Nonlinear Analysis 75 (2012) 4884–4895 Contents lists available at SciVerse ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Existence of traveling waves in compressible Euler equations with 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 September 2011 Accepted April 2012 Communicated by Enzo Mitidieri Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of firstorder, which admits an asymptotically stable equilibrium point and a saddle point We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point This gives us a traveling wave of the viscous–capillary compressible Euler equations © 2012 Elsevier Ltd All rights reserved Keywords: Compressible Euler equations Traveling wave Shock Viscosity Capillarity Equilibria Asymptotical stability Lyapunov function LaSalle’s invariance principle Attraction domain Introduction Naturally, it is interesting to see whether a shock wave admissible under a certain criterion can be approximated by some kind of smooth solutions when allowing viscosity and capillarity to the system This leads to the study of the existence of traveling waves, which has attracted many authors for many years over the past In this paper, we are interested in the global existence of traveling waves in compressible Euler equations when viscosity and capillarity are present Precisely, let us consider the following compressible Euler equations with viscosity and capillarity in Lagrange coordinates, see [1]: vt − ux = 0, ut + px = α(ν ux )x − β(µvx )xx + Et + (up)x = α(ν uux )x + β µ v β (µv vx2 )x , uvx2 − u(µvx )x (1.1) x + β (µux vx )x , x ∈ R , t > 0, where v, S , p, ε and T denote the specific volume, entropy, pressure, internal energy, and temperature, respectively; u is the velocity, and E is the total energy The non-negative functions µ = µ(v, S ) and ν = ν(v, S ) represent the viscosity and capillarity coefficients of the fluid, respectively The positive numbers α and β measure the scale of the viscosity and the ∗ Tel.: +84 2211 6965; fax: +84 3724 4271 E-mail addresses: mdthanh@hcmiu.edu.vn, hatothanh@yahoo.com 0362-546X/$ – see front matter © 2012 Elsevier Ltd All rights reserved doi:10.1016/j.na.2012.04.003 M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 4885 capillarity, respectively We note that the Lagrangian coordinates are chosen so that the calculations are simple only, since similar results hold for the Eulerian coordinates Without viscosity and capillarity, the system (1.1) is the usual gas dynamics equations in Lagrange coordinates vt − ux = 0, ut + px = 0, Et + (up)x = 0, (1.2) x ∈ R , t > Solutions of the compressible Euler equations (1.2), and more general, hyperbolic systems of conservation laws, are found in the weak forms In general, weak solutions are not unique due to the presence of discontinuities, or shock waves To select a physical, unique solution, one implements the system with a certain admissibility criterion for shock waves Various admissibility criteria have been known The concept of Lax shocks refers to shock waves that satisfy the Lax shock inequalities; see [2] This criterion is widely used when the characteristic field of the system is genuinely nonlinear For systems of conservation laws where the characteristic fields are not genuinely nonlinear, one uses a more strict Liu’s entropy condition, see [3], to define classical shocks The concept of nonclassical shocks refers to the shock waves that violate Liu’s entropy condition, but satisfy a single entropy inequality and a kinetic relation; see [4] Nonclassical shock may appear only when the characteristic field of the system fails to be genuinely nonlinear; see [5] and the references therein Traveling waves are of a special kind of smooth solutions when viscosity and capillarity are added to the system, such as (1.1) If a traveling wave connecting a left-hand state U− and a right-hand state U+ exists, then its point-wise limit by vanishing viscosity and capillarity is a shock wave connecting these two states; see [5] Therefore, an admissibility criterion for shock waves could be justified physically in establishing the existence of the traveling waves To simplify the calculations, from now on, we assume that the viscosity and capillarity coefficients ν and µ are constants Furthermore, we still use the symbols α and β to denote the quantities αν and βµ in (1.1), respectively We observe that the analysis in this paper can be extended to more general classes of viscosity and capillarity in which these quantities may depend on the specific volume and the entropy as in [1] The system (1.1) is therefore reduced to the following system vt − ux = 0, ut + px = α uxx − βvxxx , Et + (up)x = α(uux )x + β (ux vx − uvxx )x , (1.3) x ∈ R , t > In our earlier work [6], a method of estimating attraction domain of an asymptotically stable equilibrium point for traveling waves of a single conservation law with viscosity and capillarity was presented This method is relied on the Lyapunov stability techniques, in particular LaSalle’s invariance principle, and the characterization of admissible shock waves of conservation laws under consideration In this paper, we will develop this method to the case of a3 × system (1.3) Note that a complete set of compressible Euler equations with viscosity and capillarity effects, though has attracted attention of many authors, but has rarely been fully investigated due to the complexity of the system As seen in the review below, most works in the literature have reduced to either the case of a single equation, or the isothermal case of a × system where the equation of conservation of energy is ignored Our main result is the following: given any Lax shock of the compressible Euler equations (1.2), assuming that the fluid is polytropic and ideal, we will show that there exists a corresponding traveling wave of (1.3) with a suitable choice of the viscosity and the capillarity The existence of traveling waves corresponding to a given nonclassical shock for viscous–capillary models was established by LeFloch and his collaborators and students; see [4,7–9,1,10,11] The existence of traveling waves corresponding to a given Lax shock,or a classical shock for viscous–capillary models was obtained in [6,12–14] These two approaches could provide the reader with a larger vision and better understanding of traveling waves for the same or analogous viscous–capillary models Recall that traveling waves for diffusive–dispersive scalar equations were earlier studied by Bona and Schonbek [15], and Jacobs et al [16] Traveling waves of the hyperbolic–elliptic model of phase transition dynamics were also studied by Slemrod [17,18] and Fan [19,20], and Shearer and Yang [21] Traveling waves in Korteweg models in the isothermal case, Eulerian and Lagrangian capillarity models, and related topics were studied by Benzoni-Gavage and her collaborators; see [22–24] We note that a pioneering work on the related shock layers of the gas dynamics equations with viscosity and heat conduction effects (with zero capillarity) was presented by Gilbarg [25] See also the references therein The organization of this paper is as follows In Section 2, we first recall the basic concepts of hyperbolicity, jump relations, and Lax shock inequalities of the Euler equations (1.2) Second, we present the concept of traveling waves and derive a system of differential equations for these traveling waves Third, we present a result on the stability characteristics of the two equilibria of the differential equations satisfied by the traveling waves corresponding to a given Lax shock, where one equilibrium point is shown to be asymptotically stable, and the other one is shown to be a saddle point In Section 3, we first define a Lyapunov function corresponding to the asymptotically stable equilibrium point This function is then shown to possess a very desirable property: its level sets could provide us with a sharp estimation of the attraction domain of the asymptotically stable equilibrium point Finally, we establish the existence of a traveling wave for any given Lax shock, providing that the viscosity and capillarity are suitably chosen 4886 M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 Preliminaries 2.1 Hyperbolicity, jump relations and Lax shocks Throughout, the fluid is assumed to be polytropic and ideal so that the equation of state is given by ε= pv γ −1 = RT γ −1 , (2.1) where R > 0, < γ < 5/3 are constant Let us choose the independent thermodynamic variables v and S From the thermodynamical identity dε = TdS − pdv, (2.2) one has εv = −p, εS = T and one can express the pressure p as a function of the specific volume v and the entropy S: p = p(v, S ) = (γ − 1)v −γ exp S − S0 Cv , Cv = R γ −1 For smooth solutions U = (v, u, S ), it is easy to see that the system (1.3) can be rewritten as vt − ux = 0, ut + px = α uxx − βvxxx , (2.3) TSt = α u2x In a similar way, Euler equations (1.2) can be written as vt − ux = 0, ut + pv (v, S )vx + pS (v, S )Sx = 0, St = (2.4) Let us recall the basic concepts for the system (1.2) The Jacobian matrix of the system (2.4) and therefore of (1.2) is given by A(v) = pv −1 0 pS , which admits three distinct real eigenvalues λ1 = − −pv (v, S ) < λ2 = < λ3 = −pv (v, S ) Thus, the system (1.2) is strictly hyperbolic Moreover, it has been known that the first and the third characteristic fields are genuinely nonlinear, while the second characteristic field is linearly degenerate Consider a Lax i-shock wave of the hyperbolic system (1.2), connecting a given left-hand state U− = (v− , u− , S− ) to some right-hand state U+ = (v+ , u+ , S+ ) and propagating with the speed s = si (U− , U+ ), i = 1, These states and s satisfy the Rankine–Hugoniot jump relations −s[v] − [u] = 0, −s[u] + [p] = 0, −s[E ] + [pu] = 0, (2.5) where [v] = v+ − v− , [u] = u+ − u− , etc., and the Lax shock inequalities λi (U+ ) < si (U− , U+ ) < λi (U− ), i = 1, (2.6) For polytropic ideal gas, it is known that the Lax shock inequalities, for 3-shock for example, are equivalent to each of the following conditions (i) (ii) (iii) (iv) v− ≤ v+ , p− ≥ p+ , S− ≥ S+ , u− ≥ u+ For a 1-shock, the inequalities in (i)–(iii) are reversed, while the inequality in (iv) is the same M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 4887 2.2 Traveling waves and properties Let us now turn to traveling waves We call a traveling wave of (1.3) connecting the left-hand state U− and the right-hand state U+ a smooth solution of (1.3) depending on the variable U = U (y) = (v(y), u(y), S (y)), y = x − st , where s is a constant, and satisfying the boundary conditions lim U (y) = U± y→±∞ d lim y→±∞ dy U (y) = lim (2.7) d2 U (y) = y→±∞ dy2 Substituting U = U (y), y = (x − st ), into (2.3), we get −sv ′ − u′ = 0, −su′ + p′ = α u′′ + βv ′′′ , (2.8) −sTS = α(u ) , ′ ′ where (·)′ = d(·)/dy Substituting u′ from the first equation to the remaining two equations of (2.8) we can reduce the size of the system as, assuming s ̸= 0, −s2 v ′ − p(v, S )′ = sαv ′′ + βv ′′′ , (2.9) −TS ′ = α s(v ′ )2 Integrating the first equation on the interval (−∞, y), using the boundary conditions (2.7), we obtain p(v, S ) − p(v− , S− ) + s2 (v − v− )2 = −α sv ′ − βv ′′ (2.10) Thus, by setting w = v′ , and using (2.10), we can rewrite the system (2.9) as a system of first-order differential equations v ′ = w, −s α w− p(v, S ) − p(v− , S− ) + s2 (v − v− ) , w′ = β β −s α w S′ = (2.11) T We will simplify further the system (2.11) as follows Multiplying (2.10) by v ′ , and using (2.2) and the third equation in (2.11), we obtain β ((v ′ )2 )′ = ε ′ (v, S ) + p(v− , S− )v ′ − s2 (v − v− )v ′ (2.12) Integrating (2.12) over (−∞, y) and using the first equation of (2.11), we get β w2 = ε − ε− + p(v− , S− )(v − v− ) − s2 (v − v− )2 , ε− = ε(v− , S− ) (2.13) Using equation of state (2.1), we can resolve for the pressure p as a function of (v, w) from (2.13) as p= (γ − 1)βw + f (v), 2v (2.14) where f (v) = γ −1 s2 ε− − p− (v − v− ) + (v − v− )2 v (2.15) Observe that f (v− ) = p− , f (v+ ) = p+ , (2.16) 4888 M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 where the second identity follows from (2.5) Eq (2.14) also determines the entropy as a function of the two variables v and w : S = S (v, w) Therefore, the system (2.11) is reduced to the following system of two differential equations of first-order v ′ = w, −s α w′ = w − h(v, w), β β (2.17) where h(v, w) = (γ − 1)βw2 + f (v) − p− + s2 (v − v− ) 2v (2.18) 2.3 Equilibria and their stability properties Let (v, w) be an equilibrium point of the system (2.17) Then, by definition, it holds that w = 0, g (v) := h(v, 0) = f (v) − p− + s2 (v − v− ) = 0, (2.19) where f is defined by (2.15) It is derived from (2.5), (2.16) and (2.19) that the points (v± , 0) are equilibria of the system (2.17) so that g (v± ) = The function g is strictly convex, since it can be expressed as the sum of strictly convex functions and linear functions: v2 ε− p − v− s2 − p− + + v − 2v− + − − p− + s2 (v − v− ) v v v Thus, the fact that g (v± ) = implies that the strictly convex function g has exactly two zeros v± The system (2.17) therefore has exactly two equilibria (v± , 0) g (v) = (γ − 1) The following lemma provides us with the stability properties of the equilibria Lemma 2.1 Given a 3-shock between the left-hand state (v− , u− , S− ) and the right-hand state (v+ , u+ , S+ ) with shock speed s satisfying the Lax shock inequalities (2.6) The following conclusions hold (a) The equilibrium point (v− , 0) is a saddle point of the system of differential equations (2.17) (b) The point (v+ , 0) is an asymptotically stable equilibrium point of (2.17) Similar results hold for 1-shock waves Proof First, it follows from the equation of state p = p(v, S ) = (γ − 1)v −γ exp S − S0 Cv that pv (v, S ) = −γ p(v, S ) v (2.20) The Jacobian matrix of the system (2.17) at these equilibria is given by B± = −(f ′ (v± ) + s2 )/β −sα/β , (2.21) which has the characteristic equation as ξ2 + sα β ξ+ f ′ (v± ) + s2 β = (2.22) We have γ −1 (v − v ) + −p− + s2 (v − v− ) − v v s 1−γ 2 = ε − p (v − v ) + (v − v ) + p v − s v(v − v ) − − − − − − v2 f ′ (v) = 1−γ ε− − p− (v − v− ) + s2 (2.23) M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 4889 Substituting v = v− into (2.23), using (2.20), we obtain f ′ (v− ) = 1−γ (ε− + p− v− ) v− −γ p− = pv (v− , S− ) = v− (2.24) Now, the Lax shock inequalities (2.6) and (2.24) imply that −pv (v− , S− ) = λ23 > s2 Thus, f ′ (v− ) + s2 = pv (v− , S− ) + s2 < 0, so that the characteristic equation (2.22) admits two real roots having opposite signs: ξ1 < < ξ2 This establishes (a) To prove (b), we estimate f ′ (v+ ) as follows Using the jump relations (2.5), we get − s2 [v] = [p] (2.25) The relation (2.25) yields [ε] + p− [v] − s2 [v]2 = [ε] + p− + p+ [v] = Thus, 1−γ γ −1 −p− + s2 [v] v+ v+ γ −1 p+ − p+ =− v+ v+ −γ p+ = = pv (v+ , S+ ) v+ f ′ (v+ ) = ε+ (2.26) It is derived from the Lax shock inequalities (2.6) and (2.26) that f ′ (v+ ) + s2 = pv (v+ , S+ ) + s2 > The last inequality implies that the characteristic equation (2.22) either has two real negative roots, or has two complex roots with the real part negative This establishes (b) Existence of traveling waves As above, we consider an i-shock wave solution of the hyperbolic system (1.2), connecting a given left-hand state U− = (v− , u− , S− ) to some right-hand state U+ = (v+ , u+ , S+ ) and propagating with the speed s − s(U− , U+ ), i = 1, For definitiveness, we consider a 3-shock such that conditions (i)–(iv) are fulfilled Similar argument can be made for 1-shocks Let us recall from Lemma 2.1 that the point (v− , 0) is a saddle point, and the point (v+ , 0) is an asymptotically stable equilibrium point of the autonomous system (2.17) Our purpose, however, is to estimate the attraction domain of the asymptotically stable node (v+ , 0) This will be done in this section using Lyapunov stability techniques 3.1 Lyapunov function Let us define a Lyapunov function candidate L(v, w) = β v v+ g (z )dz + w2 , (3.1) where g (v) = f (v) − p− + s2 (v − v− ) γ −1 s2 = ε− − p− (v − v− ) + (v − v− ) − p− + s2 (v − v− ) v We now investigate properties of the function L as follows First, we have L(v+ , 0) = (3.2) 4890 M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 Thus, to show that L is positive definite, we will point out that L(v, w) > 0, (v, w) ̸= (v+ , 0), v ≥ v− (3.3) Indeed, let us show that g (v) < 0, g (v) > 0, v− < v < v+ , v > v+ (3.4) The function g is strictly convex, since it can be expressed as the sum of strictly convex functions and linear functions: p − v− s2 ε− − p− + + g (v) = (γ − 1) v v v2 v − 2v− + − v − p− + s2 (v − v− ) Moreover, g (v− ) = g (v+ ) = 0, v− < v+ , where the second identity follows from (2.16) and the jump relations (2.5) This establishes (3.4) Therefore, v g (z )dz > 0, v+ v− < v < v+ If v > v+ then g (v) > and therefore v g (z )dz > v+ This establishes (3.3) Moreover, it holds that L˙ (v, w) = ∇ L(v, w) · ⟨v ′ , w ′ ⟩ =− whenever αs + β α s (γ − 1)w + w2 ≤ β 2v (3.5) (γ − 1)w ≥0 2v or w≥ −2α s v (γ − 1)β (3.6) The above argument yields the following lemma Lemma 3.1 The function L defined by (3.1) is a Lyapunov function in the domain (v, w) | v > v− , w > D= −2α s v (γ − 1)β (3.7) 3.2 Estimating attraction domain It is not difficult to check that lim v v→∞ v + g (z )dz = ∞ Thus, one can always select a value ν > w+ such that ν v+ g (z )dz > v− v+ g (z )dz > (3.8) For example, one can take ν = w∗ + 1, where w∗ > w+ is the (unique) value such that w∗ v+ g (z )dz = v− v+ g (z )dz The inequality (3.8) yields L(ν, 0) > L(v− + ε, 0) > 0, (3.9) M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 4891 Fig The sets Gε defined by (3.11) and Ωδ defined by (3.14) for any < ε < (w+ − w− )/2 On the other hand, since the function f is strictly convex, f ′ is strictly increasing Thus, max |f ′ (v)| = max{|f ′ (v− )|, |f ′ (ν)|} v∈[v− ,ν] By (2.24), f ′ (v− ) = −γ p− /v− Define a number M as follows: 1/2 M := max{γ p− /v− , |f ′ (ν)|} + s2 + > max |f ′ (v)| + s2 1/2 v∈[v− ,ν] (3.10) Now, we fix an arbitrary value < ε < (v+ − v− )/2, and set 2 2 Gε = (v, w) ∈ R |(v − v+ ) + w ≤ |v+ − (v− + ε)| , v ≤ v+ M (v, w) ∈ R |(v − v+ )2 + |v+ − ν|2 2 w ≤ |v − ν| , v ≥ v + + , (M |v+ − (v− + ε)|)2 (3.11) (see Fig 1) One needs to implement a condition to make sure that these regions Gε are included in the domain of the Lyapunov function (3.1) In fact, one has Gε ⊂ D, providing that the tangent line (∆) passing through the origin to the curve C : w = |v+ − v− |2 , v ≤ v+ , w < 0, M2 lies above the ‘‘boundary line’’ (Γ ) that determines the boundary of D: (v − v+ )2 + w= −2α s v, (γ − 1)β v > See Fig This can be done by requiring that the slope of the tangent line (∆) is larger than the one of the boundary line (Γ ), i.e., M (v+ − v− ) −2 α s − > , (γ − 1)β v+ − (v+ − v− )2 or α M (γ − 1)(v+ − v− ) > β 2s v+ − (v+ − v− )2 (3.12) Lemma 3.2 Let Gε be the set defined by (3.11) and let ∂ Gε denote its boundary It holds that (v,w)∈∂ Gε L(v, w) = L(v− + ε, 0) Moreover, the minimum value is achieved at the unique point (v− + ε, 0), i.e L(v, w) > L(v− + ε, 0), for all (v, w) ∈ ∂ Gε \ {(v− + ε, 0)} (3.13) 4892 M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 Fig In order that the set Gε defined by (3.11) is always a subset of the domain of the Lyapunov function (3.1) given by (3.7), the tangent line (∆) must lie above the ‘‘boundary line’’ Γ for v ≥ v− Proof We need only to establish (3.13) On the semi-ellipse ∂ Gε , v ≤ v+ , one has w2 = M (|v+ − (v− + ε)|2 − (v − v+ )2 ) Thus, along this semi-ellipse, it holds that L(v, w)|(v,w)∈∂ Gε ,v≤v+ = v v+ g (z )dz + := ϕ(v), M2 (|v+ − (v− + ε)|2 − (v − v+ )2 ) v ∈ [v− + ε, v+ ] Besides, it is derived from (2.16) and (2.25) that g (v) = f (v) − f (v+ ) + s2 (v − v+ ) Therefore, one can estimate the derivative of the function ϕ as follows dϕ(v) dv = g (v) − M (v − v+ ) = −(v − v+ ) M − f (v) − f (v+ ) + s2 v − v+ = (v+ − v) M − f ′ (ξ ) − s2 , v < ξ < v+ , > 0, v ∈ (v− + ε, v+ ) where the last inequality follows from (3.10) The function g is therefore strictly increasing for v ∈ [v− + ε, v+ ] and attains its strict minimum on this interval at the end-point v = v− + ε , i.e ϕ(v) > ϕ(v− + ε), v ∈ (v− + ε, v+ ] This yields L(v, w) > L(v− + ε, 0), for all (v, w) ∈ ∂ Gε \ {(v− + ε, 0)}, v ≤ v+ Arguing similarly, we can see that L(v, w) > L(ν, 0), for all (v, w) ∈ ∂ Gε \ {(ν, 0)}, v ≥ v+ The last two inequalities and (3.9) establish (3.13) The proof of Lemma 3.2 is complete The following lemma provides us with properties of the level sets of the Lyapunov function (3.1) Lemma 3.3 Fix an arbitrary < ε < (w+ − w− )/2, and let Gε be defined by (3.11) Then, for any positive number < δ < L(v− + ε, 0), the set Ωδ := {(v, w) ∈ Gε |L(v, w) ≤ δ} (3.14) is a compact set, lies entirely inside Gε , positively invariant with respect to (2.17), and has the point (v+ , 0) as an interior point; see Fig In addition, the initial-value problem for (2.17) with initial condition (u(0), v(0)) = (v0 , v0 ) ∈ Ωδ admits a unique global solution (v(y), w(y)) for all y ≥ 0, which converges to (v+ , 0) as y → +∞ Consequently, the set Ωδ is an attraction set of the asymptotically stable equilibrium point (v+ , 0) M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 4893 Proof The proof is similar to the one of Lemma 3.2, [6] However, for the sake of completeness, we will present the proof First, it is clear that Ωδ is a compact set Next, we will show that the set Ωδ is in the interior of Gε Assume the contrary, then there is a point U0 ∈ Ωδ which lies on the boundary of Gε Then, it follows from (3.13) that L(U0 ) ≥ L(v− + ε, 0) > δ which is a contradiction, since U0 ∈ Ωδ , L(U0 ) ≤ δ Thus, the closed curve L(u, v) = δ lies entirely in the interior of Gε Moreover, it is derived from Lemma 3.1 that dL(u(y), v(y)) dy ≤ This yields L(u(y), v(y)) ≤ L(u(0), v(0)) ≤ δ, ∀y > The last inequality means that any trajectory starting in Ωδ cannot cross the closed curve L(v, w) = δ Therefore, the compact set Ωδ is positively invariant with respect to (2.17) Therefore, a standard theory of differential equations implies that the system (2.17) has a unique global solution for y ≥ whenever U (0) ∈ Ωδ Next, let us define a set E = {(v, w) ∈ Ωδ | L˙ (v, w) = 0} = {(v, 0)| v > v− } We will show that the set M = {(v+ , 0)} is the largest invariant set in E It is sufficient to show that no solution can stay identically in E, except the constant solution v ≡ v+ , w ≡ Indeed, let (v, w) be a solution that stays identically in E Then, v ′ = w ≡ 0, f (v) − p− + s2 (v − v− ) ≡ which implies (v, w) ≡ (v+ , 0) Applying LaSalle’s invariance principle, we can see that any trajectory U starting in Ωδ converges to (u+ , 0) as y → ∞ The proof of Lemma 3.3 is complete It follows from Lemma 3.3 that we can express the set Ωδ as Ωδ := {(v, w) ∈ R |L(v, w) ≤ δ} (3.15) And therefore, we can deduce that the set Ω= Ωδ=L(v− +2ε,0) , (3.16) 0 0}, and the other leaves the saddle point in the quadrant Q2 = {(v, w)|v < v− , w < 0} We will therefore show that the stable trajectory entering Q1 converges to the asymptotically stable equilibrium point (v+ , 0) Reversing the sides of the first equation of (2.17) and multiplying it by the second equation of (2.17) side-by-side, and then integrating the resulting equation from (−∞, y), we get y w(y)w ′ (y)dy = −∞ y −∞ −s α (γ − 1)w w− − g (v) v ′ (y)dy β 2v β or w2 v = v− (γ − 1)w2 −s α w− − g (z ) dz β 2z β (3.20) Eq (3.20) gives us w2 + β v v− g (z )dz = − sα β w(v − v− ) − (γ − 1)w 2 ln(v/v− ) < 0, (3.21) where the last inequality holds since w > and v > v− It follows from (3.17) and (3.21) and the last inequality that the stable trajectory leaving the saddle point enters the attraction domain of the asymptotically stable equilibrium point (v+ , 0): (v(y), w(y)) ∈ Ω , y < This establishes a saddle-to-stable connection The proof of Theorem 3.4 is complete 3.4 Numerical illustrations Let us illustrate in the following example the presence of a traveling wave connecting the states U− and U+ by a trajectory of (2.17) that starts near the saddle point (w− , 0) and approaches the stable node (w+ , 0) at +∞ We use the solver ‘‘ODE45’’ in MATLAB to solve numerically the solution of (2.17) Example 3.1 Let us choose α = 1, v− = 5, β = 0.5, p− = 10, γ = 1.4, v+ = 10 The trajectory of (2.17) starting at (v0 , w0 ) = (v− + 0.01, 0) approaches the asymptotically stable node (v+ , 0) = (10, 0); see Fig M.D Thanh / Nonlinear Analysis 75 (2012) 4884–4895 4895 Acknowledgments The author would like to thank the reviewers for their very constructive discussions This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2011.36 References [1] N Bedjaoui, P.G LeFloch, Diffusive–dispersive traveling waves and kinetic relations IV Compressible Euler equations, Chin Ann Math Ser B 24 (2003) 17–34 [2] P.D Lax, Shock waves and entropy, in: E.H Zarantonello (Ed.), Contributions to Nonlinear Functional Analysis, 1971, pp 603–634 [3] T.P Liu, The Riemann problem for general × 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Thanh, Existence of traveling waves in elastodynamics with variable viscosity and capillarity, Nonlinear Anal RWA 12 (2011) 236–245 [14] M.D Thanh, Traveling waves of an elliptic–hyperbolic model of