On one system of the Burgers equations arising in the two-velocity hydrodynamics

Transcription

1 Journal of Physics: Conference Series PAPER OPEN ACCESS On one system of the Burgers equations arising in the two-velocity hydrodynamics To cite this article: Kholmatzhon Imomnazarov et al 216 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - A Probabilistic Connection between the Burger and a Discrete Boltzmann Equation M.-O. Hongler and L. Streit - The time-dependent Ginzburg Landau equation for the two-velocity difference model Wu Shu-Zhen, Cheng Rong-Jun and Ge Hong-Xia - Spectral functions in a two-velocities Tomonaga-Luttinger Liquid E Orignac, M Tsuchiizu and Y Suzumura This content was downloaded from IP address on 1/1/218 at 22:16

2 Journal of Physics: Conference Series ) 1224 doi:1.188/ /697/1/1224 On one system of the Burgers equations arising in the two-velocity hydrodynamics Kholmatzhon Imomnazarov 1, Baxtier Mamasoliyev 2 Georgy Vasiliev 1, Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of the RAS, Novosibirsk, 2 National University of Uzbekistan, Mechanics and Mathematics Faculty, Tashkent imom@omzg.sscc.ru Abstract. A system of the Burgers equations of the two-velocity hydrodynamics is constructed. We consider the Cauchy problem in the case of a one-dimensional system. We have obtained a formula for solving the Cauchy problem and the estimate of the stability of this solution. It is shown that with disappearance of the kinetic friction coefficient, which is responsible for the energy dissipation, this formula turns to the famous Cauchy problem for the one-dimensional Burgers equation. The existence and uniqueness of solutions to the Cauchy problem for the one-dimensional systems of the Burgers type are proved using the method of weak approximation. 1. Introduction In recent decades, mathematicians have become increasingly interested in the problems associated with the behavior of solutions to systems of partial differential equations with a small parameter in high derivatives with allowance for the kinetic parameters. These problems arose from the physical applications, mostly from contemporary hydrodynamics compressible multiphase fluids with low viscosity). An analog to the Burgers equation arises, for example, in studying a weak nonlinear one-dimensional acoustic wave moving with a linear velocity of sound. In this case, nonlinear velocity terms in the system of the Burgers equations come from the sound velocity depending on the amplitude of the sound wave, on the second derivative terms and on the difference in the velocities representing the attenuation of sound waves associated with energy dissipation. In other words, these terms provide the continuity of solutions and are dissipative processes associated with the production of entropy. These terms in turn provide non-roll waves 9. The system under consideration is a special case of a two-velocity system of hydrodynamics equations 4-6,1, 1. A one-dimensional analog of the Navier-Stokes equations for a compressible fluid can be considered as a system of the Burgers equations which is a system of nonlinear convectiondiffusion equations 8. u t + u u x = ν u xx b u ũ), 1) ũ t + ũ ũ x = ν ũ xx + b u ũ), 2) where the quantities u and ũ can be regarded as velocity subsystems with the dimension x/t, constituting two-velocity components with the corresponding partial continuum densities ρ and Content from this work may be used under the terms of the Creative Commons Attribution. licence. Any further distribution of this work must maintain attribution to the authors) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 Journal of Physics: Conference Series ) 1224 doi:1.188/ /697/1/1224 ρ, ρ = ρ + ρ is the common density of the continuum, b = ρ ρ b, b is the coefficient of friction with the dimension 1/t, which is the analog to the Darcy factor for porous media. The positive constants ν and ν play the role of kinematic viscosities subsystems of the dimension x 2 /t. A two-velocity system of hydrodynamic equations and a system of the Burgers equations have much in common. For example, the quadratic nonlinearity with respect to the terms u and ũ with advective terms corresponding to the sound, depending on the amplitude of the sound waves and linear viscosities ν, ν, the coefficient of friction b of the right-hand side is responsible for the attenuation of the sound waves 9. With regard to the properties of the solutions, they are totally different. The system of the Burgers equations in the vanishing coefficients ν, ν, b is formed of both strong shock waves), and weak discontinuities, while the solution of the system of two-velocity hydrodynamics does not have such features. However, the range of applicability of this system is by no means limited to the above example. These systems arise in many problems, thus confirming their significance. 2. The Cauchy problem for the system of the Burgers type equations For system 1),2) in the domain Γ, T = {t, x) : t T, x R} let us consider the Cauchy problem with the following initial data u t= = u x), ũ t= = ũ x), x R. ) We are interested in the classical solution of the Cauchy problem for the system of the Burgers equations 1),2), namely, ut, x), ũt, x) C 1,2 Γ, T ) is the class of functions once continuously differentiable with respect to t and twice continuously differentiable with respect to x. Further assume that the inequality ρ < ρ. Theorem 2.1 Let u x), ũ x) C 2 R). Then the Cauchy problem 1) - ) has in the class C 1,2 Γ, T ) a unique solution, and the following estimate of stability holds Ω u 2 t, x) + ũ 2 t, x)) dxdt 1 2 b m) e2 b T. 4) Proof. Multiply both sides of equations 1) and 2) by u and ũ, respectively. After integration with respect to x and after simple transformations we obtain 1 2 t u 2 dx = ν u x ) 2 dx b u 2 u ũ) dx, 5) 1 2 t ũ 2 dx = ν ũ x ) 2 dx + b u ũ ũ 2 ) dx. 6) Hence, multiplying 5) by ρ, 6) by ρ and summing the result obtained, after simple transformations we arrive at mt) ρ ρ t m) + 2 b mτ)dτ, where mt) = u2 t, x) + ũ 2 t, x)) dx. From this and the Gronwall-Bellman inequality 2 follows mt) ρ ρ m) e2 b t. Hence, integrating from to t, obtain 4). The uniqueness is proved in a standard way. 2

4 Journal of Physics: Conference Series ) 1224 doi:1.188/ /697/1/1224. The formula for solving the Cauchy problem for the Burgers type system of equations It is convenient to use the Florin-Hopf-Cole transformation 9 ϕ = exp 1 u dx, ψ = exp 1 ũ dx, herewith the functions u and v are expressed in terms of the functions ϕ and ψ by the formulas u = 2 ν ϕ x ϕ, ũ = 2 ν ψ x ψ. In terms of the functions ϕ and ψ the system of dynamic equations 1) and 2) takes the form ) ϕt = ν ϕ ) xx ϕ x ϕ x ) ψt = ψ x ν ψ ) xx + b ν ϕ x Hence, after the integration with respect to x, we obtain b ) ln ϕν ν ψ ν, x ) ln ϕν ψ ν ϕ t = ν ϕ xx b ν ϕ ν ln ϕ ν ln ψ) + C 1t) ϕ, 7) x. ψ t = ν ψ xx + b ν ψ ν ln ϕ ν ln ψ) + C 2t) ψ, 8) where C 1 t) and C 2 t) are arbitrary temporal functions. Introducing the new functions ϕ and ψ by the formulas ϕ = ϕ exp C 1 dt, ψ = ψ exp C 2 dt. from 7) and 8) we obtain ϕ t = ν ϕ xx b ν ϕ ν ln ϕ exp ψ t = ν ψ xx + ψ b ν ν ln ϕ exp ) C 1 dt ν ln ψ exp ) C 1 dt ν ln ψ exp )) C 2 dt, 9) )) C 2 dt. 1) For simplicity, assume the functions C k t) k = 1, 2) to be equal to zero. The Cauchy problem for system 9), 1) with the data has the form ϕt, x) = b ν t ϕ t= = ϕ x), ψ t= = ψ x) G ν x, ξ, t) ϕ ξ)dξ G ν x, ξ, t τ) ϕξ, τ) ν ln ϕξ, τ) ν ln ψξ, τ)) dξ dτ, 11)

5 Journal of Physics: Conference Series ) 1224 doi:1.188/ /697/1/1224 ψt, x) = G ν x, ξ, t) ψ ξ)dξ where + b ν t G ν x, ξ, t τ) ψξ, τ) ν ln ϕξ, τ) ν ln ψξ, τ)) dξ dτ, 12) G a x, ξ, t) = 1 e x ξ)2 4 a t 4π a t is the fundamental solution of the one-dimensional heat equation. Next, assume as in 7 that the Cauchy data u x), ũ x) for large x satisfy the following conditions x u ξ)dξ = ox 2 ), Let us introduce the following functions x ũ ξ)dξ = ox 2 ). 1) F u, x, y, t) = x y)2 2t y + ut, η)dη. 14) F 1 u, x, y, t, τ) = x y)2 2t τ) + y Note that the following equalities for any x, y, t, τ are valid F u, x, y, t) F ũ, x, y, t) = F 1 u, x, y, t, τ) F 1 ũ, x, y, t, τ) = y y uτ, η)dη. 15) u η) ũ η)dη. uτ, η) ũτ, η)dη. Differentiate both sides of equalities 11) and 12) with respect to x. transformations we obtain ϕ x t, x) = 1 u ξ)g ν x, ξ, t) exp 1 ξ u η)dη dξ + b t x ξ t τ Gν x, ξ, t τ) exp 1 ψ x t, x) = 1 ũ ξ)g ν x, ξ, t) exp 1 b t x ξ t τ G ν x, ξ, t τ) exp 1 After simple ξ ξ uτ, η)dη uτ, η) ũτ, η)dη dξ dτ, ξ ũ η)dη dξ ξ ξ ũτ, η)dη uτ, η) ũτ, η)dη dξ dτ. Hence, taking into account 14), 15) and the definition of the fundamental solution of the operator conductivity we obtain 1 ϕ x t, x) = x ξ exp 1 4πν t t F u, x, ξ, t) dξ + b t 1 4πν t τ) x ξ t τ F 2u, ũ, x, ξ, t τ, τ) dξ dτ, 16) 4

6 Journal of Physics: Conference Series ) 1224 doi:1.188/ /697/1/ ψ x t, x) = 4π ν t b t x ξ t In formulas 16) and 17) the following notation is used Exp 1 F ũ, x, ξ, t) dξ 1 x ξ 4π ν t τ) t τ G 2u, ũ, x, ξ, t τ, τ) dξ dτ. 17) F 2 u, v, x, ξ, t τ, τ) = exp 1 ξ F 1u, x, ξ, t τ, τ) uτ, η) vτ, η)dη G 2 u, v, x, ξ, t τ, τ) = exp 1 ξ F 1v, x, ξ, t τ, τ) uτ, η) vτ, η)dη. Thus, we come to the following theorem. Theorem.1 Let u x), ũ x) be measurable functions that satisfy relation 1). Then for the solution of the Cauchy problem 1) - ) we have the formula ut, x) = ũt, x) = Exp 1 F u, x, ξ, t) Exp 1 F u, x, ξ, t) x ξ t Exp 1 F ũ, x, ξ, t) Exp 1 F v, x, ξ, t) x ξ t dξ t dξ t dξ + t dξ + b t b t x ξ F 2u, ũ, x, ξ, t τ, τ) dξ dτ, b t F 2 u, ũ, x, ξ, t τ, τ) dξ dτ b t x ξ Corollary.2 Let u x), ũ x) satisfy the conditions of the theorem. Cauchy problem 1) - ) we have the formula ut, x) = + b t ũt, x) = b t x ξ t Exp τ x ξ t G 2u, ũ, x, ξ, t τ, τ) dξ dτ. t G 2 u, ũ, x, ξ, t τ, τ) dξ dτ exp 1 F u, x, ξ, t) dξ F u, x, ξ, t) dξ ) ut, x) x ξ F 2 u, ũ, x, ξ, t τ, τ) dξ dτ Exp 1 F u, x, ξ, t) dξ Exp τ exp 1 F ũ, x, ξ, t) dξ F ũ, x, ξ, t) dξ ) ũt, x) x ξ G 2 u, ũ, x, ξ, t τ, τ) dξ dτ Exp 1 F ũ, x, ξ, t) dξ Then for solving the, 18). 19) Remark. With disappearance of the friction coefficient b in the absence of dissipation energy due to friction) solution 18), 19) turns to the famous Cauchy problem for the Burgers equation 7. 5

7 Journal of Physics: Conference Series ) 1224 doi:1.188/ /697/1/ The method of weak approximation for the Cauchy problem for the Burgers type system of equations Let us consider problem 1) - ) relative to the Cauchy data u, ũ, assuming that u, ũ C 2 R) and d n u x) dx n c n, d n ũ x) dx n c n, x R, n =, 1, 2, 2) where c n, c n are some non-negative constants. First, we consider the case of infinitely differentiable Cauchy data. Let us assume that u, ũ C R) and d n u x) dx n c n, d n ũ x) dx n c n, x R, n =, 1,..., 21) Following 1,, we use a weak approximation of the Cauchy problem 1) - ) with the help of a series of problems u τ t = ν u τ xx, ũ τ t = ν ũ τ xx, nτ < t u τ t + u τ u τ x =, ũ τ t + ũ τ ũ τ x =, u τ t = b u τ ũ τ ), ũ τ t = b u τ ũ τ ), n + 1 ) τ, 22) n + 1 ) τ < t n + 2 ) τ, 2) n + 2 ) τ < t n + 1)τ, 24) u τ, x) = u x), ũ τ, x) = ũ x) 25) where N τ = t, provided that N is integer exceeding 1, n =, 1,..., N 1, and the constant t satisfies equality ). Remark 4.1 When building the solution of 22) - 25) at the first fractional step we solve the Cauchy problem for the heat equation, and at the second fractional step the Cauchy problem for the transport equation v t + v v x =, 26) and at the third fractional step we solve the Cauchy problem for systems of ordinary differential equations. It is known that in the case of the Cauchy problem for equation 26) with the initial data v, x) = v x) 27) bounded together with their derivatives, the solution may be a gradient catastrophe, that is, there can exist t 1 >, such that the classical solution v of this problem would exist in the domain Γ, t1 ), be bounded in this domain, but the derivative v x in the neighborhood of a point t 1, x ) is becoming unbounded: v x t, x) with t t 1, x x 9, 11, 2. It is easy to show that if dv x) dx c 1, 28) 6

8 Journal of Physics: Conference Series ) 1224 doi:1.188/ /697/1/1224 the classical solution of problem 26), 27) is in the domain Γ, t, it is bounded and the following estimate holds where t satisfies the inequality v x t, x) c 1 1 c 1 t, t, x) Γ, t, 29) 1 c 1 t >. Let relations 21) be performed, and the constants c 1, c 1 and t satisfy the conditions 1 c 1 t >, 1 c 1 t >. ) Then the solution of u τ and ũ τ in the domain Γ, t exists and is bounded together with all its derivatives with respect to the variables t, x. It is obvious that for any fixed τ, the solution of u τ and ũ τ of problem 22)-25) is bounded regardless of the value τ: u τ t, x) c, ũ τ t, x) c. 1) Repeating the argument from 1, we can show the boundedness of the derivative solutions u τ and ũ τ of any order with respect to x : k u τ t, x) x k C k, k ũ τ t, x) x k C k, t, x) Γ, t, k =, 1,..., 2) where C k, Ck are some positive constants such that C = c, C = c. Independent of τ, from expressions 1), 2) and from equations 22) - 24) follow the estimates: k+1 u τ t, x) t x k s k, k+1 ũ τ t, x) t x k s k, t, x) Γ, t, k =, 1,..., These estimates suggest that u τ, ũ τ and their derivatives with respect to x of any order are uniformly bounded and equicontinuous in Γ, t. By the Arzela Theorem 2, using the diagonal method we can choose subsequences {u τ k}, {ũ τ k} sequences {u τ }, {ũ τ }, converging in Γ, t to the functions u and ũ, respectively, together with all the derivatives with respect to x, uniform in every bounded region of the domain Γ, t, whereby the functions u and ũ have derivatives of any order with respect to x and satisfy the following relations u, x) = u x), ũ, x) = ũ x), k ut, x) x k C k, k ũt, x) x k C k, t, x) Γ, t, k =, 1,.... The uniqueness of the solution is proved in a standard way. Consequently, the sequences of the functions {u τ }, {ũ τ } with τ, uniformly converge in the domain Γ, t to u and ũ, respectively, together with all their derivatives. The case when u, ũ C 2 R) is proved using the average functions 12. Acknowledgements The authors are grateful to a reviewer for his valuable remarks, which, in our opinion, appeared to be useful for improving the text of the manuscript. 7

9 Journal of Physics: Conference Series ) 1224 doi:1.188/ /697/1/1224 References 1 Belov Yu Ya and Demidov G V 197 In book: Chislennye metody mekhaniki sploshnoy sredy Novosibirsk: Comp. Center USSR Acad. Sci.) 1 in Russian) 2 Belov Yu Ya and Kantor S A 1999 The method of weak approximation Krasnoyarsk: Krasnoyarsk state University Press) in Russian) Demidov G V and Yanenko N N 1978 Proc. All-Union Conf. on Partial Diff. Eqs Moscow) 1 1 in Russian) 4 Dorovsky V N 1989 Geologiya i Geofisika. No Dorovsky V N and Perepechko Yu V 1992 J. Appl. Mech. and Techn. Phys Dorovsky V N and Perepechko Yu V 1989 Geologiya i Geofisika Hopf E 195 Communs Pure and Appl. Math Imomnazarov Kh Kh and Mamasoliyev B Zh 215 In book: Abst. International Sci. Conf. Modern Methods of Math. Phys. Appl. Tashkent) 167 in Russian) 9 Kulikovskii A G, Sveshnikov E I and Chugaynova A P 21 Mathematical methods for studying discontinuous solutions of nonlinear hyperbolic systems of equations Moscow) in Russian) 1 Perepechko Yu V, Sorokin K E and Imomnazarov Kh Kh 214 Proc. of XVII International conference Modern Problems of Continuum Mechanics Rostov-on-Don) 166 in Russian) 11 Raputa V F 1975 In book: Chislennye metody mekhaniki sploshnoy sredy. Novosibirsk: Computing Center of the USSR Academy of Sciences 6 9 in Russian) 12 Sobolev S L 1991 Some Applications of Functional Analysis to Mathematical Physics rd edition. American Mathematical Siciety) 1 Zhabborov N M and Imomnazarov Kh Kh 212 Some initial boundary value problems of mechanics of twovelocity media Tashkent: National University of Uzbekistan Press) in Russian) 8