DSpace at VNU: Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion

Nonlinear Analysis 72 (2010) 3136–3149 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion Mai Duc Thanh ∗ Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam article info Article history: Received September 2009 Accepted December 2009 MSC: 35L65 74N20 76N10 76L05 Keywords: Traveling wave Fluid Diffusion Dispersion Nonclassical shock Equilibria Lyapunov stability Attractor abstract This work completes the description of the method of estimating attraction domain for traveling waves for several systems of conservation laws with viscosity and capillarity effects proposed in our earlier work Thanh (2010) [26] Precisely, we establish the global existence of traveling waves for an isentropic fluid with nonlinear diffusion and dispersion coefficients The shock wave can be classical or nonclassical Interestingly, we in particular show the existence of a traveling wave for a given Lax shock but rather nonclassical when the straight line connecting the two left-hand and right-hand states crosses the graph of the pressure function two more times in the middle region Furthermore, we also discuss all the possibilities of saddle–stable, saddle–saddle, or stable–stable connections © 2009 Elsevier Ltd All rights reserved Introduction The interest for the study of nonclassical traveling waves has been justified by the general theory of nonclassical solutions (Riemann problem, initial-value problem, numerical approximations) developed by LeFloch and his collaborators for many years (see [1] and the references therein) On the other hand, nonlinear diffusion and dispersion have been found useful in many applications of fluid dynamics and material sciences This paper addresses the global existence of traveling waves of an isothermal fluid flow with nonlinear diffusion and dispersion coefficients ∂t v − ∂x u = 0, ∂t u + ∂x p(v) = ε(a(v)|vx |q ux )x − δvxxx , (1.1) where u, v > and p denote the velocity, specific volume, and the pressure, respectively The system (1.1) are conservation laws of mass and momentum of gas dynamics equations in Lagrange coordinates The system can be obtained from the common gas dynamics equations in Lagrange coordinates by writing the equation of state of the form p = p(v, S ), where S is the entropy and assume that S is constant The diffusion and dispersion terms are similar to those in [2], except the involvement of the small positive constants ε > 0, δ > which mean that the sizes of the diffusion and dispersion are ∗ Tel.: +84 2211 6965; fax: +84 3724 4271 E-mail addresses: hatothanh@yahoo.com, mdthanh@hcmiu.edu.vn 0362-546X/$ – see front matter © 2009 Elsevier Ltd All rights reserved doi:10.1016/j.na.2009.12.003 M.D Thanh / Nonlinear Analysis 72 (2010) 3136–3149 3137 small and that we are interested in the limit when these quantities go to zero The function a > is smooth, indicating a nonlinear diffusion and the constant q ≥ emphasizes the nonlinearity of the diffusion to the system Without viscosity and capillarity effects, the model (1.1) takes the usual form of two conservation laws of mass and momentum: ∂t v − ∂x u = ∂t u + ∂x p(v) = (1.2) It is interesting that whenever a traveling wave of (1.1) connecting the left-hand state (v− , u− ) and the right-hand states (v+ , u+ ) exists, its point-wise limit when ε, δ → gives a shock wave of (1.2) of the form (v, u)(x, t ) = (v− , u− ), (v+ , u+ ), if x < st , if x > st , (1.3) (see [1], for example) Furthermore, a shock wave (1.3) obtained in this way is admissible under the admissibility condition of viscosity–capillarity zero limit, according to Slemrod [3–5] Conversely, given a shock wave of the form (1.3), we would like to know whether there is a traveling wave, and this is the goal of the current paper In the case where the pressure p is convex, Slemrod [3] showed that given a Lax shock (i.e a shock satisfies the Lax shock inequalities), then there exists a corresponding traveling waves See [6–9] and the references therein for Lax shocks in gas dynamics equations and related topics LeFloch, Bedjaoui and their collaborators, see [10,1,11–14,2] have paid lots of contributions on the existence of traveling waves, focusing mainly on traveling waves associate with nonclassical shocks (the one violating Liu’s entropy condition) In these works, existence results basically come from the saddle-to-saddle connection between the two stable trajectories leaving a saddle point at −∞ and approaching the other saddle point at +∞ See [10,15,16,1,17–20] for nonclassical shocks Traveling waves for diffusive–dispersive scalar equations were earlier studied by Bona and Schonbek [21], Jacobs, McKinney, and Shearer [22] Traveling waves of the hyperbolic–elliptic model of phase transition dynamics were also studied by Slemrod [3, 4] and Fan [23,24], Shearer and Yang [25] See also the references therein In exerting on the model (1.1), this paper will complete the description of the attractor method for estimating attraction domain of a set of equilibria to construct traveling waves for hyperbolic conservation laws with viscosity and capillarity coefficients The method was proposed in our earlier work [26] exploring the applications of LaSalle’s invariance principle An additional argument taking a stable trajectory into this domain of attraction establishes a connection We observe that our analysis only requires the shock satisfies the one-sided relaxed Lax shock inequalities The relaxation means that the shock may emerge from or arrive at a state an elliptic region The shock can also jump across an elliptic region (phase transitions) Moreover, the shock may be nonclassical and violates Liu’s entropy condition In particular, the line between the left-hand and the right-hand states can cross the graph of the pressure function up to four times as in the case of van der Waals fluids, or more times The organization of this paper is as follows In Section we provide basic facts concerning shock waves and traveling waves for (1.1) and (1.2) and we describe briefly the relation between these two kinds of waves Section is devoted to the proof of the asymptotical stability of an equilibrium point and the estimation of its attraction domain In Section we will establish the stable–saddle or saddle–saddle connection that gives a traveling wave We also provide numerical illustrations of the traveling waves Finally, in Section we show that there is no stable-to-stable connection in the nonlinear case of diffusion (as in the linear case) by pointing out that the corresponding equilibrium point does not admit any asymptotically stable trajectory Preliminaries: Shock waves and traveling waves Let us recall basic concepts for the system (1.2) The Jacobian matrix of the system (1.2) is given by A(v) = p (v) −1 (2.1) which has the characteristic equation λ2 + p (v) = Thus, if p (v) ≤ 0, A(v) admits two real eigenvalues λ+ (v) = − −p (v) ≤ ≤ λ2 (v) = −p (v) Otherwise, it has two distinct complex conjugate eigenvalues λ+ (v) = −i p (v), λ2 (v) = i p (v), where i2 = −1 Consider a shock wave solution of the hyperbolic system (1.2), connecting a given left-hand state (u− , v− ) to some righthand state (u+ , v+ ) and propagating with the speed s These states and s satisfy the Rankine–Hugoniot relations s(v+ − v− ) + u+ − u− = 0, s(u+ − u− ) − (p(v+ ) − p(v− )) = (2.2) 3138 M.D Thanh / Nonlinear Analysis 72 (2010) 3136–3149 Eqs (2.2) determine the shock speed s as s2 = − p(v+ ) − p(v− ) v+ − v− , (2.3) and thus define the Hugoniot set of the form u+ = u− − s(v −v− ) Therefore, providing that (p(v+ )− p(v− ))/(v+ −v− ) ≤ 0, the shock speed s=∓ − p(v+ ) − p(v− ) v+ − v− is well defined and independent of u− and u, so we simply set s = s(v− , v+ ), where the 1- and 2-shocks correspond to the minus and the plus sign, respectively The Hugoniot set is thus composed of two Hugoniot curves corresponding to s ≤ and s ≥ Let us recall standard entropy criteria for hyperbolic systems of conservation laws The Lax shock inequalities require that any discontinuity connecting the left-hand state (v− , u− ) and the right-hand state (v+ , u+ ) satisfies λi (v+ ) < si (v+ , v− ) < λi (v− ), i = 1, 2, (2.4) where si stands for the i-shock speed, i = 1, Since we not assume the hyperbolicity, a shock may start from or arrive at a state where the system may fail to be hyperbolic Therefore, we require a relaxed version of the Lax shock inequalities for a shock connecting the left-hand state (v− , u− ) and a right-hand state (v+ , u+ ): p (v+ ) < −s2 < p (v− ) for 1-shocks p (v+ ) > −s > p (v− ) for 2-shocks (2.5) For a hyperbolic system conservation laws with non-genuinely nonlinear characteristic fields, Lax shock inequalities are usually replaced by Liu’s entropy condition to ensure the uniqueness Liu’s entropy condition is the one that imposes along Hugoniot curves: s(v− , v) ≥ s(v+ , v− ) for any v between v+ and v− , where s(v+ , v) denote the speed of the discontinuity connecting v and v+ Thus, Liu’s entropy condition means that any discontinuity connecting the left-hand state (v− , u− ) and the right-hand state (v+ , u+ ) fulfils –for 1-shocks p(v) − p(v− ) p(v+ ) − p(v− ) ≥ , for any v between v+ and v− v − v− v+ − v− –for 2-shocks p(v) − p(v− ) v − v− ≤ p(v+ ) − p(v− ) v+ − v− , for any v between v+ and v− (2.6) Observe that the Liu strict entropy condition (where the inequalities ‘‘≥’’ and ‘‘≤’’ above are replaced by the strict inequalities ‘‘>’’ and ‘‘ v− , without restriction, since similar argument can be made for 1-shocks and/or v+ < v− Given a 2-shock wave (v, u)(x, t ) = (v− , u− ), (v+ , u+ ), if x < st , if x > st , (3.1) satisfying a one-sided relaxed Lax shock inequality p (v+ ) + s2 > (3.2) In this case, as shown by Proposition 2.1, the point (v+ , 0) is a focus for the linearized system of (2.11) We aim to prove that the point (v+ , 0) is actually a stable node of the differential equations dv dy dz dy = z, (3.3) = −γ sa(v)|z |q z − h(v), −∞ < y < +∞, where h(v) = p(v) − p(v− ) + s2 (v − v− ), s2 = − p(v+ ) − p(v− ) v+ − v− , γ = ε δ (q+1)/2 (3.4) Moreover, we will use the level sets of a Lyapunov-type function to estimate its domain of attraction Let us define a Lyapunov-type function L(v, z ) = v v+ h(ξ )dξ + z2 The function in (3.5) satisfies L(v+ , 0) = 0, ∇ L(v, z ) = h(v), z (3.5) M.D Thanh / Nonlinear Analysis 72 (2010) 3136–3149 3141 Fig The sets G defined by (3.11) and Ωβ defined by (3.14) The derivative of L along trajectories of (3.3) can be estimated by L˙ (v, z ) = ∇ L(v, z ) · dv dz , dy dy = h(v), z z , −γ sa(v)|z |q z − h(v) = −γ sa(v)|z |q+2 < 0, for z = 0, (3.6) since γ > 0, s > 0, a > The Rankine–Hugoniot relations (2.2) yield h(v) = p(v) − p(v+ ) + s2 (v − v+ ) p(v) − p(v+ ) = (v − v+ ) v − v+ + s2 Moreover, the relaxed Lax shock inequality (3.2) implies that there exists a value θ > such that p(v) − p(v+ ) v − v+ + s2 > 0, |v − v+ | ≤ θ Thus, v v+ h(ξ )dξ = v v+ (ξ − v+ ) p(ξ ) − p(v+ ) ξ − v+ + s2 dξ > 0, |v − v+ | ≤ θ (3.7) Setting ν1 = v+ + θ , from (3.7) we have ν1 v+ h(ξ )dξ > Then, by continuity, it is derived from (3.7) that we can always take a point ν2 < v+ such that ν1 v+ h(ξ )dξ > ν2 v+ h(ξ )dξ > 0, (3.8) or L(ν1 , 0) > L(ν2 , 0) > (3.9) Fix these values ν1 , ν2 Let us take a sufficiently large number M so that M > max |p (v)| + s2 (3.10) v∈[ν2 ,ν1 ] Define the set G = (v, z ) ∈ R2 |(v − v+ )2 + ∪ (v, z ) ∈ R2 |(v − v+ )2 + (see Fig 2) M2 z ≤ |v+ − ν2 |2 , v ≤ v+ |v+ − ν1 |2 z ≤ |v+ − ν1 |2 , v ≥ v+ , (M |v+ − ν2 |)2 (3.11) 3142 M.D Thanh / Nonlinear Analysis 72 (2010) 3136–3149 Lemma 3.1 Let G be the set defined by (3.11) and let ∂ G denote its boundary It holds that L(v, z ) = L(ν2 , 0) (3.12) (v,z )∈∂ G Moreover, the minimum value in (3.12) is achieved at the unique point (ν2 , 0), i.e L(v, z ) > L(ν2 , 0), for all (v, z ) ∈ ∂ G \ {(ν2 , 0)} (3.13) Proof We need only establish (3.13) On the semi-ellipse ∂ G, v ≤ v+ , one has z = M (|v+ − ν2 |2 − (v − v+ )2 ) Thus, along this semi-ellipse, it holds that L(v, z )|(v,z )∈∂ G,v≤v+ = v v+ h(ξ )dξ + := g (v), M2 (|v+ − ν2 |2 − (v − v+ )2 ) v ∈ [nu2 , v+ ] We have dg (v) dv = h(v) − M (v − v+ ) = −(v − v+ ) M − p(v) − p(v+ ) v − v+ = (v+ − v) M − p (ξ ) − s2 , > 0, v ∈ (ν2 , v+ ) + s2 v+ < ξ < v, where the last inequality follows from the definition of M in (3.10) The function g is therefore strictly increasing for v ∈ [ν2 , v+ ] and attains its strict minimum on this interval at the end-point v = ν2 , i.e L(v, z ) > L(ν2 , 0), for all (v, z ) ∈ ∂ G \ {(ν2 , 0)}, v ≤ v+ Arguing similarly, we can see that L(v, z ) > L(ν1 , 0), for all (v, z ) ∈ ∂ G \ {(ν1 , 0)}, v ≥ v+ The last two inequalities and (3.9) establish (3.13) The proof of Lemma 3.1 is complete Properties of the level sets of the Lyapunov-type function (3.5) can be seen in the following lemma Lemma 3.2 Let ν1 , ν2 be defined as in (3.9) and G be defined by (3.11) For any positive number < β < L(ν2 , 0), the set Ωβ := {(v, z ) ∈ G|L(v, z ) ≤ β} (3.14) is a compact set, lies entirely inside G, positively invariant with respect to (3.3), and has the point (v+ , 0) as an interior point, (see Figs and 3) In addition, assume that ν1 , ν2 in (3.9) be chosen such that the point (v+ , 0) is the sole equilibrium point in the domain ν2 < v < ν1 Then, the initial-value problem for (3.3) with initial condition (u(0), v(0)) = (v0 , v0 ) ∈ Ωβ admits a unique global solution (v(y), z (y)) for all y ≥ Moreover, this trajectory converges to (v+ , 0) as y → +∞, i.e., lim (v(y), z (y)) = (v+ , 0) y→+∞ This means that the equilibrium point (v+ , 0) is asymptotically stable and Ωβ is a subset of the domain of attraction of (v+ , 0) We will omit the proof, since it is similar to the one of Lemma 3.2, [26] The stable trajectory, multiple equilibria and traveling waves In this section we will establish the existence of traveling waves by finding out when the stable trajectory of a saddle point enters the attraction domain of the stable node For definitiveness, we still assume that we are still concerned with a 2-shock and that v+ > v− , since the argument for the other cases are similar 4.1 Traveling waves associate with classical shocks First, let us establish the existence of traveling waves when the shock satisfying the relaxed Lax shock inequalities (2.5) and Liu’s entropy condition (2.6) Recall that a shock of this kind is known as a classical shock M.D Thanh / Nonlinear Analysis 72 (2010) 3136–3149 3143 Fig The level set Ωβ of a van der Waals fluid Theorem 4.1 (i) Given a 2-shock wave (3.1) satisfying the relaxed Lax shock inequalities (3.2) Suppose that there exists a value ν > v+ such that ν v+ h(ξ )dξ > v− v+ h(ξ )dξ , (4.1) and that (v+ , 0) is the unique equilibrium point of (3.3) in the range v− < v < ν Then, there exists a unique traveling wave of (1.1) connecting the states (v− , u− ) and (v+ , u+ ) (ii) Similar result holds for 1-shock waves Proof By applying Lemma 3.2 where ν1 = ν, ν2 = v− , we can see that the level sets Ωβ , < β < L(v− , 0) are subsets of the domain of attraction of the stable node (v+ , 0) Their union Ω = ∪0 0, we will establish the existence of a unique traveling wave connecting to some state on the right (v+ , u+ ) determined by the Rankine–Hugoniot relations (2.2) Although the argument can be applied for a general case, we only consider the case of a van der Waals fluid p(v) = 8e3(γ −1)S0 /8 (3v − 1)γ − v2 , γ = 1.4, (4.4) where S0 = 0.1 We are interested in the situation where the straight line between (v± , p(v± )) cuts the graph of p at exactly other two points (vi , p(vi )), i = 1, with v− < v1 < v2 < v+ Clearly, the two points (vi , 0), i = 1, are also the equilibria, together with (v± , 0) of the differential equations (2.10) Let us denote by M the set of the three equilibria: M = {(v+ , 0), (vi , 0), i = 1, 2} (4.5) See Fig Theorem 4.2 (a) Given a left-hand state (v− , u− ) and a 2-shock speed s > such that the straight line through (v− , p(v− )) with the slope −s2 cuts the graph of the pressure function at exactly four points (v± , p(v± )), (vi , p(vi )), i = 1, where v− < v1 < v2 < v+ Let (vi , ui ), i = 1, be the corresponding states in the (v, u)-phase domain There exists a unique traveling wave leaving (v− , u− ) at −∞ and converging to the state corresponding to one of the remaining three equilibria (v+ , 0), (vi , 0), i = 1, Thus, there are the following possibilities: the traveling wave (i) Either associate with a classical shock between (v− , u− ) and (v1 , u1 ) This case exhibits a saddle-to-stable connection, (ii) or associate with a nonclassical shock between (v− , u− ) and (v2 , u2 ) This case exhibits a saddle-to-saddle connection, (iii) or associate with a nonclassical shock between (v− , u− ) and (v+ , u+ ) This case exhibits a saddle-to-stable connection (b) Similar result holds for 1-shock waves Proof It is easy to check that the pressure function is convex for v ≥ v+ Thus, v v+ h(ξ )dξ → +∞, as v → +∞ and the condition (4.1) clearly holds Arguing similarly as in the proof of Theorem 4.1, we can see that exactly one stable trajectory leaving the saddle (v− , 0) at −∞ enters the domain of attraction of equilibria Ω , where Ω = (v, z ) ∈ R2 , v ∈ (v− , ν) : L(v, z ) − L(v− , 0) < = (v, z ) ∈ R2 , v ∈ (v− , ν) : v v− h(ξ )dξ + z2 0, s > 0, (5.1) then the equilibrium point (v− , 0) is no longer a saddle, but a focus of the linearized system It is therefore natural to raise the question whether there is a stable-to-stable connection If this holds, the attraction domains of the two equilibria would intersect and any choice for the intermediate state in their intersection would give a traveling wave In this section, we will show that if (5.1) occurs, then there is no trajectory other than the trivial one leaving the equilibrium point (v− , 0) at −∞ Thus, there will be no connection between (v± , 0) M.D Thanh / Nonlinear Analysis 72 (2010) 3136–3149 3147 Fig van der Waals fluid: Illustration of the right part of the traveling wave in (y, v)-plane (above) and the trajectory in the (v, z )-plane (below) Fig van der Waals fluid: Illustration of the right part of the traveling wave in (y, v)-plane (above) and the trajectory in the (v, z )-plane (below) for longer time Proposition 5.1 Assume that (5.1) holds for a shock wave ((v± , u± ); s) The equilibrium point (v− , 0) of the differential equations (2.10) then does not admit any asymptotically stable trajectory other than the trivial one (v, z ) ≡ (v− , 0) for y < In other words, there is no non-trivial trajectory of (2.10) approaching the equilibrium point (v− , 0) as y tends to −∞ Consequently, there does not exist any traveling wave between (v± , u± ) Proof By changing variable y → −y for simplicity, we consider the stability of the equilibrium point (v− , 0) in the positive direction The differential equations (2.10) then become dv dy dz dy = −z , (5.2) = γ sa(v)|z |q z + h(v), −∞ < y < +∞, where h(v) = p(v) − p(v− ) + s2 (v − v− ), s2 = − p(v+ ) − p(v− ) v+ − v− , γ = ε δ (q+1)/2 (5.3) 3148 M.D Thanh / Nonlinear Analysis 72 (2010) 3136–3149 We also define a Lyapunov-type function L(v, z ) = v h(ξ )dξ + v− z2 (5.4) We will show that L is positive definite Indeed, L(v− , 0) = (5.5) Moreover, the condition (5.1) implies that there exists a value θ > such that p(v) − p(v− ) v − v− + s2 < 0, |v − v− | ≤ θ This yields L(v, z ) = v v− v ≥ v− h(ξ )dξ + (ξ − v− ) z2 p(ξ ) − p(v− ) ξ − v− + s2 (v − v− ) dξ > 0, |v − v− | ≤ θ , v = v− (5.6) (5.5) and (5.6) show that L is positive definite Furthermore, we have ∇ L(v, z ) = h(v), z Thus, the derivative of L along trajectories of (5.2) is given by L˙ (v, z ) = ∇ L(v, z ) · dv dz , dy dy = h(v), z −z , γ sa(v)|z |q z + h(v) = γ sa(v)|z |q+2 > 0, for z = Let (v(y), z (y)) be an arbitrary trajectory of (5.2) 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Non-monotone traveling waves in van der Waals fluids, Anal Appl (2005) 419–446 [3] M Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch Ration Mech Anal 81... the case of van der Waals fluids, or more times The organization of this paper is as follows In Section we provide basic facts concerning shock waves and traveling waves for (1.1) and (1.2) and. .. Fig van der Waals fluid: Illustration of the right part of the traveling wave in (y, v)-plane (above) and the trajectory in the (v, z )-plane (below) Fig van der Waals fluid: Illustration of the