THÔNG TIN TÀI LIỆU
Applied Mathematics and Computation 234 (2014) 127–141 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On traveling waves in viscous-capillary Euler equations with thermal conductivity Mai Duc Thanh a,⇑, Nguyen Huu Hiep b a b Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam Faculty of Applied Science, University of Technology, 268 Ly Thuong Kiet str., District 10, Ho Chi Minh City, Viet Nam a r t i c l e i n f o Keywords: Gas dynamics equations Traveling wave Shock Viscosity Capillarity Thermal conductivity Equilibria Stability a b s t r a c t This work establishes the stability of the equilibrium states corresponding to traveling waves in viscous-capillary Euler equations when a standard thermal conductivity coefficient is present Due to the presence of the heat conduction, the associated system of ordinary differential equations is much more involved and complicated Given a shock wave, we can obtain exactly the sign of the real part of each eigenvalue of the Jacobian matrix of the corresponding system of ODEs at the two equilibria associated with the left-hand and right-hand states of the shock It turns out that one equilibrium point is asymptotically stable, why the other point is unstable and admits eigenvalues whose real parts have opposite signs Suitable approximate connections between the unstable and stable equilibria are obtained by numerical tests for various ranges of thermal conductivity: low, medium, and high values Moreover, numerical tests also suggest that a trajectory could leave the unstable equilibrium point and enters the attraction domain of the asymptotically stable equilibrium point This work therefore may motivate for further study on the existence of the traveling waves of the viscous-capillary gas dynamics equations with the presence of heat conduction Ó 2014 Elsevier Inc All rights reserved Introduction Heat conduction causes a tough obstacle for studying traveling waves in gas dynamics equations Therefore, the existence of traveling waves for gas dynamics equations with thermal conductivity is a very interesting and challenging problem This work concerns with the traveling waves of the following model of fluid dynamics equations with viscosity, capillarity, and heat conduction v t À ux ¼ 0; l ux lv x ịxx ỵ v v 2x ; v x x l j k Et ỵ upịx ẳ uux ỵ v uv x ulv x ịx ỵ lux v x ịx ỵ T v v x x x x ut ỵ px ẳ k Corresponding author E-mail addresses: mdthanh@hcmiu.edu.vn (M.D Thanh), nguyenhuuhiep@hcmut.edu.vn (N.H Hiep) http://dx.doi.org/10.1016/j.amc.2014.02.004 0096-3003/Ó 2014 Elsevier Inc All rights reserved ð1:1Þ 128 M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 for x R and t > Here, v ; S; p; e; T denote the specific volume, entropy, pressure, internal energy, temperature, respectively; u is the velocity, and Eẳeỵ u2 l ỵ v 2 x 1:2ị is the total energy The non-negative quantities k; l; j represent the viscosity, capillarity, and the heat conduction, respectively Observe that the Lagrangian coordinates are chosen so that the calculations are simple only, since similar results hold for the Eulerian coordinates It has been known that traveling waves can be used to justify an admissibility criterion for shock waves of different types, such as the Lax shocks [13] and nonclassical shocks [11] However, the study of traveling waves has been restricted mostly to the case where additional terms are merely viscosity and capillarity Heat conduction is commonly ignored The obstacle for the study of traveling waves when heat conduction is present is that this quantity causes tough inconvenience to establish a system of ordinary differential equations whose equilibria are used to characterize the traveling waves Recently, the system of ordinary differential equations associated with traveling waves of (1.1) was derived by [21], where the equilibrium points of the system were shown to correspond to the left-hand and right-hand states of the given shock In this work, we first establish the stability of these corresponding equilibrium points Precisely, by investigating the sign of the real part of each eigenvalue of the Jacobian matrix of the resulted system of ODEs at the two equilibrium points, we can show that one equilibrium point is asymptotically stable, and the other equilibrium point is unstable and it furthermore admits eigenvalues whose real parts are of opposite signs Thus, existence of a traveling wave can be expected and could be a topic for further study For example, whenever an unstable trajectory leaves the unstable equilibrium point at À1 and enters the attraction domain of the asymptotically stable equilibrium point, it will converge to the asymptotically stable equilibrium point at 1, and so a traveling wave is obtained Then, we present numerical tests, which all show that the trajectory starting very closed to the unstable equilibrium point converges to the asymptotically stable equilibrium point Traveling waves have attracted attention of many authors An early work on the related shock layers of the gas dynamics equations with viscosity and heat conduction effects (with zero capillarity) was presented in [10] Traveling waves for diffusive-dispersive scalar equations were earlier studied in [7,12] Traveling waves of the hyperbolic-elliptic model of phase transition dynamics were considered in [16,17,8,9,15] The existence of traveling waves associated with Lax shocks for viscous-capillary models was considered in our recent works [19–23] These works developed the method of estimating attraction domain of the asymptotically stable equilibrium point to establish the existence of traveling waves The existence of traveling waves corresponding to nonclassical shocks for viscous-capillary models was considered in [11,3,4,2,5,6,1] See also the references therein The organization of this paper is as follows Section provides basic concepts and properties of the fluid dynamics equations and the autonomous system of ordinary differential equations for a given traveling wave Section is devoted to the stability of the resulted equilibria of the corresponding system of ODEs In Section we present several numerical tests, where approximate traveling waves are computed for both cases of positive and negative shock speeds, with large and small thermal conductivity In Section we will provide some conclusions and discussions Preliminaries 2.1 Shock waves and Lax shock inequalities If one lets the viscosity, the capillarity and the thermal conductivity coefficients in (1.1) tend to zero, one obtains the fluid dynamics equations in the Lagrangian coordinates v t À ux ¼ 0; ut ỵ px ẳ 0; 2:1ị Et ỵ upịx ẳ 0; x R; t > 0: As well-known, the system (2.1) can be written in terms of the variable U ¼ ðv ; u; SÞ by v t À ux ẳ 0; ut ỵ pv v ; Sịv x ỵ pS v ; SịSx ẳ 0; St ẳ 0: The Jacobian matrix of the last system is given by 0 B B A ¼ B pv @ À1 0 C C pS C; A M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 129 which admits three distinct real eigenvalues pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p k1 ẳ pv v ; Sị < k2 ¼ < k3 ¼ Àpv ðv ; SÞ; so that the system (2.1) is strictly hyperbolic, whenever pv ðv ; SÞ < The value ki is often referred to as the characteristic speed of the ith characteristic field, i ¼ 1; 2; Let U À and U þ be given constant states A shock wave of (2.1) connecting the left-hand state U À and the right-hand state U ỵ is a weak solution U of the form & Ux; tị ẳ U ; if x < st; Uỵ ; if x > st; 2:2ị with a constant shock speed s The shock speed s can be determined via the following RankineHugoniot relations for this shock sv ỵ v ị ỵ uỵ u ị ẳ 0; suỵ u ị ỵ pỵ p ẳ 0; 2:3ị p ỵp eỵ e ỵ ỵ v ỵ v ị ẳ 0; that is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p À pÀ s ¼ sðU À ; U ỵ ị ẳ ặ ỵ ; vỵ v 2:4ị whenever pỵ p vỵ vÀ P 0: It has been known that a negative shock speed corresponds to the first characteristic field, and a positive shock speed corresponds to the third characteristic field of (2.1) A shock wave is admissible under a certain admissibility criterion The most standard admissibility criterion is the Lax shock inequalities which requires the shock speed s ¼ sðU À ; U ỵ ị to satisfy ki U ỵ ị < sU ; U ỵ ị < ki U ị; i ẳ 1; 3: 2:5ị 2.2 System of ODEs for traveling waves A traveling wave of (1.1) connecting a left-hand state, denoted by U À , and a right-hand state, denoted by U ỵ , is a smooth solution of (1.1) of the form U ẳ Uyị; y ¼ x À st, where s is a constant, and satises the following boundary conditions limy!ặ1 Uyị ẳ U ặ limy!ặ1 d d Uyị ẳ limy!ặ1 Uyị ẳ 0: dy dy ð2:6Þ The derivation of the systems of ordinary differential equations for traveling waves of (1.1) written in the ðv ; u; SÞ variable was obtained in [24] Here, we present another form of systems of ODEs for traveling waves of (1.1) as follows Arguing as in [24], we have the following differential equations for traveling waves of (1.1) u0 ẳ sv ; lv ị ẳ s k v v0 ỵ lv 2 v Þ À ðp À pÀ Þ À s2 ðv À v À Þ; ð2:7Þ s s3 j lðv ị2 ẳ se e ị ỵ sp v v ị v v ị2 ỵ T : 2 v Let us choose the temperature variable as the unknown and by re-writing the second equation of (2.7) in terms of the temperature, we obtain from (2.7) lv ị ẳ s k lv ðv Þ À ðp À pÀ Þ À s2 ðv À v À Þ; sv s2 lðv Þ2 À ðe À eÀ ị ỵ p v v ị v v ị2 : T0 ẳ j 2 v v0 ỵ 2:8ị 130 M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 The equations (2.8) can be written as the following system of first-order ordinary differential equations v0 ¼ w l ; Á l ðv ; Sv ; Tịị wỵ v w p p ỵ s2 v v ị ; 2l2 sv s2 lðv ị2 e e ị ỵ p v v À Þ À ðv À v À Þ2 ; T0 ¼ j 2 w0 ¼ À sk lv ð2:9Þ in terms of the variable ðv ; w; TÞ Observe that the system (2.9) is common for a general fluid Equilibrium points of (2.9) satisfy w ¼ 0; p p ỵ s2 v v ị ẳ 0; s2 e e ị ỵ p v v ị v v ị2 ẳ 0: 2:10ị Moreover, integrating the equation u0 ẳ sv from to gives us uỵ u ị ẳ sv ỵ v ị: 2:11ị From (2.10) and (2.11), we obtain the Rankine-Hugoniot relations (2.3) Thus, U ặ ẳ v ặ ; uặ ; pặ ị can be connected with each other by a shock wave of (2.1) The system of ODEs (2.9) has the advantage to be common for every fluid However, investigating the stability of its equilibrium points (2.10) would need other forms which rely on the specific equation of state of the fluid In [24], the system of ODEs for ideal fluids, stiffened gas EOS, and van der Waals fluids were obtained as follows 2.3 System of ODEs for ideal fluids An ideal fluid is governed by the equation of state p ¼ ðc À 1Þ qe; ð2:12Þ where c > is the adiabatic constant Using the thermodynamic identity, we can write p ẳ pv ; Sị ẳ c 1ịv c exp Sà À S c ð2:13Þ for some constant Sà , where c is the specific heat coefficient at constant volume Any traveling wave of (1.1) satisfies the relation (2.11), and the following system of ordinary differential equations in terms of the variable ðv ; w; SÞ v0 ¼ w l w0 ¼ À S0 ¼ ; sk lv wỵ c 1ịc lv lv 2l2 wỵ Á w2 À pðv ; SÞ À pðv À ; S ị ỵ s2 v v ị ; ð2:14Þ c2 sðc À 1Þ s2 : w ev ; Sị e ỵ pÀ ðv À v À Þ À ðv À v À Þ2 jpðv ; SÞ 2l 2.4 System of ODEs for a stiffened gas EOS A stiffened gas EOS is given by p ẳ c 1ịe e Þ=v À cpà ; where c > and ð2:15Þ eà ; pà are parameters Another form of stiffened gas EOS is p ẳ pv ; Sị ẳ c 1Þv Àc exp for some constant Sà S À Sà À pà c ð2:16Þ M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 131 An arbitrary traveling wave of (1.1) for a stiffened gas satisfies the relation (2.11), and the following system of ordinary differential equations v0 ¼ w l w0 ¼ À S0 ẳ ; sk lv wỵ c 1ịc lv 2l2 wỵ lv w2 pv ; Sị pv ; S ị ỵ s2 v v À Þ ; ð2:17Þ c2 sðc À 1Þ s2 : w À eðv ; SÞ e ỵ p v v ị v v ị2 jpv ; Sị ỵ p1 Þ 2l 2.5 System of ODEs for van der Waals fluids A van der Waals fluid is defined by the equation of state p¼ RT À a v À b v2 ð2:18Þ ; where a > 0; b > and R > are constants A van der Waals fluid can be fully determined by considering the following Helmholtz free energy function " ðv À bÞT 3=2 Fðv ; Tị ẳ RT ỵ ln c !# a v ð2:19Þ ; where c > is a parameter, see [18] for example The Helmholtz free energy determines the specic entropy as v bịT 3=2 S ẳ @ T Fv ; Tị ẳ R ln c ! ! ỵ : Thus, one can resolve the specific entropy from the last equation by T ¼ Tðv ; Sị ẳ d v bị 2=3 2S ; À 3R exp d ¼ c2=3 : The last equality and the equation of state (2.18) imply that the pressure can be expressed by p ¼ pv ; Sị ẳ Rd v bị 5=3 exp 2S a À 2: À 3R v ð2:20Þ Every traveling wave of (1.1) for a fluid of van der Waals (2.18) satisfies the relation (2.11), and the following system of ordinary differential equations v0 ¼ w l w0 ẳ ; sk lv wỵ lv À w À pðv ; SÞ À pðv À ; S ị ỵ s2 v v ị ; 2l ð2:21Þ Rw c v sv s : w À eðv ; SÞ e ỵ p v v ị v v ị2 ỵ S0 ẳ lv bÞ jTðv ; SÞ 2l 2 Stability of the equilibria of traveling waves In this section, let us investigate the stability of the two equilibria of (2.9) under the condition (2.10) As mentioned in Section 2, the corresponding states U Ỉ can be connected by a shock wave of (2.1) And we expect that the stability property of these equilibria could yield the existence of a traveling wave of (1.1) connecting both states of the shock 3.1 Stability of the equilibria of traveling waves for ideal fluids Theorem 3.1 (Stability of equilibria for ideal fluids) Consider a shock of (2.1) of the form (2.2) satisfying the Lax shock inequalities (2.5) for an ideal fluid (i) For s > 0: The equilibrium point v ỵ ; 0; Sỵ ị of (2.14) is asymptotically stable; the point ðv À ; 0; SÀ Þ is unstable, where the Jacobian matrix of (2.14) at ðv À ; 0; SÀ Þ has one positive eigenvalue, and the other two eigenvalues are either real and negative, or complex and have negative real parts 132 M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 (ii) For s < 0: The equilibrium point ðv À ; 0; SÀ Þ of (2.14) is asymptotically stable in the negative direction (y ! À1); the point v ỵ ; 0; Sỵ ị is unstable, where the Jacobian matrix of (2.14) at v ỵ ; 0; Sỵ Þ has one negative eigenvalue, and the other two eigenvalues are either real and positive, or complex and have positive real parts Proof We need only prove for the case (i), since the argument for the case (ii) is similar A straightforward calculation shows that the Jacobian matrix of the system (2.14) at V ặ ẳ v ặ ; 0; Sặ ị is given by 0 l B B Aặ ẳ B B pv v ặ ; Sặ ị s @ lsk vặ c1ịc lv ặ C C pcặ C C; A csjv ặ 3:1ị where pặ ẳ pv ặ ; Sặ ị The characteristic equation of the matrix Aặ can be written in the form X ỵ aặ2 X ỵ aặ1 X ỵ aặ0 ẳ 0; where aặ0 ẳ aặ1 ẳ aặ2 ẳ l l s2 ỵ pv v ặ ; Sặ ịị cs j vặ; c 1ịpặ cs2 k ; s2 ỵ pv v ặ ; Sặ ị ỵ ỵ vặ sk lv ặ ỵ cs j j 3:2ị v ặ: First, consider the point v ỵ ; 0; Sỵ ị The Lax shock inequalities imply that s2 > Àpv ðv ỵ ; Sỵ ị: This yields aỵ0 ẳ aỵ1 ẳ aỵ2 ẳ l l s2 ỵ pv v þ ; Sþ ÞÞ cs j s2 þ pv ðv þ ; Sþ Þ þ sk lv þ þ cs j v ỵ > 0; c 1ịpỵ vỵ ỵ cs2 k j ! > 0; v ỵ > 0: So, all the coefficients of the characteristic equation X þ aþ2 X þ aþ1 X þ aþ0 ¼ ð3:3Þ are positive and its all real roots must therefore be negative It would be a contradiction if the contrary is assumed, since the left-hand side of (3.3) would be positive Thus, if all three roots of (3.3) are real, then they must be negative Otherwise, (3.3) must have exactly one real root, denoted by X , and two complex conjugate roots, denoted by X ; X This real root r ¼ X must also be negative, as argued above We remain to show that the real parts of X and X are negative Indeed, one can verify easily that the factorization of the cubic polynomial aX ỵ bX ỵ cX ỵ d ẳ X rịaX ỵ b þ arÞX þ c þ br þ ar Þ gives the other two roots of the cubic equation by b raị ặ 2a p D ; D ¼ b À 4ac À 2abr À 3a2 r2 : The two complex roots of the cubic equation (3.3) are given by X 2;3 ẳ b ỵ rị ặ pffiffiffiffi D ; D ¼ b À 4c À 2br 3r ; where b ẳ aỵ2 ; c ẳ aỵ1 ; d ẳ aỵ0 : Since X ; X are complex, we have D < 0, and ReX ị ẳ ReX ị ẳ b ỵ rị=2 Set f Xị ẳ X ỵ aỵ2 X ỵ aỵ1 X ỵ aỵ0 : 3:4ị M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 133 It is not difficult to check that f bị ẳ aỵ2 aỵ1 ỵ aỵ0 < 0; f 0ị ẳ aỵ0 > 0: This means that there is a real root in the interval ðÀb; 0Þ and since r < is the unique real root of the cubic function f ðXÞ, it holds that r > Àb: Thus ReX ị ẳ ReX ị ẳ b þ rÞ=2 < 0: This establishes the stability of the point v ỵ ; 0; Sỵ ị Next, consider the point ðv À ; 0; SÀ Þ The Lax shock inequalities imply that s2 < Àpv ðv À ; SÀ ị: This yields a0 ẳ l s2 ỵ pv ðv À ; SÀ ÞÞ cs j v À < 0; a2 ẳ sk lv ỵ cs j v À > 0: It is derived from Viète’s theorem that the roots X ; X and X (real or complex) of the cubic equation X ỵ a2 X ỵ a1 X ỵ a0 ẳ satisfy X ỵ X ỵ X ẳ ÀaÀ2 < 0; ð3:5Þ X X X ¼ ÀaÀ0 > 0: A cubic equation must have at least one real root The other two roots must be either both real or both complex and conjugate If these two roots are also real, then the third inequality in (3.5) means that one root is positive, and the other two roots have the same sign Then, we deduce from the first inequality of (3.5) that one root is real and two roots are negative Otherwise, let X be the real root and let X and X be the two complex conjugate roots Then, the first inequality in (3.5) implies that ReX ị ẳ ReX Þ < ÀX < This completes the proof of Theorem 3.1 h As seen by Theorem 3.1, it is possible that an unstable trajectory could leave for s > the unstable equilibrium point ðv À ; 0; S ị at y ẳ and may enter the attraction domain of the asymptotically stable equilibrium point v ỵ ; 0; Sỵ ị If so, the unstable trajectory will then converge to the point v ỵ ; 0; Sỵ Þ as y ! This establishes a traveling wave connecting the corresponding points ðv Ỉ ; ; SỈ Þ Similar observations can be made for the case s < 3.2 Equations for traveling waves for stiffened gas EOS The following theorem establishes the stability of the equilibria of (2.17) for stiffened gas EOS Theorem 3.2 (Stability of equilibria for stiffened gas EOS) Consider a shock of (2.1) of the form (2.2) satisfying the Lax shock inequalities (2.5) for an ideal fluid (i) For s > 0: The equilibrium point v ỵ ; 0; Sỵ ị of (2.17) is asymptotically stable; the point ðv À ; 0; SÀ Þ is unstable, where the Jacobian matrix of (2.17) at ðv À ; 0; SÀ Þ has one positive eigenvalue, and the other two eigenvalues are either real and negative, or complex and have negative real parts (ii) For s < 0: The equilibrium point ðv À ; 0; SÀ Þ of (2.17) is asymptotically stable in the negative direction (y ! 1); the point v ỵ ; 0; Sỵ Þ is unstable, where the Jacobian matrix of (2.17) at v ỵ ; 0; Sỵ ị has one negative eigenvalue, and the other two eigenvalues are either real and positive, or complex and have positive real parts Proof It is not difficult to verify that the Jacobian matrix of the system (2.17) at V ặ ẳ v ặ ; 0; Sặ ị is given by B Bặ ẳ B @ pv v ặ ; Sặ ị s l lskv ặ c1ịc lv ặ C C pặ ỵp c A; 3:6ị csjv ặ ; where pặ ẳ pv ặ ; Sặ ị The characteristic equation of Aặ is given by ặ ặ ặ X ỵ b2 X ỵ b1 X ỵ b0 ẳ 0; 3:7ị 134 M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127141 where ặ b0 ẳ ặ b1 ẳ l s2 ỵ pv v ặ ; Sặ ịị l j s2 ỵ pv v ặ ; Sặ ị ỵ sk ặ b2 ẳ cs lv ặ ỵ cs j v ặ; c 1ịpặ ỵ p1 ị vặ ỵ cs2 k j ; 3:8ị vặ: Thus, the remaining of the proof can be made similarly as the one of Theorem 3.1 h 3.3 Equations for traveling waves for van der Waals fluids For a van der Waals fluid (2.18), the corresponding system of ODEs (2.21) may admit up to four equilibria Nonclassical shocks may appear in a van der Waals fluid A shock wave of this kind may violate the Lax shock inequalities For both kinds of classical and nonclassical shocks, the shock speed is often compared with the characteristic speeds at the left-hand and right-hand states of the shock The reader is referred to the standard material [14] for the concept of classical and nonclassical shocks in hyperbolic systems of conservation laws So, in order to include both kinds of classical and nonclassical shocks, we need to state the result on the stability of the equilibria of (2.21) in a more general form as in the following theorem Theorem 3.3 (Stability of equilibria for van der Waals fluids) Consider a shock (2.2) of (2.1) for a van der Walls fluid with the shock speed s > 0, and the corresponding system of ODEs (2.21) of traveling waves of (1.1) If V ¼ ðv ; 0; S0 Þ is an equilibrium point of (2.21) resulted from the given shock, then the following conclusions hold (i) If s2 ỵ pv v ; S0 ị > 0, then the equilibrium point ðv ; 0; S0 Þ of (2.21) is asymptotically stable (ii) If s2 þ pv ðv ; S0 Þ < 0, then the point ðv ; 0; S0 Þ is unstable, where the Jacobian matrix of (2.21) at ðv ; 0; S0 Þ has one positive eigenvalue, and the other two eigenvalues are either real and negative, or complex and have negative real parts Similar conclusions also hold for the case of negative shock speeds Proof It is derived from (2.20) that @ v pv ; Sị ẳ @ S pv ; Sị ẳ 5Rd 3v bị 8=3 exp 2S 2a ỵ 3; 3R v ð3:9Þ 2d 5=3 3ðv À bÞ 2S : exp À 3R Since the equilibrium point V is resulted from the shock between the left-hand and the right-hand states U Ỉ , the state U ¼ ðv ; u0 ¼ uÀ À sðv À v À Þ; S0 Þ, when considered as the right-hand state, satisfies the Rankine–Hugoniot relations (2.3) So, a straightforward calculation gives the Jacobian matrix of the system (2.21) at the equilibrium point V ¼ ðv ; 0; S0 Þ by B B CẳB B pv v ; S0 ị s @ 0 l À lskv R lðv ÀbÞ C C ÀpS ðv ; S0 Þ C C: A À jTðcvvs0v;S0 Þ eS ð3:10Þ It is not difficult to check that the characteristic equation of C can be written as X ỵ c2 X ỵ c1 X ỵ c0 ẳ 0; 3:11ị where c0 ¼ c1 ¼ c2 ¼ l ðpv v ; S0 ị ỵ s2 ị cv sv j ; cv s2 k RpS ðv ; S0 Þ ; pv ðv ; S0 ị ỵ s2 ỵ ỵ l j v0 b sk lv ỵ c v sv j : ð3:12Þ 135 M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 Consider the case (i) It is derived from (3.9) that pS ðv ; SÞ is always positive Thus, if s2 ỵ pv v ; S0 Þ > 0, then all the coefficients c0 ; c1 and c2 given by (3.12) of the characteristic polynomial (3.11) are positive The remaining proof of (i) and the proof of (ii) are therefore similar to the ones of Theorem (3.1) h Numerical tests Since the point of the paper is to study traveling waves when a thermal conductivity coefficient is available, the following tests will be based on different ranges of thermal conductivity: low, medium, and large values Moreover, we also include the two cases of positive and negative shock speeds 4.1 Test 1: Traveling wave approximating a shock with positive speed Let us consider numerical approximations of a traveling wave of (1.1) for an ideal fluid This traveling wave can be used to approximate a given shock wave between a left-hand state denoted by U À and a right-hand state denoted U ỵ with a positive shock speed The parameters for this test are chosen by c ¼ 1:4; Sà ¼ 0:3; c ¼ 1=2; k ¼ 1=5; l ¼ 5k; j ¼ 1=3; ð4:1Þ ðv ; u ; S ; p ị ẳ 2; 1; 3; 33:558937ị; v ỵ ; uỵ ; Sỵ ; pỵ ị ẳ 3; 2:8317144; 2:9961435; 18:876902ị; m ẳ c 1ị=c ỵ 1ị; U ặ ẳ v ặ ; uặ ; Sặ ị; V ặ ẳ v ặ ; 0; Sặ ị: As indicated by Theorem 3.1, the equilibrium point V À of (2.14) is unstable and the point V ỵ is asymptotically stable The point V is chosen to be very closed to the point V À : V ẳ V ỵ 0:001; 0; 0:0001ị: We will compute the trajectory of (2.14) with the parameters given by (4.1) starting at V using the function ODE45 of MATLAB from y ¼ to y ¼ 50 Figs and show that this trajectory converges to the asymptotically stable equilibrium point V ỵ as y becomes larger and larger Furthermore, Fig reveals that the trajectory may enter the domain of attraction of the asymptotically stable equilibrium point V ỵ This test indicates that the traveling wave may exist If it exists, it is approximated by a reasonable trajectory which starts near the left-hand state U À and converges to the right-hand state U þ 4.2 Test 2: Positive shock speed and high thermal conductivity Let us consider numerical approximations of a traveling wave of (1.1) for an ideal fluid, where the shock is peed is positive and the thermal conductivity is low This traveling wave can be used to approximate a given shock wave between a left-hand state denoted by U À and a right-hand state denoted U ỵ The parameters for this test are chosen by c ¼ 1:4; Sà ¼ 1; c ¼ 1; k ¼ 1=10; l ¼ 5k; j ¼ 100; ðv À ; uÀ ; SÀ ; p ị ẳ 1; 1; 1:5; 0:65948851ị; v ỵ ; uỵ ; Sỵ ; pỵ ị ẳ 2; 0:3521771; 1:4588051; 0:239814ị; m ẳ c 1ị=c ỵ 1ị; U ặ ẳ v ặ ; uặ ; Sặ ị; V ặ ẳ v ặ ; 0; Sặ ị: In this case, the equilibrium point V À of (2.14) is unstable and the point V ỵ is asymptotically stable 4:2ị 136 M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 Fig Test 1: approximate connection between the unstable equilibrium point V À and the asymptotically stable equilibrium point V ỵ A trajectory starting near the unstable equilibrium point V À approaches the asymptotically stable equilibrium point V ỵ Fig Test 1: numerical approximation of a traveling wave connecting U and U ỵ specific volume, velocity, pressure, and specific entropy M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 137 Set V ẳ V ỵ 0:001; 0; 0:001ị; which is very closed to V À The trajectory of (2.14) with the parameters given by (4.2) starting at V is computed using the function ODE45 of MATLAB from y ¼ to y ¼ 500 Figs and show that this trajectory converges to the asymptotically stable equilibrium point V ỵ as y becomes larger and larger Possibly, the trajectory may enter the domain of attraction of the asymptotically stable equilibrium point V ỵ This test also indicates that the traveling wave connecting U Ỉ may exist and it can be approximated by a trajectory which starts near the left-hand state U À and converges to the right-hand state U ỵ 4.3 Test 3: Negative shock speed and medium thermal conductivity Let us consider numerical approximations of a traveling wave of (1.1) for an ideal fluid, where the shock is peed is negative and the thermal conductivity is medium The left-hand state U À and the right-hand state U ỵ and the other parameters are given as follows c ¼ 1:9; Sà ¼ 0:5; c ¼ 1; k ¼ 2; l ¼ 10k; j ¼ 1; ð4:3Þ ðv À ; uÀ ; SÀ ; p ị ẳ 2; 1; 1; 0:39758753ị; v ỵ ; uỵ ; Sỵ ; pỵ ị ẳ 1; 0:17195743; 1:1769454; 1:7710717ị; U ặ ẳ v ặ ; uặ ; Sặ ị; V ặ ẳ v ặ ; 0; Sặ ị: For s < 0, as indicated by Theorem 3.1, the equilibrium point V À of (2.14) is unstable and the point V ỵ is asymptotically stable in the negative direction y ! 1) Set V ẳ V ỵ ỵ ð0:001; 0; À0:001Þ; Fig Test 2: approximate connection between the unstable equilibrium point V À and the asymptotically stable equilibrium point V ỵ A trajectory starting near the unstable equilibrium point V À approaches the asymptotically stable equilibrium point V ỵ 138 M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 Fig Test 2: numerical approximation of a traveling wave connecting U À and U ỵ specic volume, velocity, pressure, and specic entropy which is chosen to be very closed to the point V ỵ The trajectory of (2.14) with the parameters given by (4.3) starting at V is computed using the function ODE45 of MATLAB from y ¼ back to y ¼ À500 One can see from Figs and that this trajectory converges in the ‘‘backward time’’ direction to the asymptotically stable equilibrium point V À as Ày becomes larger and larger It is possible that the trajectory may enter the domain of attraction of the asymptotically stable equilibrium point V À This test shows a suitable approximation of a traveling wave connecting the left-hand state U and the right-hand state U ỵ 4.4 Test 4: Negative shock speed and low thermal conductivity Let us consider numerical approximations of a traveling wave of (1.1) for an ideal fluid, where the shock is speed is negative and the thermal conductivity is low The left-hand state U and the right-hand state U ỵ and the other parameters are given as follows c ¼ 1:9; Sà ¼ 0:5; c ¼ 2; k ¼ 1=5; l ¼ 5k; j ¼ 0:3; ð4:4Þ ðv À ; uÀ ; S ; p ị ẳ 2; 1; 1; 0:30964148ị; v ỵ ; uỵ ; Sỵ ; pỵ ị ẳ 1; 2:0342488; 1:3538908; 1:3793121ị; U ặ ẳ v ặ ; uặ ; Sặ ị; V ặ ẳ v ặ ; 0; Sặ ị: The equilibrium point V of (2.14) is unstable and the point V ỵ is asymptotically stable in the negative direction ðy ! À1) M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 139 Fig Test 3: approximate connection between the unstable equilibrium point V ỵ and the asymptotically stable equilibrium point V À A trajectory starting near the unstable equilibrium point V ỵ approaches the asymptotically stable equilibrium point V Fig Test 3: numerical approximation of a traveling wave connecting U and U ỵ specic volume, velocity, pressure, and specific entropy 140 M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 Fig Test 4: approximate connection between the unstable equilibrium point V þ and the asymptotically stable equilibrium point V À A trajectory starting near the unstable equilibrium point V ỵ approaches the asymptotically stable equilibrium point V À Fig Test 4: numerical approximation of a traveling wave connecting U and U ỵ specic volume, velocity, pressure, and specific entropy M.D Thanh, N.H Hiep / Applied Mathematics and Computation 234 (2014) 127–141 141 Let V ¼ V ỵ ỵ 0:001; 0; 0:001ị; which is chosen to be very closed to the point V ỵ The trajectory of (2.14) with the parameters given by (4.4) starting at V is computed using the function ODE45 of MATLAB from y ¼ back to y ¼ À200 One can see from Figs and that this trajectory converges in the ‘‘backward time’’ direction to the asymptotically stable equilibrium point V À as Ày becomes larger and larger Moreover, the shape of the curve in Fig indicates that it may enter the domain of attraction of the asymptotically stable equilibrium point V À This test also shows a suitable approximation of a traveling wave connecting the left-hand state U À and the right-hand state U þ Conclusions and discussions The presence of a thermal conductivity coefficient in the system often causes difficulty for the study of traveling waves In this work we establish the stability of the equilibria corresponding to a given traveling wave of the gas dynamics equations with viscosity, capillarity, and thermal conductivity coefficients We note that the use of the stability of the equilibria corresponding to the given traveling wave connecting a left-hand state U À to a right-hand state U ỵ for viscous-capillary models depends on the kind (classical or nonclassical) of the associated shock wave The tests show very good numerical approximations of possible traveling waves These tests also suggest that a trajectory possibly leaving the unstable equilibrium point V À at À1 enters the domain of attraction of the asymptotically stable equilibrium point V ỵ and so converges to V ỵ at This connection, if conrmed, could give a traveling wave connecting the lefthand state U with the right-hand state U ỵ Thus, this work may motivate for future study on the estimation of the attraction domain of the asymptotically stable equilibrium point and therefore the existence of traveling waves in models with heat conduction Future researches motivated from this work are expected to proceed in two separate directions, depending on the kind of the corresponding shock wave If the shock satisfies the Lax shock inequalities, the study of attraction domain of the asymptotically stable equilibrium point could be expected, and one can get the existence from unstable-to-stable and/or stable-tounstable connection as in [19–23] Otherwise, one may expect an unstable-to-unstable connection as in [11,3,4,2,5,6,1] Acknowledgment This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) References [1] N Bedjaoui, C Chalons, F Coquel, P.G LeFloch, Non-monotone traveling waves in van der Waals fluids, Ann Appl (2005) 419–446 [2] N Bedjaoui, P.G LeFloch, Diffusive-dispersive traveling waves and kinetic relations III An hyperbolic model from nonlinear elastodynamics, Ann Univ Ferrara Sci Mat 44 (2001) 117–144 [3] N Bedjaoui, P.G LeFloch, Diffusive–dispersive traveling waves and kinetic relations I Non-convex hyperbolic conservation laws, J Differ Equ 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Diffusive-dispersive traveling waves and kinetic relations IV Compressible Euler equations, Chin Ann Math 24B (2003) 17– 34 [6] N Bedjaoui, P.G LeFloch, Diffusive-dispersive traveling waves and kinetic relations... of traveling waves In this section, let us investigate the stability of the two equilibria of (2.9) under the condition (2.10) As mentioned in Section 2, the corresponding states U Ỉ can be connected... Thanh, Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion, Nonlinear Anal.: T.M.A 72 (2010) 3136–3149 [21] M.D Thanh, Existence of traveling waves in elastodynamics with
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