Existence of traveling waves associated with lax shocks which violate oleiniks entropy criterion

Applicable Analysis An International Journal ISSN: 0003-6811 (Print) 1563-504X (Online) Journal homepage: http://www.tandfonline.com/loi/gapa20 Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion Nguyen Huu Hiep, Mai Duc Thanh & Nguyen Dinh Huy To cite this article: Nguyen Huu Hiep, Mai Duc Thanh & Nguyen Dinh Huy (2016): Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion, Applicable Analysis, DOI: 10.1080/00036811.2016.1157864 To link to this article: http://dx.doi.org/10.1080/00036811.2016.1157864 Published online: 09 Mar 2016 Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=gapa20 Download by: [New York University] Date: 12 March 2016, At: 08:38 APPLICABLE ANALYSIS, 2016 http://dx.doi.org/10.1080/00036811.2016.1157864 Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion Nguyen Huu Hiepa,c , Mai Duc Thanhb and Nguyen Dinh Huya Downloaded by [New York University] at 08:38 12 March 2016 a Faculty of Applied Science, University of Technology, Ho Chi Minh City, Vietnam; b Department of Mathematics, International University, Vietnam National University-HCM, Ho Chi Minh City, Vietnam; c Department of Mathematics and Computer Science, University of Science, Vietnam National University-HCM, Ho Chi Minh City, Vietnam ABSTRACT ARTICLE HISTORY This paper answers to the question whether a shock wave in conservation laws satisfying the Lax shock inequalities but not Oleinik’s entropy criterion is admissible under the vanishing viscosity-capillarity effects Such a shock appears in van der Waals fluids when a secant line meets the graph of the flux function at four distinct points, and the shock jumps between the two farthest points The existence of the corresponding traveling waves would justify the admissibility of the shock For this purpose, we will first show that the corresponding traveling waves satisfy a system of differential equations with two saddle points and two asymptotically stable points Second, we estimate the domains of attraction of the asymptotically stable equilibrium points, relying on Lyapunov’s stability theory Third, we investigate the circumstances when an unstable trajectory leaving the saddle point corresponding to the left-hand state of the shock will ever enter the domain of attraction of each of the two asymptotically stable equilibrium points Finally, we establish the existence of traveling waves associated with a Lax shock but violating the Oleinik’s entropy criterion Received 19 September 2015 Accepted 21 February 2016 COMMUNICATED BY M Shearer KEYWORDS Shock wave; traveling wave; Lax shock inequalities; Oleinik’s entropy criterion; viscosity; capillarity; Lyapunov stability AMS SUBJECT CLASSIFICATIONS 35L65; 74N20; 76N10; 76L05 Introduction In this paper, we study the existence of traveling waves associated with a Lax shock but violating the Oleinik entropy criterion (see Oleinik [1], Lax [2], Liu [3]) of the following diffusive–dispersive conservation law ∂t u + ∂x f (u) = β(b(u)ux )x + γ (c1 (u)(c2 (u)ux )x )x , x ∈ IR, t > 0, (1.1) where the unknown u = u(x, t) > a0 , a0 is either a finite constant, or a0 = −∞, the positive constants β and γ indicate small-scaled quantities, the function b(u) > 0, u > a0 , represents the diffusion, and the functions c1 (u) > 0, c2 (u) > 0, u > a0 , represent the dispersion We assume that the functions b(u), c1 (u), and c2 (u) are differentiable, and there is a positive constant c0 so that c1 (u) ≥ c0 , c2 (u) ≥ c0 , for all u ∈ (a0 , ∞) The flux function f : (a0 , ∞) → IR is a twice differentiable function whose graph changes the concavity twice and has two inflection points Precisely, we assume that there are two CONTACT Mai Duc Thanh © 2016 Taylor & Francis mdthanh@hcmiu.edu.vn Downloaded by [New York University] at 08:38 12 March 2016 N H HIEP ET AL Figure Flux function and slope of secant line determines the shock speed s constants a0 < a1 < a2 such that f (u) > 0, for u ∈ (a0 , a1 ) ∪ (a2 , +∞) for u ∈ (a1 , a2 ) f (u) < 0, lim f (u) = −∞, (1.2) u→a0 where (.) = d(.)/du, (.) = d (.)/du2 The motivation for considering this kind of flux functions comes from the shape of the pressure p = p(v, S0 ), v > a0 > of van der Waals fluids as a function of the specific volume v for each fixed entropy S = S0 We note that the topic of shocks waves in van der Waals fluids is interesting and attracts the attention of many authors For simplicity, in the following we always assume that (1.3) a0 = −∞ We note that the argument in this work may be applied for the case where a0 is finite Admissible shock waves of the conservation law ut + f (u)x = (1.4) can be obtained as the limit of traveling waves of (1.1) when β → 0+ , γ → 0+ Consider a straight line (d) passing through a given point (u− , f (u− )) on the graph of the flux function z = f (u) with a slope s, see Figure Under the assumptions (1.2), the line (d) may also meet the graph of the flux function f at three other points (ui , f (ui )), i = 1, 2, Accordingly, we can have three shock waves of (1.4) between the left-hand state u− and the right-hand state u+ with the same shock speed s, where u+ can be u1 , u2 or u3 Recall that a weak solution of (1.4) of the form u(x, t) = u− , if x < st, u+ , if x > st, (1.5) is called a shock wave between the left-hand state u− and the right-hand states u+ with the shock speed s As seen in van der Waals fluids, all of these three shock waves may be admissible in the sense that they may all satisfy an entropy inequality and a kinetic relation The reader is referred to Hayes and LeFloch [4], LeFloch [5] and LeFloch and Thanh [6,7] for nonclassical shock waves Furthermore, these three shocks can be classified into three kinds as follows APPLICABLE ANALYSIS (i) The shock wave between u− and u+ = u3 is called a classical shock, as it satisfies Oleinik’s entropy criterion f (u) − f (u− ) f (u+ ) − f (u− ) ≥ u − u− u+ − u− for all u between u− and u+ (1.6) (ii) The shock wave between u− and u+ = u2 is called a nonclassical shock, because it violates Oleinik’s entropy criterion (1.6) Note that this shock does not satisfy the Lax shock inequalities: (1.7) f (u+ ) < s < f (u− ), Downloaded by [New York University] at 08:38 12 March 2016 where (.) = d(.)/du (iii) The shock wave between u− and u+ = u1 is a nonclassical Lax shock, since it violates Oleinik’s entropy criterion (1.6), but it satisfies the Lax shock inequalities (1.7) The existence of traveling waves of (1.1) associated with a classical shock such as the one in (i) was studied in [8–12] The existence of traveling waves of (1.1) associated with a non-Lax, nonclassical shock such as the one in (ii) was studied in Bedjaoui and LeFloch [13–17], and Bedjaoui et al [18] See also the references therein Thus, admissible shock waves of the kinds (i) and (ii) (i.e they satisfy an entropy inequality as observed in [5,6]) have been shown to be also admissible under the vanishing viscosity-capillarity effects The question is whether the shock wave of the kind (iii) may also be admissible under the vanishing viscosity-capillarity effects This paper will give a positive answer to this question, by establishing the existence of traveling waves of (1.1) associated with a nonclassical Lax shock as in the case (iii) mentioned above We note that traveling waves for diffusive–dispersive scalar equations were studied by Bona and Schonbek [19], Jacobs et al [20] Admissibility criteria by vanishing viscosity and capillarity effects using traveling waves were investigated by Slemrod [21,22] and Fan [23,24], Shearer and Yang [25] See also [26–28] for related works The organization of this paper is as follows In Section 2, we study the stability of the equilibria of a system of ordinary differential equations associated with shock waves satisfying the Lax shock inequality, but violating Oleinik’s entropy criterion In Section 3, we investigate the attraction domains of the asymptotically stable equilibrium points, and we will provide estimates of these domains In Section 4, we will establish the existence of traveling waves Finally, in Section 5, we will present numerical simulations of these traveling waves Traveling waves, equilibria and their stability A traveling wave of (1.1) is a smooth solution of the form u = u(y), y = x − st, where s is a constant, and satisfies the following boundary conditions lim u(y) = u± , y→±∞ (2.1) lim u (y) = lim u (y) = 0, y→±∞ y→±∞ where (.) = d(.)/dy, (.)" = d (.)/dy Substituting u = u(y), y = x − st into (1.1) and integrating over (−∞, y), using the boundary conditions (2.1), we obtain −s(u − u− ) + f (u) − f (u− ) = βb(u)u + γ c1 (u)(c2 (u)u ) (2.2) By letting y → +∞ in (2.2), and using the boundary conditions (2.1), we have −s(u+ − u− ) + f (u+ ) − f (u− ) = 0, or s= f (u+ ) − f (u− ) u+ − u− (2.3) N H HIEP ET AL This shows that s is the shock speed of the shock wave between the left-hand state u− and the right-hand state u+ Setting v = c2 (u)u , we can re-write the second-order differential Equation (2.2) as a system of two first-order differential equations as follows where du v = dy c2 (u) dv b(u) β = h(u) − v, dy γ c1 (u) γ c1 (u)c2 (u) (2.4) h(u) = −s(u − u− ) + f (u) − f (u− ) (2.5) Downloaded by [New York University] at 08:38 12 March 2016 The boundary conditions (2.1) become lim u(y) = u± , y→±∞ (2.6) lim v(y) = lim v (y) = y→±∞ y→±∞ Thus, the problem (2.2)–(2.1) is reduced to (2.4)–(2.6) Let us study the behavior of trajectories of the autonomous system (2.4) Equilibrium points of (2.4) have the form (u, 0), where u satisfies h(u) = 0, which means that u, u− , s satisfy the Rankine–Hugoniot relation (2.3) for u = u+ Let us define β b(u) v , h(u) − v F(u, v) = c2 (u) γ c1 (u) γ c1 (u)c2 (u) Then, the Jacobian matrix of (2.4) is given by ⎛ ⎜ A(u, v) = DF(u, v) := ⎜ ⎝ c2 (u) v c22 (u) b(u) β − γ c1 (u)c2 (u) − h(u) γ c1 (u) v ⎞ ⎟ c2 (u) ⎟, b(u) ⎠ β − γ c1 (u)c2 (u) where (.) = d(.)/du So, the Jacobian matrix at any equilibrium point (u0 , 0) of (2.4) is given by ⎛ ⎜ A(u0 ) = A(u0 , 0) := ⎜ ⎝ −s + f (u0 ) γ c1 (u0 ) ⎞ ⎟ c2 (u0 ) ⎟, ⎠ β b(u0 ) − γ c1 (u0 )c2 (u0 ) (2.7) since h(u0 ) = 0, and h (u) = −s + f (u) The boundary conditions (2.6) can be deduced using the stability of the equilibrium points (u± , 0) The following proposition characterizes the stability of an equilibrium point (u0 , 0) of (2.4) Proposition 2.1: The point (u0 , 0) is an equilibrium point of (2.4) iff h(u0 ) = 0, where h is defined by (2.5) Moreover, (i) If f (u0 ) > s, then the matrix A(u0 ) defined by (2.7) has two real eigenvalues of opposite signs, so that (u0 , 0) is a saddle point; (ii) If f (u0 ) < s, then the matrix A(u0 ) defined by (2.7) has two eigenvalues of negative real parts, so that (u0 , 0) is asymptotically stable, either a stable node or a stable focus APPLICABLE ANALYSIS Proof: The characteristic equation of the matrix A(u0 ) defined by (2.7) is given by λ2 + b(u0 ) s − f (u0 ) β λ+ = γ c1 (u0 )c2 (u0 ) γ c1 (u0 )c2 (u0 ) (2.8) Since the left-hand side of (2.8) is a quadratic polynomial, the following conclusions hold s − f (u0 ) < 0, the characteristic Equation (2.8) has two real roots of opposite signs γ c1 (u0 )c2 (u0 ) (ii) Denote by λ1 and λ2 the two roots of the characteristic Equation (2.8) Then (i) Since s − f (u0 ) > 0, γ c1 (u0 )c2 (u0 ) b(u0 ) β < λ1 + λ2 = − γ c1 (u0 )c2 (u0 ) Downloaded by [New York University] at 08:38 12 March 2016 λ1 · λ2 = Thus, either λ1 , λ2 are two negative eigenvalues, or λ1 and λ2 are complex and conjugate and have a negative real part Precisely, if := b(u0 ) β γ c1 (u0 )c2 (u0 ) −4 s − f (u0 ) ≥ 0, γ c1 (u0 )c2 (u0 ) then the two eigenvalues λ1 , λ2 are the same real and: λ1 ≤ λ2 < If and have a negative real part to be < 0, then λ1 , λ2 are complex b(u0 ) −1 β < γ c1 (u0 )c2 (u0 ) This completes the proof of Proposition 2.1 As indicated by Proposition 2.1, whenever f (u0 ) < s, the equilibrium point (u0 , 0) of (2.4) is asymptotically stable So, it has the domain of attraction In the following we will find an estimate for this domain of attraction, which is a subset of it For this purpose, we will use a Lyapunov function, and an estimation of attraction domain of the asymptotically stable equilibrium point (u0 , 0) will be made through level sets of this Lyapunov function Precisely, we define u0 L(u, v) = u v2 c2 (ξ ) h(ξ )dξ + γ c1 (ξ ) It holds that L(u0 , 0) = 0, ∇L(u, v) = − (2.9) c2 (u) h(u), v γ c1 (u) The derivative of L along trajectories of (2.4) is given by c2 (u) h(u).u + v.v γ c1 (u) v β b(u) c2 (u) h(u) + v h(u) − v =− γ c1 (u) c2 (u) γ c1 (u) γ c1 (u)c2 (u) b(u) β v < 0, for all v = =− γ c1 (u)c2 (u) ˙ v) = − L(u, On the other hand, since h (u0 ) = −s + f (u0 ) < 0, there exists a number θ = θ (u0 ) > such that h (u) < 0, u ∈ (u0 − θ , u0 + θ ), N H HIEP ET AL then h(ξ ) > h(u0 ) = 0, ξ ∈ [u0 − θ , u0 ), and h(ξ ) < h(u0 ) = 0, ξ ∈ (u0 , u0 + θ ] which yields u0 L(u, 0) = u c2 (ξ ) h(ξ )dξ > 0, γ c1 (ξ ) u ∈ [u0 − θ , u0 ) ∪ (u0 , u0 + θ ] This means that L is a Lyapunov function for the equilibrium point (u0 , 0) in u ∈ (u0 − θ , u0 + θ ) u c2 (ξ ) h(ξ )dξ > 0, and that the function u → h(u) is continuous, Now, set p = u0 − θ Since p γ c1 (ξ ) there exists a number q > u0 such that u0 Downloaded by [New York University] at 08:38 12 March 2016 p c2 (ξ ) h(ξ )dξ ≥ γ c1 (ξ ) or u0 q c2 (ξ ) h(ξ )dξ > 0, γ c1 (ξ ) L(p, 0) ≥ L(q, 0) > (2.10) Let us choose a sufficiently large positive number M = M(p, q) such that c2 (u) Lf , u∈[p,q] c1 (u) M > |s| + max where Lf is the Lipschitz constant of f over [p, q] Then, we define a region v ≤ |u0 − q|2 , u ≥ u0 , M2 |u0 − p|2 (u, v) ∈ IR2 |(u − u0 )2 + v ≤ |u0 − p|2 , (M|u0 − q|)2 Gp,q (u0 ) = (u, v) ∈ IR2 |(u − u0 )2 + (2.11) u ≤ u0 , which is a compact set The following proposition provides us with an estimate of the attraction domain of the asymptotically stable equilibrium point (u0 , 0) of (2.4) Lemma 2.2: Let (u0 , 0) be an asymptotically stable equilibrium point of (2.4) satisfying f (u0 ) < s, and let Gp,q (u0 ) be defined by (2.11) Then, for any positive number β < L(q, 0), the set β = {(u, v) ∈ Gp,q (u0 )|L(u, v) ≤ β} (2.12) is compact, positively invariant and lies entirely in G, and contains (u0 , 0) as an interior point Moreover, every trajectory of (2.4) starting in (2.13) = β β∈(0,L(q,0)) must approach the set of equilibria in as y → +∞ The proof of Lemma 2.2 is similar to the one in [11, Lem 3.2], so it is omitted Attraction domain of the equilibria Under the assumptions (1.2), we consider the case where the straight line (d) passing through a given point (u− , f (u− )) with the slope s meets the graph at four distinct points at u− , u1 , u2 , u3 For simplicity we may assume in the sequel that u1 < u2 < u3 < u− , (3.1) APPLICABLE ANALYSIS see Figure This intersection determines three shock waves, where u− is the left-hand state, s is the shock speed, and the right-hand state u+ can be u1 , u2 or u3 It is not difficult to check that h(u− ) = h(u1 ) = h(u2 ) = h(u3 ) = 0, where h is defined by (2.5) Moreover, it holds that Downloaded by [New York University] at 08:38 12 March 2016 h(u) > 0, for u ∈ ( − ∞, u1 ) ∪ (u2 , u3 ) ∪ (u− , +∞), h(u) < 0, for u ∈ (u1 , u2 ) ∪ (u3 , u− ), h (u1 ) = −s + f (u1 ) < 0, h (u3 ) = −s + f (u3 ) < 0, h (u2 ) = −s + f (u2 ) > 0, h (u− ) = −s + f (u− ) > (3.2) It follows from Proposition 2.1 and (3.2) that (u− , 0) and (u2 , 0) are saddle points, while (u1 , 0) and (u3 , 0) are asymptotically stable equilibrium points of (2.4) We define a Lyapunov function for each of these asymptotically stable equilibrium points as follows For (u1 , 0), we set u1 L1 (u, v) = u c2 (ξ ) v2 h(ξ )dξ + γ c1 (ξ ) (3.3) v2 c2 (ξ ) h(ξ )dξ + γ c1 (ξ ) (3.4) and for (u3 , 0), we set u3 L3 (u, v) = u Lemma 3.1: Consider a Lax shock connecting the left-hand u− and the right-hand state u+ = u3 with the shock speed s If L3 (u2 , 0) ≥ L3 (u− , 0), (3.5) then there exists a traveling wave connecting u− and u3 Proof: Let us define an attraction domain of the asymptotically stable equilibrium point (u3 , 0) by = {(u, v) ∈ Gp,u− (u3 )|L3 (u, v) < L3 (u− , 0)}, (3.6) where Gp,q (u0 ) is defined by (2.11) This attraction domain has the point (u− , 0) as a boundary point Let λ− > be an eigenvalue of A(u− ) corresponding to an eigenvector V− = (1, c2 (u− )λ− ) Then, every trajectory of (2.3) starting at (u− , 0) has to approach the line through (u− , 0) with direction V− So, this trajectory leaves (u− , 0) in one of the two quadrants Q1 (u− ) = {(u, v) ∈ IR2 |u < u− , v < 0}, Q2 (u− ) = {(u, v) ∈ IR2 |u > u− , v > 0} (3.7) Let us consider the trajectory leaving (u− , 0) in the quadrant Q1 Since L(u, v) is smooth, the tangent of at (u− , 0) is vertical This shows that the line through (u− , 0) with direction V− intersects with So, the trajectory leaving (u− , 0) at −∞ comes into the attraction domain According to Lemma 2.2, this trajectory approaches the set of equilibria of (2.4) as y → +∞ in , which contains exactly one point (u3 , 0) Now, we assume that L3 (u2 , 0) < L3 (u− , 0) It holds that N H HIEP ET AL u1 L1 (u− , 0) = u− c2 (ξ ) h(ξ )dξ = γ c1 (ξ ) u1 u2 c2 (ξ ) h(ξ )dξ − γ c1 (ξ ) u3 u2 c2 (ξ ) h(ξ )dξ + γ c1 (ξ ) u3 u− c2 (ξ ) h(ξ )dξ γ c1 (ξ ) = L1 (u2 , 0) − L3 (u2 0) + L3 (u− , 0) > L1 (u2 , 0) > By (1.2), L1 (u, 0) = u1 u c2 (ξ ) u→−∞ h(ξ )dξ −−−−→ +∞ So, there exists ν < u1 such that γ c1 (ξ ) L1 (ν, 0) = L1 (u− , 0) > Downloaded by [New York University] at 08:38 12 March 2016 We can now define = {(u, v) ∈ Gν,u− (u1 ) : L1 (u, v) < L1 (u− , 0)} It holds that L1 (ui , 0) < L1 (u− , 0), (3.8) (3.9) i = 1, 2, This implies that contains three equilibria (ui , 0), i = 1, 2, It is easy to see that the points (u− , 0) and (ν, 0) belong to the closure of Lemma 3.2: Given an equilibrium point (u− , 0) of (2.4) and a constant s Assume that the straight line through (u− , f (u− )) with the slope s cuts the graph of f at four distinct points: u1 < u2 < u3 < u− Then, there is a trajectory of (2.4) leaving (u− , 0) at −∞ approaches one of the two asymptotically stable equilibrium points (u1 , 0) and (u3 , 0) Proof: If L3 (u2 , 0) ≥ L3 (u− , 0), the conclusion is established by Lemma 3.1 Assume that L3 (u2 , 0) < L3 (u− , 0) Arguing similarly as in the proof of Lemma 3.1, we can show that there is a trajectory of (2.4) leaving (u− , 0) at −∞ goes to the attraction domain Thus, this trajectory must approach the set of equilibria of (2.4) in Furthermore, contains three equilibria {(ui , 0) : i = 1, 2, 3}, so this trajectory tends to one of these three points as y → +∞ It is now sufficient to show that every trajectory of (2.4) leaving (u− , 0) cannot approach (u2 , 0) Indeed, since L3 (u2 , 0) < L3 (u− , 0), there is a number ν3 ∈ (u3 , u− ) such that L3 (ν3 , 0) = L3 (u2 , 0) (3.10) Then, we define a domain of attraction of (u3 , 0) by = {(u, v) ∈ Gu2 ,ν3 (u3 ) : L3 (u, v) < L3 (u2 , 0)} (3.11) It is easy to see that has the point (u2 , 0) on its boundary (Figure 2) On the other hand, it holds that L1 (u2 , 0) > Thus, there exists a number ν1 < u1 such that L1 (ν1 , 0) = L1 (u2 , 0) > (3.12) Then, we define a domain of attraction of (u1 , 0) by = {(u, v) ∈ Gν1 ,u2 (u1 ) : L1 (u, v) < L1 (u2 , 0)} (3.13) The point (u2 , 0) also lie on the boundary of If a trajectory of (2.4) tending to (u2 , 0) as y → +∞, it must approach (u2 , 0) in the direction of the eigenvector V2 = (1, c2 (u2 )λ2 ) of A(u2 ), where λ2 < APPLICABLE ANALYSIS is the corresponding eigenvalue This implies that the trajectory would go to one of the attraction domains or This is a contradiction, since any trajectory going to an attraction set must remain in that set Lemma 2.2 is completely proved Next, we will study the properties of an arbitrarily trajectory of (2.4) leaving the saddle point (u− , 0) at −∞ and going to the attraction domain of (u1 , 0) Let y be the first time this trajectory cuts the line v = at u, that is y = sup{α ∈ IR|v(y) < 0, ∀y ∈ ( − ∞, α)}, u = u(y) (3.14) This also means that y = +∞ if (u(y), v(y)) does not meet the line v = If a trajectory (u(y), v(y)) cuts the lines u = ui , i = 1, 2, 3, then we define Downloaded by [New York University] at 08:38 12 March 2016 yi < y : u(yi ) = ui , i = 1, 2, (3.15) Consider a trajectory of (2.4) leaving (u− , 0) in the quadrant Q1 (u− ) = {(u, v) ∈ IR2 |u < u− , v < 0} v(y) < So, the trajectory part corresponding to y < y cuts the For y < y, it holds that u (y) = c2 (u(y)) lines u = ui , i = 1, 2, at most once This shows that the definition of the points yi , i = 1, 2, in (3.15) are reasonable Proposition 3.3: Let (u(y), v(y)) be a trajectory of (2.3) which comes to the domain defined by (3.9) Let y, u be given by (3.14) Then u ∈ (ν, u1 ] ∪ (u2 , u3 ] Proof: Since is an attraction domain of (u3 , 0), any trajectory comes into (3.16) must stay in So, ν < u < u− For every y < y, there exists ξ ∈ (y, y) such that v (ξ ) = Letting y → y, we obtain −v(y) v(y) − v(y) = > y−y y−y v (y) ≥ Substituting y = y into (2.4), we have v (y) = h(u) ≥ γ c1 (u) This yields h(u) ≥ This together with (3.2) imply that u ∈ (ν, u1 ] ∪ [u2 , u3 ] Finally, by Lemma 3.2, any trajectory of (2.4) cannot approach (u2 , 0), so u = u2 This completes the proof Existence of traveling waves In this section, we still assume the conditions (1.2), where u1 < u2 < u3 < u− , and L3 (u2 , 0) < L3 (u− , 0) Downloaded by [New York University] at 08:38 12 March 2016 10 N H HIEP ET AL Figure Attraction domains of the asymptotically stable equilibrium points We define the function L : IR → IR by u L(u) = u− c2 (ξ ) h(ξ )dξ γ c1 (ξ ) (4.1) c2 (u) h(u) By (3.2), the function L(u) is increasing in the intervals γ c1 (u) ( − ∞, u1 ), (u2 , u3 ), and (u− , +∞), and decreasing in the intervals (u1 , u2 ) and (u3 , u− ) Proposition 4.1: Suppose the straight line through (u− , 0) with the slope s cuts the graph of the flux function f at four distinct points u1 , < u2 < u3 < u− If u ∈ (u2 , u3 ], then It holds that L (u) = L(u2 ) < k L(u3 ), (4.2) where L is defined by (4.1), and √ β k= γ u− u2 b(ξ ) dξ c1 (ξ ) (4.3) Proof: From the first equation of (2.4) we imply that v = c2 (u)u Multiplying the second equation of (2.4) by v, we have vv = c2 (u) β b(u) h(u)u − vu γ c1 (u) γ c1 (u) For y < y, u(y) > u > u2 Integrating (4.4) over ( − ∞, y] gives us u c (ξ ) β u b(ξ ) v2 = h(ξ )dξ − v(ξ )dξ γ u− c1 (ξ ) u− γ c1 (ξ ) u c (ξ ) < h(ξ )dξ = L(u) u− γ c1 (ξ ) (4.4) Downloaded by [New York University] at 08:38 12 March 2016 APPLICABLE ANALYSIS 11 Figure A case where a trajectory does not enter the attraction domains Since u ∈ (u2 , u− ), L(u) ≤ L(u3 ) This yields v > − 2L(u3 ), u ∈ (u2 , u− ) (4.5) Integrating again (4.4) over ( − ∞, y] gives us u 0= u− β c2 (ξ ) h(ξ )dξ − γ c1 (ξ ) γ u u− b(ξ ) v(ξ )dξ c1 (ξ ) Using (4.5), we deduce that √ β u b(ξ ) β u b(ξ ) v(ξ )dξ ≤ − 2L(u3 ) dξ γ u− c1 (ξ ) γ u− c1 (ξ ) √ √ β u2 b(ξ ) dξ = k L(u3 ) < − 2L(u3 ) γ u− c1 (ξ ) L(u) = (4.6) On the other hand, the function L(u) is increasing on (u2 , u3 ) So, L(u) > L(u2 ) (4.7) From (4.6) and (4.7) we get (4.3) This completes the proof Theorem 4.2: Given an equilibrium point (u− , 0) of (2.4) and a constant s Assume that the straight line through (u− , f (u− )) with the slope s cuts the graph of f at four distinct points u1 , < u2 < u3 < u− so that √ k L(u3 ) ≤ L(u2 ), (4.8) mL(u3 ) ≤ S1 where L is defined by (4.1), and β = m 2γ b(u) u∈[ν1 ,u2 ] c1 (u) >0 (4.9) 12 N H HIEP ET AL and S1 is the area of the attraction domain Then, there exists a traveling wave of (1.1) corresponding to the nonclassical Lax shock connecting the left-hand state u− and the right-hand state u+ = u1 Proof: As seen by Proposition 3.3, u ∈ ( − ∞, u1 ] ∪ (u2 , u3 ] Moreover, by Proposition 4.1, since √ / (u2 , u3 ] Thus, u ∈ ( − ∞, u1 ] On the other hand, a trajectory k L(u3 ) ≤ L(u2 ), it holds that u ∈ of (2.4) whenever it comes to the attraction domain will never exit So, u ∈ (ν, u1 ] Integrating (4.4) over (y2 , y), y ≤ y, and denoting v2 := v(y2 ), we have Downloaded by [New York University] at 08:38 12 March 2016 v − v22 = u u2 c2 (ξ ) β h(ξ )dξ − γ c1 (ξ ) γ u u2 b(ξ ) v(ξ )dξ c1 (ξ ) (4.10) If a trajectory of (2.4) does not approach (u1 , 0), it then never enters the attraction domain This shows that (u(y), v(y)) ∈ / , ∀y ∈ [y2 , y) This is equivalent to the condition L1 (u, v) = L1 (u, 0) + Substituting v2 ≥ L1 (u2 , 0), ∀u ∈ (u, u2 ] (4.11) v2 from (4.10) into (4.11), we have u L1 (u, 0) + u2 β c2 (ξ ) h(ξ )dξ − γ c1 (ξ ) γ This yields β γ u u2 u u2 v2 b(ξ ) v(ξ )dξ + ≥ L1 (u2 , 0) c1 (ξ ) v2 b(ξ ) v(ξ )dξ ≤ c1 (ξ ) (4.12) From (4.5), (4.9), and (4.12) we deduce that m ν1 u2 β v(ξ )dξ ≤ γ or ν1 u2 b(ξ ) v(ξ )dξ ≤ L(u3 ), c1 (ξ ) ν1 v(ξ )dξ ≤ mL(u3 ) (4.13) u2 On the other hand, if the trajectory does not enter , it always lies below (Figure 3) So, the area of the region bounded by the trajectory (u(y), v(y)), y ∈ [ν1 , u2 ] and the lines u = ν1 , u = u2 , v = is larger than half of the area of the region In other words, ν1 v(ξ )dξ ≤ mL(u3 ) S1 < (4.14) u2 However, the second inequality of (4.8) prevents this to happen So, any trajectory of (2.4) leaving (u− , 0) enters and tends to (u1 , 0) as y → +∞ This completes the proof of Theorem 4.2 Downloaded by [New York University] at 08:38 12 March 2016 APPLICABLE ANALYSIS 13 Figure Graph of the flux function given by (4.2) intersects the straight line z = − u at four distinct points Numerical Illustration Consider the simplified model from (1.1) where all the coefficient functions are constant: ∂t u(x, t) + ∂x f (u(x, t)) = β∂xx u(x, t) + γ ∂xxx u(x, t), where f (u) = (u + 3)(u − 2)(u2 − u/2) − u + 1, x ∈ IR, t > 0, u ∈ IR (5.1) (5.2) The straight line (d) : z = − u cuts the graph of z = f (u) at four distinct points corresponding to the u-values u1 = −3, u2 = 0, u3 = 1/2, and u− = 2, see Figure The shock speed between u− and any state ui , i = 1, 2, is equal to the slope of the straight line (d), that is s= f (ui ) − f (u− ) = −1, ui − u− i = 1, 2, The system of ordinary differential Equations (2.4) for s = −1 now becomes du =v dy dv β = (u − u− + f (u) − f (u− )) − v dy γ γ (5.3) Set w0 = (u− − 0.01, −0.001), which is closed to the saddle point (u− , 0) of (5.3) We will use the command “ode45” in MATLAB to generate the approximate solution of (5.3) with the initial condition (u(0), v(0)) = w0 for different values of β and γ First, let us take β = 1/10, γ = 1/300, (5.4) Then, the trajectory of (5.3) starting at w0 converges to the stable node (u3 , 0) as y → ∞, see Figures and This trajectory corresponds to a traveling wave associated with a classical shock between the left-hand state u− and the right-hand state u+ = u3 Downloaded by [New York University] at 08:38 12 March 2016 14 N H HIEP ET AL Figure Trajectory of (4.3) starting near the saddle point (u− , 0) converges to the stable node (u+ = 1/2, 0) for the choice of (4.4) This is an approximation of a traveling wave corresponding to a classical shock Figure Approximation of the right part of the traveling wave corresponding to a classical shock in the (y, u)-plane for the choice of (4.4) However, when we take β = 1/10, γ = 1/50, (5.5) then the trajectory starting at the same initial state w0 as above converges to the stable node (u1 , 0) as y → ∞, see Figures and This trajectory corresponds to a traveling wave associated with a nonclassical Lax shock between the left-hand state u− and the right-hand state u+ = u1 Downloaded by [New York University] at 08:38 12 March 2016 APPLICABLE ANALYSIS 15 Figure Trajectory of (4.3) starting near the saddle point (u− , 0) converges to the stable focus (u+ = −3, 0) for the choice of (4.5) This is an approximation of a traveling wave corresponding to a nonclassical Lax shock Figure Approximation of the right part of the traveling wave corresponding to a nonclassical Lax shock in the (y, u)-plane for the choice of (4.5) 16 N H HIEP ET AL Acknowledgements The authors would like to thank the reviewers for their very constructive comments and fruitful discussions Disclosure statement No potential conflict of interest was reported by the authors Funding This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) [grant number B2015-28-02] Downloaded by [New York University] at 08:38 12 March 2016 References [1] Oleinik OA Construction of a generalized solution of the Cauchy problem for a quasi-linear equation of first order by the introduction of vanishing viscosity Amer Math Soc Trans Ser 1963;33:277–283 [2] Lax PD Shock waves and entropy In: Zarantonello EH, editor Contributions to nonlinear functional analysis New York (NY): Academic Press; 1971 p 603–634 [3] Liu TP The Riemann problem for general × conservation 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viscosity-capillarity J Differ Equ 2011;251:439–456  ... 2016 http://dx.doi.org/10.1080/00036811.2016.1157864 Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion Nguyen Huu Hiepa,c , Mai Duc Thanhb and Nguyen... nonclassical Lax shock, since it violates Oleinik’s entropy criterion (1.6), but it satisfies the Lax shock inequalities (1.7) The existence of traveling waves of (1.1) associated with a classical... In this paper, we study the existence of traveling waves associated with a Lax shock but violating the Oleinik entropy criterion (see Oleinik [1], Lax [2], Liu [3]) of the following diffusive–dispersive