DSpace at VNU: Existence of traveling waves in elastodynamics with variable viscosity and capillarity tài liệu, giáo án,...
Nonlinear Analysis: Real World Applications 12 (2011) 236–245 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Existence of traveling waves in elastodynamics with variable viscosity and capillarity Mai Duc Thanh ∗ Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam article abstract info Article history: Received 17 May 2010 Accepted June 2010 Motivated by our earlier works, Thanh (2010) [3,4], we study the global existence of traveling waves associate with a Lax shock of a model of elastodynamics where the viscosity and capillarity are functions of the strain The system is hyperbolic and may not be genuinely nonlinear The left-hand and right-hand states of a Lax shock correspond to a stable node and a saddle point By defining a Lyapunov-type function and using its level sets, we estimate the attraction domain of the stable node Then we show that the saddle point lies on the boundary of the attraction domain of the stable node Moreover, exactly one stable trajectory enters this attraction domain This gives a stable-to-saddle connection for 1-shocks (a saddle-to-stable connection for 2-shocks), and therefore defines exactly one traveling wave connecting the two states of the Lax shock © 2010 Elsevier Ltd All rights reserved Keywords: Conservation law Traveling wave Shock wave Lax shock inequalities Elastodynamics Viscosity Capillarity Diffusion Dispersion Equilibria Lyapunov function Attraction domain Introduction We are interested in the global existence of traveling waves associated with a Lax shock for the general model of nonlinear elastodynamics and phase transitions with a nonlinear viscosity and capillarity The model consists of the conservation law of momentum and the continuity equation in elastodynamics describing the longitudinal deformations of an elastic body with negligible cross-section with variable nonlinear viscosity µ(w) and variable nonlinear capillarity λ(w): w2 ∂t v − ∂x σ (w) = λ′ (w) x − (λ(w)wx )x + (µ(w)vx )x , x (1.1) ∂t w − ∂x v = Here, the unknown v and w > −1 represent the velocity and deformation gradient (the strain), respectively The constraint w > −1 follows from the principle of impenetrability of matter, however, it is useless in this paper and we therefore not impose this condition The stress σ = σ (w) is a function of the strain w The function µ(w) characterizes the viscosity inducing diffusion effect and the function λ(w) represents the positive capillarity inducing dispersion effect The reader is referred to LeFloch [1] for the derivation of the model (1.1) Throughout, the stress function σ is assumed to be differentiable and σ′ > such that the corresponding system of conservation laws without viscosity and capillarity ∗ Tel.: +84 2211 6965; fax: +84 3724 4271 E-mail addresses: mdthanh@hcmiu.edu.vn, hatothanh@yahoo.com 1468-1218/$ – see front matter © 2010 Elsevier Ltd All rights reserved doi:10.1016/j.nonrwa.2010.06.010 (1.2) M.D Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245 ∂t v − ∂x σ (w) = 0, ∂t w − ∂x v = 0, 237 (1.3) is strictly hyperbolic, see [2], for example In addition, the continuous viscosity and smooth capillarity are required to satisfy the conditions µ(w) > κ, λ(w) > 0, |λ′ (w)| < m for all w, where m > 0, κ > are constants (1.4) In our recent paper [3], the global existence of traveling waves of a single conservation law with constant viscosity and capillarity was established In this work, we propose a method of estimating attraction domain for the stable node of the resulted differential equations The argument of this method is completed by our second work [4], where by considering an isothermal fluid with nonlinear diffusion and dispersion coefficients, we went further by proving that the saddle point is in fact lying on the boundary of the stable node Moreover, we also pointed our that exactly one stable trajectory of the differential equations leaves the saddle point and enters the domain of attraction of the stable node This gives a complete description of a method of estimating the attraction domain for several systems of conservation laws with viscosity and capillarity effects Precisely, the method consists of several steps: Step Derive a system of nonlinear first-order differential equations corresponding to the given shock; Step Showing that one state of the shock corresponds to a stable node, the other state corresponds to a saddle point of the system of differential equations given by Step 1; Step Define a suitable Lyapunov-type function for the stable node obtained in Step Use the level sets of this function to estimate the domain of attraction based on certain invariant properties of the level sets; Step Point out that there is one trajectory leaving the saddle and entering the attraction domain obtained in Step This gives a traveling wave connecting the two states of the given shock Besides, the rate of change of a small quantity need not be small! In particular, if one can speak of small variable nonlinear viscosity µ(w) and capillarity λ(w), their effects may not be negligible when the rates of change µ′ (w), λ′ (w) are quite large Our goal in this paper is to show that the above method still works even for the general model (1.1) Clearly, the arguments in [3,4] have to be reconsidered or improved Many important contributions for the study of traveling waves for viscous–capillary models such as (1.1) have been carried out by Hayes and LeFloch [5], then by Bedjaoui and LeFloch [6–8], and a recent joint work by Bedjaoui et al [9] Traveling waves for diffusive–dispersive scalar equations were earlier studied by Bona and Schonbek [10], Jacobs et al [11] Traveling waves of the hyperbolic–elliptic model of phase transition dynamics were also studied by Slemrod and Fan [12–15], Shearer and Yang [16] Shock waves and entropy solutions of the hyperbolic system of conservation laws such as (1.3) were considered in [2,17–19] When the cross-section is taken into account, the Lax shocks and moreover the Riemann problem for the model of fluid flows in a nozzle with variable cross-section and the shallow water equations were considered by LeFloch and Thanh [20,21], Kröner et al [22] and Thanh [23] See also the references therein for related works This paper is organized as follows In Section we recall the basic properties of Lax shocks of the system (1.3) and the derivation of a system of ordinary first-order differential equations obtained by substituting a traveling wave to the viscous–capillary model (1.1) We then point our that given a Lax shock of (1.3), there corresponds two equilibria of the system of differential equations in which one is a stable node, and the other is a saddle In Section we estimate the domain of attraction of the stable node which is large enough such that the saddle point belongs to the boundary of this domain In Section we establish the existence of traveling waves by indicating that exactly one trajectory leaves the saddle point and enters the domain of attraction of the stable node Finally, we also include in Section some numerical illustrations for the traveling waves Basic concepts and results on Shock waves and traveling waves 2.1 Hyperbolicity and Shock waves of (1.3) The Jacobian matrix of the system (1.3) is given by A(v, w) = −1 −σ ′ (w) which gives the characteristic equation det(A − λI ) = λ2 − σ ′ (w) = Since σ ′ (w) > 0, the Jacobian matrix A admits two real and distinct eigenvalues λ1 (v, w) = − σ ′ (w) < < λ2 (v, w) = σ ′ (w) A discontinuity of (1.3) connecting two given states u− = (v− , w− ), u+ = (v+ , w+ ) with the propagation speed of discontinuity s is a weak solution of (1.3) of the form u(x, t ) = (v− , w− ) (v+ , w+ ) if x < st , if x > st , 238 M.D Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245 and satisfies the Rankine–Hugoniot relations s(v+ − v− ) + (σ (w+ ) − σ (w− )) = 0, s(w+ − w− ) + (v+ − v− ) = (2.1) The Eqs (2.1) yield s2 = σ (w+ ) − σ (w− ) w+ − w− An admissible Lax shock, or a Lax shock for short, connecting the left-hand and the right-hand states u− = (v− , w− ) and u+ = (v+ , w+ ), respectively, with the shock speed s = s(u− , u+ ) is a discontinuity of (1.3) satisfying the Lax shock inequalities λi (u+ ) < s(u− , u+ ) < λi (u− ), i = 1, (2.2) 2.2 Traveling waves of (1.1) Let us now turn to traveling waves We call a traveling wave of (1.1) connecting the left-hand state (v− , w− ) and the right-hand state (v+ , w+ ) a smooth solution of (1.1) of the form (v, w) = (v(y), w(y)), y = x − st where s is a constant, and satisfies the boundary conditions lim (v, w)(y) = (v± , w± ) y→±∞ lim y→±∞ d dy (v(y), w(y)) = lim y→±∞ d2 dy2 (2.3) (v(y), w(y)) = (0, 0) Substituting (v, w) = (v, w)(y), y = x − st into (1.1), and re-arranging terms, we get sv ′ + (σ (w))′ = λ′ (w) w ′2 + λ(w)w′′ ′ − (µ(w)v ′ )′ , sw ′ + v ′ = 0, where (.)′ = d(.)/dy Integrating the last equations on the interval (−∞, y), using the boundary conditions (2.1), we obtain w s(v − v− ) + (σ (w) − σ (w− )) = λ′ (w) + λ(w)w′′ − µ(w)v ′ , s(w − w− ) + (v − v− ) = ′2 By letting y → +∞, we can see that s and (v± , w± ) satisfy the Rankine–Hugoniot relations (2.1) Substituting v − v− = −s(w − w− ), v ′ = −sw ′ from the second equation in (2.1) into the second one, we obtain a second-order differential equation for the unknown function w : −s2 (w − w− ) + (σ (w) − σ (w− )) = λ′ (w) w ′2 + λ(w)w ′′ + sµ(w)w ′ or w′′ = − λ′ (w) ′2 sµ(w) ′ σ (w) − σ (w− ) − s2 (w − w− ) w − w + 2λ(w) λ(w) λ(w) (2.4) Setting z = w′ , h(w) = s2 (w − w− ) − (σ (w) − σ (w− )), we reduce the second-order differential equation (2.4) to the following × system of first-order differential equations w′ = z , z′ = − z 2λ(w) (λ′ (w)z + 2sµ(w)) − h(w) λ(w) (2.5) , or, in a more compact form dU dy = F (U ), −∞ < y < +∞ (2.6) where U = (w, z ), F (U ) = z, − z 2λ(w) (λ′ (w)z + 2sµ(w)) − h(w) λ(w) The above argument reveals that a point U in the (w, z )-phase plane is an equilibrium point of the autonomous differential equations (2.6) if and only if U = (w± , 0), where w± and the shock speed s are related by (2.1) M.D Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245 239 Since h(w± ) = 0, the Jacobian matrix of the system (2.6) is given by ′ DF (w± , 0) = σ (w± ) − s λ(w± ) − sµ(w± ) (2.7) λ(w± ) The characteristic equation of DF (v± , 0) is then given by = 0, sµ(w± ) − − β λ(w± ) −β |DF (w± , 0) − β| = σ ′ (w± ) − s2 λ(w ) ± or sµ(w± ) β2 + λ(w± ) β+ s2 − σ (w± ) λ(w± ) = (2.8) Assume that the jump satisfies the Lax shock inequalities λ2 (w− ) > s > λ2 (w+ ) which yields σ (w− ) > s2 > σ (w+ ) (2.9) If s > 0, we can see that the characteristic equation |DF (w− , 0) − β| admits two real roots with opposite sign, and that the characteristic equation |DF (w+ , 0) − β| admits two roots with negative real parts There are similar arguments for the case s < This leads us the the following conclusions Proposition 2.1 (i) Given a Lax shock associate with λ1 with the left-hand and right-hand states u− = (v− , w− ), u+ = (v+ , w+ ) and the shock speed s = s1 (u+ , u− ) Then, the point (w− , 0) is an asymptotically stable node, and the point (w+ , 0) is a saddle of (2.5) (ii) Given a Lax shock associate with λ2 with the left-hand and right-hand states u− = (v− , w− ), u+ = (v+ , w+ ) and the shock speed s = s2 (u+ , u− ) Then, the point (w+ , 0) is an asymptotically stable node, and the point (w− , 0) is a saddle of (2.5) Proposition 2.1 indicates that given a Lax shock, there is possibly a stable-to-saddle or saddle-to-stable connection Whenever such a connection is established, we obtain a traveling wave associated with the given Lax shock Estimating the attraction domain 3.1 Assumptions and examples Given a Lax 2-shock connecting the left-hand and right-hand states u− = (v− , w− ) and u+ = (v+ , w+ ), respectively, with the shock speed s = s(u− , u+ ) Suppose for definitiveness that w+ < −w− Throughout, we assume the following hypotheses (H1) The values w± satisfy |w+ − w− | ≤ 2κ , (3.1) m where κ, m are positive constants as in (1.4) (H2) There exists a value ν < w+ such that w− ∫ ν h(ξ ) λ(ξ ) dξ < 0, ∫ w and w+ h(ξ ) dξ > 0, λ(ξ ) w ∈ [ν, w− ], (3.2) where h(w) = s2 (w − w− ) − (σ (w) − σ (w− )) Example 3.1 We omit the condition w > −1, and extend the function σ = σ (w) to the whole −∞ < w < +∞ Let the function σ be twice differentiable and strictly convex: σ ′′ (w) > 0, w ∈ R Then the Lax shock inequalities λ2 (w+ ) < s(u+ , u− ) < λ2 (w− ) 240 M.D Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245 are equivalent to the condition w− > w+ And w− ∫ −∞ h(ξ ) dξ = −∞ λ(ξ ) Thus, for any pair (w+ , w− ), there is always such a ν satisfying (H2) Example 3.2 Let us take the model of elastodynamics described by the Eqs (1.1), where the stress σ is a twice differentiable function of w satisfying σ ′ (0) > 0, wσ ′′ (w) > for w ̸= 0, (which implies σ ′ (w) > for all w > −1) and lim σ (w) = −∞, lim σ ′ (w) = +∞ w→+∞ w→−1 (See [2]) Assume λ(w) ≡ λ = constant Then, as argued similarly as in [2], we can see that for each w− > 0, there exists exactly one value denoted by ϕ∞ (w− ) < such that s2 (w− , ϕ∞ (w− ))(ϕ∞ (w− ) − w− ) − (σ (ϕ∞ (w− )) − σ (w− )) = and that s2 (w− , w)(w − w− ) − (σ (w) − σ (w− )) < if w < ϕ∞ (w− ), where s2 (w− , w) = σ (w) − σ (w− ) w − w− Moreover, for each pair (w0 , w1 ), there is exactly one value denoted by ϕ # (w0 , w1 ) such that s2 (w0 , w1 ) = s2 (w0 , ϕ # (w0 , w1 )) Setting w∗ = ϕ # (w− , ϕ∞ (w− )) := ζ (w− ), and taking w+ such that ζ (w− ) < w+ < w− , and defining ν = ϕ # (w− , w+ ), we can see that the first condition of (H2) holds Moreover, since in this case the Lax shock inequalities are equivalent to the Liu entropy conditions, the second condition of (H2) also holds 3.2 Lyapunov-type function Under the hypotheses (H1) and (H2), we now consider the autonomous system obtained from the previous section dw dy dz = z, =− z (λ′ (w)z + 2sµ(w)) − h(w) , dy 2λ(w) λ(w) Let us define a Lyapunov-type function candidate L(w, z ) = ∫ w h(ξ ) w+ λ(ξ ) dξ + z2 −∞ < y < +∞ (3.3) (3.4) The following lemma indicates that the function L defined by (3.7) is a Lyapunov-type function Lemma 3.1 Setting D = (ν, w− ) × {|z | < 2sκ m } ∋ (w+ , 0) Under the hypotheses (H1), (H2), it holds that L(w+ , 0) = 0, L(w, z ) > 0, for (w, z ) ∈ D \ {(w+ , 0)}, L˙ (w, z ) < in D \ {z = 0}, L˙ (w, z ) = on D ∩ {z = 0} Proof First, we have immediately L(w+ , 0) = 0, L(w, z ) ≥ ∫ w w+ h(ξ ) dξ > 0, λ(ξ ) (w, z ) ∈ D, w ̸= w+ , (3.5) M.D Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245 241 which establishes the first line of (3.5) Second, the derivative of L along trajectories of (3.3) can be estimated as follows ˙L(w, z ) = ∇ L(w, z ) · dw , dz dy dy h(w) z h(w) = , z z, − (λ′ (w)z + 2sµ(w)) − λ(w) 2λ(w) λ(w) =− 2ε > 0, we set (w − w+ )2 z2 + ≤ 1, w ≥ w+ (w+ − (w− − ε))2 (M (w+ − (w− − ε)))2 z2 (w − w+ )2 + ≤ , w ≤ w ∪ (w, z ) | + , (w+ − ν)2 (M |w+ − (w− − ε)|)2 Gε = (w, z ) | (3.8) where M is given by (3.7) and ν is defined in (3.2) Then, it is not difficult to check that Gε ⊂ [ν, w− ) × |z | ≤ 2sκ m Now, we claim that the function ∫ w h(ξ ) w+ λ(ξ ) dξ is strictly increasing for w near w− , w ≤ w− Indeed, the Lax shock inequalities (2.2) imply that there exists a positive number < θ < |w− − w+ | such that σ (w) − σ (w− ) > s2 for |w − w− | < θ w − w− This implies that for |w − w− | < θ , h(w) > 0, which establishes the statement Fix this θ > 0, it then holds that L(w− , 0) > L(w− − ε, 0), < ε < θ (3.9) The following lemma provides us with properties of the sets Gε Lemma 3.2 For any positive number ε so that < 2ε < θ < w− − w+ , where θ is given in (3.9), let Gε be the set defined by (3.8) and let ∂ Gε denote its boundary It holds that (w,z )∈∂ Gε L(w, z ) = L(w− − ε, 0) Moreover, the minimum value in (3.9) is strict, i.e L(w, z ) > L(w− − ε, 0), for all (w, z ) ∈ ∂ Gε \ {(w− , 0)} (3.10) Proof We need only to establish the second statement, i.e., (3.10), since the first statement is a consequence of (3.10) On the semi-ellipse ∂ Gε , w ≤ w+ , one has z = M (|w+ − (w− − ε)|2 − (w − w+ )2 ) 242 M.D Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245 Thus, along this left semi-ellipse, it holds that L(w, z )|(w,z )∈∂ Gε ,w≤w+ = ∫ w h(ξ ) M2 (|w+ − (w− − ε)|2 − (w − w+ )2 ) λ(ξ ) := g (w), w ∈ [w+ , w− − ε] w+ dξ + Then, it holds for any w ∈ (w+ , w− − ε) that dg (w) dw h(w) − M (w − w+ ) σ (w) − σ (w+ ) 2 s − −M = (w − w+ ) λ(w) w − w+ s − σ ′ (ξ ) = (w − w+ ) − M , w+ < ξ < w, λ(w) < = λ(w) The function g is therefore strictly decreasing for w ∈ [w+ , w− − ε] and attains its strict minimum on this interval at the end-point w = w− − ε , i.e L(w, z ) > L(w− − ε, 0), for all (w, z ) ∈ ∂ Gε \ {(w− − ε, 0)}, w− − ε ≥ w ≥ w+ Arguing similarly, we can see that L(w, z ) > L(ν, 0), for all (w, z ) ∈ ∂ Gε \ {(ν, 0)}, ν ≤ w ≤ w+ The last two inequalities and (3.2) establish (3.10) The proof of Lemma 3.2 is complete Properties of the level sets of the Lyapunov-type function (3.5) can be seen in the following lemma Lemma 3.3 Under the assumptions and the notations of Lemma 3.2, the set Ωε := {(w, z ) | L(w, z ) ≤ L(w− − 2ε, 0)} (3.11) is a compact set, lies entirely inside Gε , positively invariant with respect to (3.3), and has the point (w+ , 0) as an interior point As a consequence, the initial-value problem for (3.3) with initial condition (u(0), v(0)) = (w0 , w0 ) ∈ Ωε admits a unique global solution (w(y), z (y)) defined for all y ≥ Moreover, this trajectory converges to (w+ , 0) as y → +∞, i.e., lim (w(y), z (y)) = (w+ , 0) y→+∞ This means that the equilibrium point (w+ , 0) is asymptotically stable and Ωε is a subset of the domain of attraction of (w+ , 0) Proof Evidently, Ωε is a compact set We claim that the set Ωε is in the interior of Gε Assume the contrary, then there is a point U0 ∈ Ωε ∩ ∂ Gε Then, as seen in Lemma 3.2, the minimum of L over ∂ Gε is attained at w = w− − ε , so L(U0 ) ≥ L(w− − ε, 0) > L(w− − 2ε, 0) which is a contradiction, since U0 ∈ Ωε , L(U0 ) ≤ L(w− − 2ε, 0) Thus, the closed curve L(w, z ) = L(w− − 2ε, 0) lies entirely in the interior of Gε Moreover, it is derived from Lemma 3.1 that dL(w(y), z (y)) dy ≤ Thus, L(w(y), w(y)) ≤ L(w(0), z (0)) ≤ L(w− − 2ε, 0), ∀y > The last inequality means that any trajectory starting in Ωε cannot cross the closed curve L(w, z ) = L(w− − 2ε, 0) Therefore, the compact set Ωε is positively invariant with respect to (4.3) As known in the standard existence theory of differential equations, (3.3) has a unique solution for y ≥ whenever U (0) ∈ Ωε On the other hand, we set E = {(w, z ) ∈ Ωε | L˙ (w, z ) = 0} = {(w, z ) ∈ Ωε | z = 0} It is derived from LaSalle’s invariance principle that every trajectory of (3.3) starting in Ωε approaches the largest invariant set M of E as y → ∞ Thus, to complete the proof, we need only to point out that M = {(w+ , 0)} This can be done by proving that no solution can stay identically in E, except the trivial solution (w, z )(y) ≡ (w+ , 0) Indeed, let (w, z ) be a solution that stays identically in E Then, dw(y) dy = z (y) ≡ 0, M.D Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245 243 which implies w ≡ w− , since (w− , 0) is the unique equilibrium point in Ωε Thus, every every trajectory of (2.6) starting any point in Ωε must approach (w− , 0) as y → ∞ The proof of Lemma 3.3 is complete It follows from Lemma 3.3 that the union Ω = ∪0 w− , then only the stable trajectory goes into Q2 may converge to the stable node And we will show that the stable trajectory goes into Q2 in fact converges to the stable node (w+ , 0) Indeed, in a neighborhood of the saddle point (w− , 0), says |z | ≤ 2sκ/m, it holds that λ′ (w)z + 2sµ(w) > (4.3) Multiplying it by the second equation of (3.3) by z = dw/dy, from (4.2) we get z dz dy =−