2 2 8ν ε (t − τ ) −∞ t ∞ Cε h(ξ, τ ) K (s − ξ, t − τ ) dξdτ ν,ε r ν (t − τ ) 0 −∞ t ∞ Cε 1 1 K (s − ξ, t − τ)dξdτ ν,ε r 2 3/4 3/2 ν (t − τ ) ν 0 −∞ t ∞ 2 1/4 Cε 1 (4νε ) |ξ − s | + K (s − ξ, t − τ ) dξdτ ν,ε r 2 3/4 ν ν (t − τ ) (t − τ )4νε 0 −∞ 2 1/4 1/4 Cε 1 1 (4νε ) Cεt + dτ for ε< 1and ν< 1. (59) 2 3/4 3/2 7/2 ν (t − τ ) ν ν ν Finally, proceeding analogously as in (59) one shows that 1/4 Cεt |I | (60) 7/2 Therefore, combining (58)–(60) with (51) and taking ε so small that t (ε) <C ν (61) r 1 one obtains that |I + I + I | . 1 2 3 The last estimate combined with (55) and (54) implies by (53) that for all suﬃciently small ε> 0 h(s ,t) for all t ∈ [0,t (ε)], r r which contradicts to the initial assumption of the rupture of the solution to (47) at the ﬁnite time t = t (ε). Remark 5.1. We note, a posteriori, that the estimate on the rupture time (61) together with the second estimate in (49) and assumption on the initial data (50) imply that 1/4 Ct (ε) 0 <ε h(s , 0) Cν . (62) 3/2 This condition on the minimal height at which the viscous term starts to dominate the curvature term is consistent with both the self-similar scalings [1] and numerical observations. 6. Discussion This article presents several interesting analytical results on the qualitative behaviour of solutions to viscous sheet system (1) and its Lagrangian counterpart (16). It poses also several interesting open questions summarised below: 79 Page 16 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP 1 2 • Theorem 2.1 allows for (H ,L ) weak solutions with non-negative initial data. It would be interesting to investigate implications of estimates (9)–(10) for possible analytical bounds from above and below on the support of such solutions in time. In particular, one could characterise a class of initial data with support less then (0, 1) with solution to (1) having full support (0, 1) for all positive times. i.e. demonstrating the immediate initial healing of the dry spots. • Theorem 4.1 shows exponential decay to constant proﬁle for solutions to (1) with σ = 0 and special initial data (35), see Remark 4.4. Numerical solutions of system (1) indicate that the class of initial data for which similar results hold can be extended (see Fig. 2, second row). Also in the case f =0 in (22) the ﬁnite decay should be then not to the constant proﬁle u = 1, but rather to the spatially inhomogeneous solution to stationary Eq. (40). In this context, it is interesting to point out the existence of the Lie-B¨ acklund transform, which maps Eq. (22) into the classical heat equation. Explicit connections between solutions to these two equations were derived in [34]. • We should point out the diﬀerence of the results of Theorem 4.1 to known ones for Eq. (22) considered on the whole real line R . In particular, in this case non-existence of L solutions and extinction phenomena (u → 0) in the ﬁnite time were proven (see [32, 35] and references therein). In contrast, while considering (22) on the ﬁnite interval (0, 1) and analysing convergence to u = 1 instead, we were able to show existence of the unique strong positive solution to (22) and to escape from these singularities. Also we believe that the exponential decay shown in estimate (28) cannot be improved to the ﬁnite time decay for the same reason. Indeed, linearising Eq. (22)at u = 1 results in the classical heat equation for u ¯ = u − 1: u ¯ = νu ¯ , t ss which suggests that close to u = 1 solutions to (22) at most exponentially decay in inﬁnite time. • The analytical argument of Sect. 5 shows the right lower bound (5) for the minimum of the height in the regime ν 1, see Remark 5.1 and Fig. 1. In order to make this argument rigorous, we suggest that a comparison principle between solutions of (46) and the approximate Eq. (47) should be proposed. At the moment, it is not obvious as both equations are nonlinear parabolic of the fourth order. • Theorem A.2 in contrast to Theorem 2.1 relies on the uniform bound from below m in (A.6)for the radially symmetric solutions to system (A.2). In the spirit of this study, it would be interesting to provide an analytical argument, similar to the one in Sect. 5 for the one-dimensional system (1), showing that solutions to (A.2) cannot rupture in a ﬁnite time. Alternatively, one could think on possible analytical estimates for m depending on the initial data (h ,v ). Such results would have 0 0 an important implication on the point rupture in 2D viscous sheets considered in [14]. Acknowledgements The authors would like to thank Jens Eggers for pointing to them out the low bound (5) for the solutions to (1) as well as for valuable comments on the results of this study. MAF was supported by grant MTM2017-89423-P. GK would like to acknowledge support from Leverhulme grant RPG-2014-226. GK gratefully acknowledges the hospitality of ICMAT during a research visit to Madrid. RT would like to acknowledge support by the Ministry of Education and Science of Ukraine, grant number 0118U003138. Part of this research was performed during participation of authors at the thematic research programme “Nonlinear Flows” of the Erwin Schr¨ odinger International Institute for Mathematics and Physics. Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International Li- cense (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. ZAMP Asymptotic decay and non-rupture of viscous sheets Page 17 of 21 79 A Asymptotic decay for the radially symmetric solutions In 2D space, the viscous sheet system (1) has the form [14]: v +(v ·∇)v = σ∇Δh + ∇· (h[∇v +(∇v) +2(∇· v)I]), (A.1a) h = −div (hv) , (A.1b) where v ∈ R is the velocity in the plane (x, y) ∈ B (0), where B (0) is the unit disc and σ, μ denote 1 1 again surface tension and viscosity. Here, we are interested in the radial symmetric solutions to (A.1)havingthe form: 2 2 h(x, y, t)= h(r, t), v(x, y, t)= v(r, t)e with r = x + y ∈ (0, 1). Under the last ansatz system (A.1) reduces to [14]: 1 4μ h vh v + vv = σ [rh ] + (rv) − , (A.2a) t r r r r r h r 2r r r (rhv) h = − . (A.2b) We consider (A.2) in the domain (0,T ) × (0, 1) with boundary conditions h (0,t)= h (1,t)= v(0,t)= v(1,t) = 0 (A.3) r r which together with (A.2b) imply the mass conservation: 1 1 h(r, t)r dr = h (r)r dr = M. (A.4) 0 0 We ﬁrst establish the energy and entropy estimates for system (A.2). Lemma A.1. For classical positive solutions to problem (A.2)–(A.3), the following energy and entropy inequalities hold: ⎡ ⎤ 1 1 2 2 2 hv |h | h|(vr) | r r ⎣ ⎦ + σ r dr −2μ dr , dt 2 2 r 0 0 (A.5a) # $ 1 1 1 2 2 2 h h |h | |(rh ) | C hv r r r r v +4μ + σ r dr −2μσ dr + dr, dt 2 h 2 r μσm r 0 0 0 (A.5b) provided h(r, t) m> 0 for all t ∈ [0,T ). (A.6) Proof of Lemma A.1. Let us multiply Eq. (A.2a)by hur and integrate it in space. After applying inte- gration by parts several times and using Eq. (A.2b) one obtains the equality ⎡ ⎤ 1 1 1 1 2 2 2 hv |h | hv d 2 ⎣ ⎦ + σ r dr = −4μ h|v | r dr + dr + hvv dr . (A.7) r r dt 2 2 r 0 0 0 0 79 Page 18 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP Note that the last integral in (A.7) can be bounded by ⎡ ⎤ 1 1 1 hv 1 2 ⎣ ⎦ − hvv dr h|v | r dr + dr . r r 0 0 0 This together with (A.7) imply (A.5a). To show (A.5b) we multiply Eq. (A.2a)by rh and integrate it in space. The left hand side of (A.2a) produces then: 1 1 1 1 (v + v )h r dr = vh r dr − vh r dr + vv h r dr t r r r rt x r dt 0 0 0 0 1 1 1 (hvr) = vh r dr − (vr) dr + vv h r dr r r x r dt 0 0 0 1 1 1 hv d 2 = vh r dr − h|v | r dr − dr r r dt 0 0 0 hv σ = vh + + |h | r dr, (A.8) r r dt 8μ 8μ where in the second equality above we used Eq. (A.2b). Next, for the curvature term at the right hand side of (A.2a) one obtains 1 1 1 1 [rh ] h r dr = − |[rh ] | dr. (A.9) r r r r r r r 0 0 In turn, for the viscous term at the right hand side of (A.2a) one obtains 1 1 1 rh h vh rh h v|h | r r r r (rv) − dr = − (rv) dr − dr r r h r 2r h r h r r 0 0 0 1 1 1 rh (rhv) rh v|h | r r r r = − dr + h v dr − dr h r h h r r 0 0 0 1 1 1 rh (rhv) h v|h | r r r r = dr + r h v dr + dr h r h h r r 0 0 0 1 1 1 2 2 h h |h | v|h | tr r r r 1 1 = − r dr − (rhv) dr + dr 2 2 h h h 0 0 0 1 1 1 2 2 h h |h | v|h | tr r r r 1 1 = − r dr + h r dr + dr 2 2 2 h h h 0 0 0 1 1 2 2 |h | v|h | r r 1 d 1 = − r dr + dr, (A.10) 2 dt 2 h h 0 0 ZAMP Asymptotic decay and non-rupture of viscous sheets Page 19 of 21 79 where in the fourth and ﬁfths equalities above we have used (A.2b) again. Next, we estimate the second integral at the right hand side of (A.10) using (A.5a) and (A.6)as ⎛ ⎞ ⎛ ⎞ 1/2 1/2 1 1 1 2 2 v|h | 1 hv ⎝ ⎠ ⎝ ⎠ dr √ dr |h | r dr ||h || , r r ∞ h r 0 0 0 ⎛ ⎞ ⎛ ⎞ 1/2 1/2 1 1 2 2 C hv |(rh ) | r r ⎝ ⎠ ⎝ ⎠ dr dr r r 0 0 1 1 2 2 C hv |(rh ) | r r dr +2μσ dr, (A.11) 8μσm r r 0 0 where in the second inequality above we have used the estimate ⎛ ⎞ 1/2 |(rh ) | r r ⎝ ⎠ ||h || dr . r ∞ Finally, by combining estimates (A.8)–(A.11) one arrives at the entropy inequality (A.5b). Similar to the one-dimensional case (Theorem 2.1) the energy and entropy inequalities (A.5a)–(A.5b) allow us to show the global exponential asymptotic decay, but now only for positive classical solutions to (A.2) having globally lower bound (A.6). We denote radial energy and entropy functionals as % & 1 2 S(u, h):= h v +4μ + σ|h | r dr, ' ( 1 2 2 E(u, h):= hv + σ|h | r dr. 1 2 Theorem A.2. (asymptotic exponential decay) Assume that initial data (h ,v ) ∈ H (0, 1) × L (0, 1), 0 0 σ> 0 and μ> 0 and h(r, t) m> 0 for all t> 0. (A.12) Then, there exist positive constants A ,B ,i =1, 2 depending only on E(h ,v ),S(h ,v ) and parame- i i 0 0 0 0 ters σ, μ and m such that −B t −B t 1 2 ||h − M || 1 A e , and ||v|| 2 A e for all t 0. (A.13) 1 2 H (B (0)) L (B (0)) 1 1 Proof of Theorem A.2. Using the Poincar´ e inequality 2 1 2 2 |∇h| dxdy |D h| dxdy, ∇h · n| =0, ∂B (0) B (0) B (0) 1 1 which reduces in the case of the radial symmetric function h(x, y, t)= h(r, t)to 1 1 |(rh ) | r r 2 1 |h | r dr dr, 0 0 79 Page 20 of 21 M. A. Fontelos, G. Kitavtsev and R. M. Taranets ZAMP from the energy and entropy inequalities (A.5a)–(A.5b) we ﬁnd that 1 T 1 μσ σ 2 2 C |h | r dr + |h | r drdt S(v ,h )+ . 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Taranets Institute of Applied Mathematics and Mechanics of the NASU Dobrovolskogo Str. 1 84100 Sloviansk Ukraine (Received: December 7, 2017; revised: May 6, 2018)
Zeitschrift für angewandte Mathematik und Physik – Springer Journals
Published: May 28, 2018
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