This article was downloaded by: [New York University] On: 13 October 2014, At: 14:37 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Applicable Analysis: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gapa20 Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws a b Mai Duc Thanh & Nguyen Huu Hiep a Department of Mathematics, International University, Vietnam National University-Ho Chi Minh City, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam b Faculty of Applied Science, University of Technology, 268 Ly Thuong Kiet str., District 10, Ho Chi Minh City, Vietnam Published online: 30 Jul 2014 To cite this article: Mai Duc Thanh & Nguyen Huu Hiep (2014): Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws, Applicable Analysis: An International Journal, DOI: 10.1080/00036811.2014.915520 To link to this article: http://dx.doi.org/10.1080/00036811.2014.915520 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Downloaded by [New York University] at 14:37 13 October 2014 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Applicable Analysis, 2014 http://dx.doi.org/10.1080/00036811.2014.915520 Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws Mai Duc Thanha∗ and Nguyen Huu Hiepb a Department of Mathematics, International University, Vietnam National University-Ho Chi Minh City, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam; b Faculty of Downloaded by [New York University] at 14:37 13 October 2014 Applied Science, University of Technology, 268 Ly Thuong Kiet str., District 10, Ho Chi Minh City, Vietnam Communicated by M Shearer (Received March 2013; accepted 13 April 2014) Given any shock wave of a conservation law where the flux function may not be convex, we want to know whether it is admissible under the criterion of vanishing viscosity/capillarity effects In this work, we show that if the shock satisfies the Oleinik’s criterion and the Lax shock inequalities, then for an arbitrary diffusion coefficient, we can always find suitable dispersion coefficients such that the diffusive-dispersive model admits traveling waves approximating the given shock The paper develops the method of estimating attraction domain for traveling waves we have studied before Keywords: Conservation laws; traveling wave; shock; diffusion; dispersion; equilibria; asymptotical stability; Lyapunov function; LaSalle’s invariance principle; attraction domain AMS Subject Classifications: 35L65; 74N20; 76N10; 76L05 Introduction Let us consider the scalar conservation law ∂t u(x, t) + ∂x f (u(x, t)) = 0, x ∈ IR, t > 0, (1.1) where the flux function f is assumed to be merely differentiable, and may not be convex or concave, see Figure Solutions of (1.1) are understood in the sense of distributions and so are called weak solutions Weak solutions are in general discontinuous as they may contain shock waves, which are discontinuous jumps As well known, weak solutions are not unique Moreover, there are several kinds of shock waves, depending on the corresponding admissibility criterion, such as classical shocks and nonclassical shocks Each of these kinds of shock waves can be suitable for a particular application A reasonable mathematical modeling of the application can be made through the inclusion of some diffusion and dispersion terms in the Equation (1.1) to get a diffusive-dispersive models ∗ Corresponding author Email: mdthanh@hcmiu.edu.vn © 2014 Taylor & Francis M.D Thanh and N.H Hiep Downloaded by [New York University] at 14:37 13 October 2014 Figure A nonconvex flux function and a shock ∂t u(x, t) + ∂x f (u(x, t)) = β(b(u)u x )x + γ (c1 (u)(c2 (u)u x )x )x , x ∈ IR, t > 0, (1.2) where β > 0, γ > are small scales, the functions b(u), c1 (u), c2 (u), andu ∈ IR are assumed to be differentiable and positive, and c2 (u), u ∈ IR, is twice differentiable The first and the second terms on the right-hand side of (1.2) represent the diffusion and dispersion coefficients, respectively It has been known that traveling waves of (1.2) connecting a given left-hand state u − and a right-hand state u + tends to the shock wave connecting these left-hand and right-hand states when β and γ tend to zero Thus, the existence of traveling wave solutions of (1.2) corresponding to a shock wave can justify the admissibility criterion used to select this shock wave In [1], given a diffusive-dispersive model with constant viscosity and capillarity coefficients, the author proposed a method of estimating attraction domain of an asymptotically stable equilibrium point to establish the existence of a traveling wave associated with a given Lax shock In this paper, we will develop the method and improve the argument in [1] to show that for any given Lax shock of (1.1) satisfying the Oleinik’s condition, there are corresponding traveling waves of (1.2), providing that the diffusion and dispersion coefficients are chosen in a convenient way Traveling waves for diffusive and/or dispersive terms have attracted many authors Traveling waves were considered earlier for diffusive-dispersive scalar equations by Bona and Schonbek [2], Jacobs, McKinney, and Shearer [3] Traveling waves and admissibility criteria of the hyperbolic-elliptic model of phase transition dynamics were also studied by Slemrod [4,5] and Fan [6,7] Traveling waves corresponding to nonclassical shocks were studied by LeFloch and his collaborators and students, see [8–16] The developments of the method of estimating attraction domain of an asymptotically stable equilibrium point to establish the existence of a traveling wave for various models were carried out in [17–21] See also [21–23] for related works Background on shock waves and traveling waves First, let us recall that a shock wave of (1.1) is a weak solution of the form u(x, t) = u−, u+, x < st, x > st, (2.1) Applicable Analysis where u − , u + are the left-hand and right-hand states, respectively, and s is the shock speed A function of the form (2.1) is a weak solution of the conservation law (1.1), the Rankine– Hugoniot relation (2.2) −s(u + − u − ) + f (u + ) − f (u − ) = holds This relationship (2.2) means that the shock speed s can be evaluated by s = s(u − , u + ) = f (u + ) − f (u − ) u+ − u− Often, weak solutions are not unique Admissibility criteria have been set to select a unique solution For a single conservation law (1.1), one may use Oleinik’s criterion: Downloaded by [New York University] at 14:37 13 October 2014 f (u) − f (u − ) f (u + ) − f (u − ) ≥ , u − u− u+ − u− for any u between u + and u − (2.3) The condition (2.3) is also stated as f (u) − f (u + ) f (u + ) − f (u − ) ≤ , u − u+ u+ − u− for any u between u + and u − Geometrically, the inequality (2.3) means that if u + < u − , the graph of f is lying below the straight line ( ) connecting the two points (u ± , f (u ± )) in the interval [u + , u − ] To deal with simultaneous conservation laws, one can make use of the Lax shock inequalities, see [25] A shock wave of (1.1) is called a Lax shock if it satisfies the Lax shock inequalities f (u − ) > s(u − , u + ) > f (u + ), u− = u+ (2.4) Next, a traveling wave of (1.2) connecting a left-hand state u − to a right-hand state u + is a smooth solution of the form u = u(y), y = x − st, where s is a constant, and satisfies the boundary conditions lim u(y) = u ± , y→±∞ du d 2u = lim = y→±∞ dy y→±∞ dy lim (2.5) Substituting the traveling wave u into (1.2), we can see that the traveling wave u satisfies the ordinary differential equation −su + ( f (u)) = β(b(u)u ) + γ (c1 (u)(c2 (u)u ) ) , (2.6) where (.) = d(.)/dy Integrating (2.6) and using the boundary condition (2.5), we obtain βb(u)u + γ c1 (u)(c2 (u)u ) = −s(u − u − ) + f (u) − f (u − ) (2.7) Furthermore, the last equation and (2.5) give us s= f (u + ) − f (u − ) , u+ − u− which means that u − , u + , and s satisfy the Rankine–Hugoniot relation (2.2) So, these quantities are, respectively, the left-hand state, right-hand state, and shock speed of a shock wave of (2.1) 4 M.D Thanh and N.H Hiep Set v(y) = c2 (u)u Then, for γ = 0, the second-order differential equation (2.7) can be re-written as a system of two differential equations of first order v , u = c2 (u) βb(u)v + (−s(u − u − ) + f (u) − f (u − )), v =− (2.8) γ c1 (u)c2 (u) γ c1 (u) where u = u(y), v = v(y), y ∈ IR satisfying lim u(y) = u ± , lim v(y) = Downloaded by [New York University] at 14:37 13 October 2014 y→±∞ y→±∞ Setting h(u) = −s(u − u − ) + f (u) − f (u − ), U = (u, v)T , F(U ) = βb(u)v v ,− + h(u) c2 (u) γ c1 (u)c2 (u) γ c1 (u) T , (2.9) we can re-write the system (2.8) in the form dU = F(U ), dy −∞ < y < +∞ (2.10) It is easy to check that a point U in the (u, v)-phase plane is an equilibrium point of the autonomous differential equations (2.8) if and only if U has the form U = (u + , 0) for some constant u + so that the states u ± and the shock speed s satisfy the Rankine–Hugoniot relation (2.2) Consequently, u = u(x, t) defined by (2.1) is a weak solution of the conservation law (1.1) Conversely, a jump of (1.1) of the form (2.1) gives equilibria (u − , 0), (u + , 0) of the differential equation (2.8) The Jacobian matrix D F(U ) of the system (2.10) at U = (u, v) is given by ⎞ ⎛ c (u)v − 22 ⎟ ⎜ c2 (u) c2 (u) ⎟, D F(U ) = ⎜ ⎠ ⎝ βd (u)v c1 (u)h(u) f (u) − s βb(u) − + − − γ γ c1 (u) γ c1 (u)c2 (u) γ c1 (u) where d(u) = b(u) At u ± , using the condition that h(u ± ) = 0, we have c1 (u)c2 (u) ⎞ ⎛ ⎟ ⎜ c2 (u ± ) ⎟ D F(u ± , 0) = ⎜ ⎠ ⎝ f (u ± ) − s βb(u ± ) − γ c1 (u ± ) γ c1 (u ± )c2 (u ± ) The characteristic equation of D F(u ± , 0) is then given by λ2 + a1 (u ± )λ − a2 (u ± ) = 0, where a1 (u ± ) = βb(u ± ) > 0, γ c1 (u ± )c2 (u ± ) a2 (u ± ) = f (u ± ) − s γ c1 (u ± )c2 (u ± ) Applicable Analysis Assume that the shock satisfies the Lax shock inequalities f (u − ) > s(u − , u + ) > f (u + ), u− = u+ Then, a2 (u + ) < 0, a2 (u − ) > Since a2 (u − ) > 0, the Jacobian matrix at (u − , 0) admits two real eigenvalues having opposite signs Downloaded by [New York University] at 14:37 13 October 2014 λ1 (u − , 0) = λ2 (u − , 0) = −a1 (u − ) − a12 (u − ) + 4a2 (u − ) −a1 (u − ) + a12 (u − ) + 4a2 (u − ) < 0, > (2.11) The point (u − , 0) is thus a saddle point As seen above, a2 (u + ) < So, the Jacobian matrix at (u + , 0) admits two eigenvalues with negative real parts λ1,2 (u + , 0) = −a1 (u + ) ± a12 (u + ) + 4a2 (u + ) (2.12) Precisely, if a12 (u + ) + 4a2 (u + ) ≥ 0, then λ1,2 (u + , 0) are real and negative Otherwise, λ1,2 (u + , 0) are complex, conjugate, and have the real negative part −a1 (u + )/2 Thus, the point (u + , 0) is asymptotically stable Existence of traveling waves 3.1 Estimate of attraction domain of the attracting equilibrium Let us re-write the system (2.8) in the form v , u = c2 (u) βb(u)v + h(u), v =− γ c1 (u)c2 (u) γ c1 (u) (3.1) where h(u) = −s(u − u − ) + f (u) − f (u − ) We define a Lyapunov function candidate corresponding to the equilibrium point (u + , 0): L(u, v) = γ u+ u v2 c2 (ξ ) h(ξ )dξ + c1 (ξ ) (3.2) Let the shock wave connecting the left-hand state u − with the right-hand state u + with the shock speed s = s(u − , u + ) satisfy the Oleinik’s criterion and the Lax shock inequalities For definitiveness, we assume that u+ < u−, without restriction 6 M.D Thanh and N.H Hiep Theorem 3.1 There always exists a value u ∗ < u + such that h(u) > 0, u∗ < u < u+, h(u) < 0, u+ < u < u− (3.3) Consequently, the function L defined by (3.2) satisfies L(u + , 0) = 0, L(u, v) > 0, , βb(u)v ˙ < 0, L(u, v) = − γ c1 (u)c2 (u) u∗ < u < u−, u = u+, for v = (3.4) This means that L is a Lyapunov function on the set Downloaded by [New York University] at 14:37 13 October 2014 D: u∗ ≤ u ≤ u− Proof Since h (u + ) = −s + f (u + ) < 0, by the Lax shock inequalities, and h(u + ) = 0, the continuity implies that h(u) > for u ∈ (u + − ε, u + ), for some ε > This establishes the first statement in (3.3) The second statement of (3.3) follows from the Oleinik criterion: h(u) = −s(u − u − ) + f (u) − f (u − ) = (u − u − ) −s + f (u) − f (u − ) u − u− < 0, for u + < u < u − Next, we have L(u + , 0) = 0, and + u c2 (ξ ) h(ξ )dξ > L(u, v) ≥ γ u c1 (ξ ) by using (3.3) This establishes the statements in the first line of (3.4) Next, the derivative of L along trajectories of (3.1) is given by du dv ˙ , > L(u, v) = ∇ L(u, v)· < dy dy c2 (u)h(u) v βb(u)v h(u) =− + − + v γ c1 (u) c2 (u) γ c1 (u)c2 (u) γ c1 (u) βb(u)v =− < 0, for v = 0, γ c1 (u)c2 (u) which establishes the second line in (3.4) Lem m a 3.2 (a) Let u m = (u ∗ + u + )/2, where u ∗ is given by (3.3) Given any continuous function c1 (u) > 0, u ∈ IR, there are always infinitely many choices of a C ∞ function c2 (u) > 0, u ∈ IR such that c2 c1 um u∗ L ∞ [u + ,u − ] ||h|| L ∞ [u + ,u − ] c2 (u) du > |u + − u − | c1 (u) min[u ∗ ,u m ] h(u) (3.5) Downloaded by [New York University] at 14:37 13 October 2014 Applicable Analysis Figure An estimate (u + , 0) (b) β of the attraction domain of the asymptotically stable equilibrium point Conversely, given any positive continuous function c2 (u), u ∈ IR, there are always infinitely many choices of a C ∞ function c1 (u), u ∈ IR satisfying (3.5) Let c1 , c2 be a pair of positive continuous functions satisfy (3.5) Then, for any positive number < β < L(u − , 0), the set β := {(u, v) ∈ D| L(u, v) ≤ β} (3.6) (see Figure 2) is a compact set, positively invariant with respect to (3.1), and has the point (u + , 0) as an interior point Proof (a) Observe that a C ∞ function has the derivative up to any order Set α to be the value of the right-hand side of the inequality (3.5): α= ||h|| L ∞ [u + ,u − ] |u + − u − | min[u ∗ ,u m ] h(u) (3.7) Obviously, α > Let us take an arbitrary positive continuous function ω(u), u ∈ IR, and define a function ⎧ 2α ⎪ ⎪ max{c1 (u), ω(u)}, if u ∈ [u ∗ , u m ], ⎪ ⎪ um − u∗ ⎪ ⎪ ⎪1 ⎪ ⎪ if u ∈ [u + , u − ], ⎪ ⎪ min{c1 (u), ω(u)}, ⎪ ⎨ c(u + ) − c(u m ) (u − u m ), if u ∈ [u m , u + ], (3.8) c(u) = c(u m ) + ⎪ u+ − um ⎪ ⎪ ⎪ 2α ⎪ ⎪ max{c1 (u ∗ ), ω(u ∗ )} = c(u ∗ ), if u ≤ u ∗ , ⎪ ⎪ u ⎪ m − u∗ ⎪ ⎪ ⎪ ⎩ min{c1 (u − ), ω(u − )} = c(u − ), if u ≥ u − It is easy to see that the function c defined by (3.8) is positive and continuous on IR, and M.D Thanh and N.H Hiep 2α c1 (u), u ∈ [u ∗ , u m ], um − u∗ (3.9) c(u) ≤ c1 (u), u ∈ [u + , u − ] Then, we will consider the mollification cε in the interval I = (u ∗ − 1, u − + 1) of c defined by (3.8) Recall that a standard mollifier is defined by c(u) ≥ C exp |u|21−1 , if |u| ≥ 1, η(u) = if |u| < 1, , Downloaded by [New York University] at 14:37 13 October 2014 where 1 = exp du C |u| − −1 Then, for each ε > 0, the function η(u/ε), ε ηε (u) = u ∈ IR, is C ∞ , has the support in (−ε, ε), and satisfies IR ηε (u)du = A mollification of c on I is defined by cε (u) = (c ∗ ηε )(u) = u − +1 u ∗ −1 ηε (u − v)c(v)dv = ε −ε ηε (v)c(u − v)dv (3.10) As well known, cε ∈ C ∞ (Iε ), where Iε = (u ∗ − + ε, u − + − ε) It is not difficult to check that cε (u) = c(u ∗ ), u ∈ (u ∗ − + ε, u ∗ − ε), cε (u) = c(u − ), u ∈ (u − + ε, u − + − ε), for < ε < 1/2 Thus, a natural extension of cε to be constant outside Iε makes sure I ) Moreover, since c is continuous, cε → c as ε → uniformly that it is of C ∞ (R on compact subsets of the interval I = (u ∗ − 1, u − + 1) Thus, since c(u), u ∈ IR, is positive, there exists ε0 > such that 2c(u) > cε (u) > c(u)/2, u ∈ [u + , u − ] ∪ [u ∗ , u m ], < ε < ε0 From (3.9) and the last inequalities, considering cε as c2 , we can estimate the left-hand side of (3.5) by um cε c1 u∗ L ∞ [u + ,u − ] cε (u) du > c1 (u) um c(u) du 2c1 (u) 2c u∗ c1 L ∞ [u + ,u − ] um 2α ≥ du 2(1/2) u ∗ 2(u m − u ∗ ) = α, < ε < ε0 , Applicable Analysis where α is the value of the right-hand side of the inequality (3.5) defined by (3.7) Therefore, (3.5) is satisfied for cε to play the role of c2 , and so we can take c2 = cε for any ε ∈ (0, ε0 ) Observe that the definition of c in (3.8) involves an arbitrary choice of a positive continuous function ω So, there are a lot of choices for c2 , given c1 Conversely, if c2 is given, by setting c¯1 = 1/c2 , c¯2 = 1/c1 , and repeating the above argument for c¯1 , c¯2 , we obtain a lot of choices for c1 This establishes the part (a) (b) First, we can see that β is closed, by continuity Next, we will show that β is bounded Let us take a constant M > such that M2 > max Downloaded by [New York University] at 14:37 13 October 2014 We will show that the set β c2 (u) u∈[u ∗ ,u − ] c1 (u) β || f || L ∞ [u ∗ ,u − ] + |s| (3.11) is contained in a bounded set: v ≤ |u + − u − |2 , u ≥ u + M2 |u + − u ∗ |2 ∪ (u, v) ∈ IR2 |(u − u + )2 + v ≤ |u + − u ∗ |2 , u ≤ u + (M|u + − u − |)2 (3.12) ⊂ G := (u, v) ∈ IR2 |(u − u + )2 + Consider the semi-ellipse ∂G, u ≥ u + , where it holds that v = M (|u + − u − |2 − (u − u + )2 ) Substituting v from the last equation to the expression of L, we have L(u, v) (u,v)∈∂G,u≥u + M2 c2 (ξ ) h(ξ )dξ + (|u + − u − |2 − (u − u + )2 ) c (ξ ) u := g(u), u ∈ [u + , u − ] = u+ It follows from (3.11) that c2 (u) dg(u) =− h(u) − M (u − u + ) du c1 (u) c2 (u) f (u) − f (u + ) −s = −(u − u + ) M + c1 (u) u − u+ < 0, u ∈ (u + , u − ) Hence, the function g is therefore strictly decreasing for u ∈ [u + , u − ] This yields L(u, v) = L(u − , 0) L(u, v) = L(u ∗ , 0) (u,v)∈ ∂G,u≥u + Similarly, it holds that (u,v)∈ ∂G,u≤u + Next, we will show that L(u ∗ , 0) ≥ L(u − , 0), or L(u ∗ , 0) − L(u − , 0) ≥ 10 M.D Thanh and N.H Hiep Actually, we can estimate as follows γ (L(u ∗ , 0) − L(u − , 0)) = > > u+ u∗ um u∗ um u∗ c2 (u) h(u)du − c1 (u) c2 (u) h(u)du − c1 (u) u− u+ c2 (u) h(u)du c1 (u) h(u) max u− c2 (u) h(u)du − c1 (u) [u ∗ ,u m ] c2 (u) [u + ,u − ] c1 (u) u+ h(u) max u− du c2 (u) [u + ,u − ] c1 (u) u+ du c2 c2 (u) du − h(u)du [u ∗ ,u m ] c1 L ∞ [u + ,u − ] u − u ∗ c1 (u) u m c (u) du ≥ h(u) [u ∗ ,u m ] u ∗ c1 (u) c2 − h L ∞ [u + ,u − ] |u + − u − | c1 L ∞ [u + ,u − ] = h(u) Downloaded by [New York University] at 14:37 13 October 2014 u+ um > 0, where the last inequality is derived from (3.5) Thus, L(u ∗ , 0) ≥ L(u − , 0), and therefore (3.13) L(u, v) = L(u − , 0), (u,v)∈∂G where ∂G denotes the boundary of G Furthermore, it is easy to check that the function u → L(u, 0) is increasing for u + < u < u − and decreasing for u ∗ < u < u + Thus, (3.13) implies β is a subset of G for any positive number β < L(u − , 0) This establishes (3.12), and we conclude that β is a compact set Now, we will show that the set β is in fact in the interior of G Indeed, assume the contrary, then there is a point (u , v0 ) ∈ β which lies on the boundary of G Then, in view of (3.13), and by definition of minimum, we would have L(u , v0 ) ≥ L(u − , 0) > β ≥ L(u , v0 ), which is a contradiction Thus, the closed curve L(u, v) = β lies entirely in the interior of G Moreover, the semi-negativity of the derivative along trajectories of (3.1) of L yields d L(u(y), v(y)) ≤ dy Thus, L(u(y), v(y)) ≤ L(u(0), v(0)) ≤ β, ∀y > 0, which shows that any trajectory starting in β cannot cross the closed curve L(u, v) = β Therefore, the compact set β is positively invariant with respect to (3.1) This completes the proof of Lemma 3.2 In what follows, we will show that trajectories of (3.1) starting in β exist for all y > Moreover, these trajectories will converge to the node (u + , 0) as y → +∞ Applicable Analysis 11 Theorem 3.3 Under (3.5), for any given (u , v0 ) ∈ β , the initial-value problem for (3.1) with initial condition (u(0), v(0)) = (u , v0 ) admits a unique global solution (u(y), v(y)) for all y ≥ Moreover, this trajectory converges to (u + , 0) as y → +∞, i.e lim (u(y), v(y)) = (u + , 0) y→+∞ This means that the equilibrium point (u + , 0) is asymptotically stable Downloaded by [New York University] at 14:37 13 October 2014 Proof As seen by Lemma 3.2, the set β defined by (3.6) is a compact set and is positively invariant with respect to (3.1) This simply means that any solution of (3.1) starting in β lies entirely in β It follows from the existence theory of differential equations that there is a unique solution starting in β defined for all y ≥ Denote E = {(u, v) ∈ ˙ v) β | L(u, = 0} Then, it follows from (3.4) that E = {(u, v) ∈ β |v = 0} (3.14) LaSalle’s invariance principle (see [26], for example) implies that every trajectory of (3.1) starting in β approaches the largest invariant set M of the set E as y → ∞ Therefore, it is sufficient to show that the set M contains only one point (u + , 0) This can be done by showing that no solution can stay in E, other than the trivial solution (u(y), v(y)) ≡ (u + , 0) Indeed, let (u(y), v(y)) be a solution that stays in E Then, v(y) du(y) = ≡ 0, dy c2 (y) which implies that u is identically equal to u + And so, (u, v) = (u(y), v(y)), y ≥ 0, is identically equal to (u + , 0) Thus, LaSalle’s invariance principle implies that every trajectory of (3.1) starting in β converges to the equilibrium point (u + , 0) as y → ∞ This completes the proof 3.2 Saddle-to-attractor connection and the traveling wave In this subsection, we will establish existence result First we consider the stable trajectories issuing from the saddle point (u − , 0) at −∞ The union of the sets β , < β < L(u − , 0) can give a reasonable estimate for the attraction domain In fact, we set = ∪0 0} Since we need the stable trajectory that goes beyond the attracting equilibrium, we will study only the stable trajectory that leaves the saddle point in the quadrant Q Multiply both sides of the second equation of (3.1) by v = c2 (u) du dy and integrating from (−∞, y), we get y −∞ or v dv dy = dy y −∞ c2 (u) du dy − βb(u)v + h(u) dy, γ c1 (u)c2 (u) γ c1 (u) u v2 β = h(w) dw c2 (w) − d(w)v + γ γ c (w) u− Since (u, v) in Q , v < So 0≤ or u− u v2 < u u− c2 (w) h(w)dw, γ c1 (w) v2 c2 (w) h(w)dw + < γ c1 (w) Applicable Analysis 13 The last inequality shows that the stable trajectory leaving the saddle point enters the attraction domain of the attracting equilibrium: (u(y), v(y)) ∈ , Downloaded by [New York University] at 14:37 13 October 2014 which establishes a saddle-to-attractor connection Thus, we have shown that a traveling wave connecting a left-hand state u − and a righthand state u + exists, provided the diffusion and dispersion coefficients are chosen suitably, for example, when (3.5) is satisfied This leads us to the following main theorem Theorem 3.5 Any shock wave (2.1) of (1.1) satisfying Oleinik’s criterion (2.3) and the Lax shock inequalities (2.4) can be approximated by a traveling wave of (1.2) under (3.5) More precisely, given any shock wave of (1.1) connecting a left-hand state u − and a right-hand state u + which satisfies Oleinik’s criterion (2.3) and the Lax shock inequalities (2.4), there are infinitely many choices of coefficients c1 , c2 such that the condition (3.5) is satisfied and the diffusive-dispersive model (1.2) therefore has a traveling wave connecting the left-hand state u − and the right-hand state u + References [1] Thanh MD Global existence of traveling wave for general flux functions Nonlinear Anal.: T.M.A 2010;72:231–239 [2] Bona J, Schonbek ME Traveling-wave solutions to the Korteweg-de Vries-Burgers equation Proc Royal Soc Edinburgh 1985;101:207–226 [3] Jacobs D, McKinney W, Shearer M Travelling wave solutions of the modified KortewegdeVries-Burgers equation J Differ Equ 1995;116:448–467 [4] Slemrod M Admissibility criteria for propagating phase boundaries in a van der Waals fluid Arch Rational Mech Anal 1983;81:301–315 [5] Slemrod M The viscosity-capillarity criterion for shocks and phase transitions Arch Rational Mech Anal 1983;83:333–361 [6] Fan H A vanishing viscosity approach on the dynamics of phase transitions in van der Waals fluids J Differ Equ 1993;103:179–204 [7] Fan H Traveling waves, Riemann problems and computations of a model of the dynamics of liquid/vapor phase transitions J Differ Equ 1998;150:385–437 [8] Hayes BT, LeFloch PG Non-classical shocks and kinetic relations: scalar conservation laws Arch Ration Mech Anal 1997;139:1–56 [9] Bedjaoui N Diffusive-dispersive traveling waves and kinetic relations I Non-convex hyperbolic conservation laws J Diff Eqs 2002;178:574–607 [10] Bedjaoui N, LeFloch PG Diffusive-dispersive traveling waves and kinetic relations II A hyperbolic-elliptic model of phase-transition dynamics Proc Roy Soc Edinburgh 2002;132:545–565 [11] Bedjaoui N, LeFloch PG Diffusive-dispersive traveling waves and kinetic relations III An hyperbolic model from nonlinear elastodynamics Ann Univ Ferrara Sc Mat 2001;44: 117–144 [12] Bedjaoui N, LeFloch PG Diffusive-dispersive traveling waves and kinetic relations IV Compressible Euler equations Chin Ann Math 2003;24:17–34 [13] Bedjaoui N, LeFloch PG Diffusive-dispersive traveling waves and kinetic relations V Singular diffusion and nonlinear dispersion Proc Roy Soc Edinburgh 2004;134:815–843 Downloaded by [New York University] at 14:37 13 October 2014 14 M.D Thanh and N.H Hiep [14] Bedjaoui N, Chalons C, Coquel F, LeFloch PG Non-monotone traveling waves in van der Waals fluids Ann Appl 2005;3:419–446 [15] LeFloch PG Hyperbolic systems of conservation laws The theory of classical and nonclassical shock waves, Lectures in mathematics Basel: ETH Zürich, Birkhäuser; 2002 [16] Liu TP The Riemann problem for general × conservation laws Trans Am Math Soc 1974;199:89–112 [17] Thanh MD Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion Nonlinear Anal.: T.M.A 2010;72:3136–3149 [18] Thanh MD Existence of traveling waves in elastodynamics with variable viscosity and capillarity Nonlinear Anal.: R.W.A 2011;12: 236–245 [19] Thanh MD Traveling waves of an elliptic-hyperbolic model of phase transitions via varying viscosity-capillarity J Differ Equ 2011;251:439–456 [20] Thanh MD Existence of traveling waves in compressible Euler equations with viscosity and capillarity Nonlinear Anal.: T.M.A 2012;75:4884–4895 [21] Thanh MD, Huy ND, Hiep NH, Cuong DH Existence of traveling waves in van der Waals fluids with viscosity and capillarity effects Nonlinear Anal.: T.M.A 2014;95:743–755 [22] Gilbarg D The existence and limit behavior of the one-dimensional shock layer Am J Math 1951;73:256–274 [23] Benzoni-Gavage S, Danchin R, Descombes S Well-posedness of one-dimensional Korteweg models Electron J Diff Eqs 2006;59:1–35 [24] Benzoni-Gavage S, Danchin R, Descombes S On the well-posedness of the Euler-Korteweg model in several space dimensions Indiana Univ Math Journal 2007;56:1499–1579 [25] Lax PD Shock waves and entropy In: Zarantonello EH, editor Contributions to nonlinear functional analysis New York (NY): Academic Press; 1971 p 603–634 [26] Khalil HK Nonlinear systems New Jersey: Prentice Hall; 2002  ... http://dx.doi.org/10.1080/00036811.2014.915520 Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws Mai Duc Thanha∗ and Nguyen Huu Hiepb a Department of Mathematics, International University,... improve the argument in [1] to show that for any given Lax shock of (1.1) satisfying the Oleinik’s condition, there are corresponding traveling waves of (1.2), providing that the diffusion and... us to the following main theorem Theorem 3.5 Any shock wave (2.1) of (1.1) satisfying Oleinik’s criterion (2.3) and the Lax shock inequalities (2.4) can be approximated by a traveling wave of