Author(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)
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1 Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL DOI Rights Osaka University
2 Huang, F., Matsumura, A. and Shi, X. Osaka J. Math. 4 4, 93 ON THE STABILITY OF CONTACT DISCONTINUITY FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY FEIMIN HUANG, AKITAKA MATSUMURA and XIAODING SHI Received June,. Introduction The one-dimensional compressible Navier-Stokes equations read in the Eulerian coordinates ρ ρ = ρ ρ. = µ ρ ρ = κ θ µ where is the velocity, ρ > the density, θ the absolute temperature, µ > the viscosity constant and κ > the coefficient of heat conduction. The pressure = ρ θ and the internal energy = ρ θ are related by the second law of thermodynamics. There have been a lot of works on the asymptotic behaviors of the solutions for the system.. Most of these results are concerned with the rarefaction wave and viscous shock wave. We refer to [ 5] for case and [4 5, 7 8] for 3 3 case and references therein. However there is no result on the contact discontinuity for the system. until now due to various difficulties. Although some progress on the contact discontinuity were obtained by Liu and Xin [9] and Xin [7] in which the asymptotic toward the contact discontinuity was investigated for the initial value problem IVP of viscous conservation laws with uniformly artificial viscosity, no result is known for the physical system, especially for the compressible N-S equations.. Therefore we really want to give a positive result on the contact discontinuity for the physical system.. To simplify our problem, we focus our attention on the perfect gas. In this situation,. ρ θ = θ ρ.3 where ρ θ = γ θ const > is the gas constant and γ > is the adiabatic exponent. As observed
3 94 F. HUANG, A. MATSUMURA AND X. SHI by [9, 7], the contact discontinuity can not be the asymptotic state, and a diffusive wave, which approximates the contact discontinuity on any finite time interval, instead dominates the large time behavior of the solutions for their viscous system. For the system., we also expect that the asymptotic behavior of the solutions is governed by a nonlinear diffusive wave. We call it viscous contact discontinuity. We observe that the sign of the first derivative of the velocity in the nonlinear diffusive wave is important to the a priori estimate and is not good in the whole space. Thus it is difficult to obtain the asymptotic stability of the viscous contact discontinuity for IVP of the system.. We further observe that the effect of the sign could exactly be neglected for initial boundary value problem IBVP of the system. because there has the inequality of Poincar type in the half space. It might be possible to obtain the a priori estimate for IBVP of the system. although IBVP is usually much more difficult than IVP in many situations. Motivated by this observation, we consider the system. in the part >, where = is a free boundary, with the following boundary conditions.4 = = θ = θ > and.5 µ = = which means the gas is attached at the free boundary = to the atmosphere with pressure see [5] and the initial data.6 ρ θ = ρ θ lim ρ θ = ρ θ where ρ, θ are positive constants and θ = θ. Because here we only consider the case of a single contact discontinuity, we require.7 = = θ ρ Since the boundary condition.4 means the particles always stay on the free boundary =, if we use the Lagrangian coordinates, then the free boundary becomes a fixed boundary. Thus we transform the Eulerian coordinates to the Lagrangian coordinates by.8 = ρ =
4 STABILITY OF CONTACT DISCONTINUITY 95 and our free boundary value problem..7 is then changed into.9 = > > θ = µ > > θ θ = κ θ µ > > θ = = θ θ µ = > θ = θ θ as where = etc. = /ρ, θ = { /γ }θ const and θ = θ/. We now construct the viscous contact discontinuity which is expected to describe the large time behavior of the solution to the system.9. To achieve our goal, we consider the following Riemann problem. = θ = θ θ = θ = θ if < θ = θ if > where ± and θ ± are given positive constants. It is known from [6] that the Riemann problem. admits a contact discontinuity. = provided that { θ < θ >. = θ = = θ From the structure of the Riemann solution, we conjecture.3 = = On the other hand, it is observed that the third equation of.9 can be reduced to.4 γ θ θ = κ θ µ
5 96 F. HUANG, A. MATSUMURA AND X. SHI by.9. Motivated by.4, we further conjecture that following equations =.5 = κ γ = θ = θ is governed by the Direct computations from.3 and.5 yields.6 = = θ = θ = κ γ γ > We seek the self similarity solution ξ, ξ = / of.6. Namely ξ is the solution of the following equation.7 ξ = = θ = θ = ξ It is known from [] that there exists a unique solution ξ of.7 which is a monotone function, increasing if θ > θ and decreasing if θ < θ. Furthermore, has the following property due to [],.8 θ θ θ θ where and are positive constants depending on θ ±. From.7, we have.9 = ξ =: ξ which yields. ξ = Ê ξ Due to [3], the asymptotic behavior of Ê ξ is. Ê ξ = 3ξ as ξ for some positive constant 3 depending on θ ±. Thus by.8., it is easy to verify the following estimates of. Lemma.. Let θ θ = δ, the following estimates hold. 4 δ 4 3/ δ 3/
6 STABILITY OF CONTACT DISCONTINUITY 97.3 δ 5/ δ After = ξ is obtained, we define.4 = = κγ γ It is easy to see has the following property.5 [κ ] / all which means the nonlinear wave approximates the contact discontinuity in norm, on any finite time interval as κ tends to zero. We call viscous contact discontinuity. Therefore, we conjecture that the solution θ of the system.9 asymptotically tends to the viscous contact discontinuity. The aim of this paper is to justify the above conjecture. Our result is, roughly speaking, as follows. If θ θ is small, then the viscous contact discontinuity is asymptotic stable. Our plan of this paper is as follows. in Sec., we reformulate the original problem and state our main result; in Sec. 3, we establish the a priori estimate and prove our main result; in Sec. 4, we show the local existence of the solution. NOTATIONS. Throughout this paper, several positive generic constants are denoted by, without confusions. For function spaces, denotes the -th order Sobolev space with its norm.6 = = / when := The domain will be often abbreviated without confusions.. Reformulated problem and main result To state our main result, we put the perturbation φ ψ ζ by. = φ = ψ θ = ζ
7 98 F. HUANG, A. MATSUMURA AND X. SHI where is the viscous contact discontinuity defined in.6 and.9. By the definition of the viscous discontinuity, we have. γ = = µ = κ µ where.3.4 κγ [ ] = ln µ ln γ = µ κγ ln = 4 γ Substituting. into.9,.9 and.4 yields φ ψ = ζ ψ φ ψ = µ φ γ ζ ζ φ ψ.5 ζ = κ κ µ ψ µ φ φ θ φ µ ψ φ = = ζ = φ ψ ζ = θ We assume.6 φ ψ = φ ψ ζ = ζ Our main result is Theorem. Stability theorem. There exist positive constants δ and ε such that for any θ θ = δ δ, if φ ψ ζ ε, then the problem.5 has a unique global solution φ ψ ζ satisfying φ ψ ; ζ ; φ ; ψ ζ ;
8 STABILITY OF CONTACT DISCONTINUITY 99 and sup φ ψ ζ as REMARK. Although Theorem. is only concerned with the perfect gas case, it is not difficult to obtain the similar result for general case by the same argument. REMARK. One would expect similar result for Dirichlet problem =, θ = θ. However, if we still use the same viscous contact discontinuity constructed in.6 and.4, then the decay rate of the boundary value ψ = / is not good. It is difficult to control the terms from the boundary. We shall prove the stability Theorem. by the local existence and the a priori estimate. We look for the solution φ ψ ζ in the solution space = >, where = {φ ψ ζ : φ ψ [ ]; ζ [ ]; φ ψ ζ inf φ > φ ;.7 ψ ζ ; } for,, >, and =. Let.8 = sup φ ψ ζ [ ] In the present paper, the a priori estimate will be investigated in the next section and the local existence will be left to the last section. Nevertheless we first state our local existence. Proposition. Local Existence. There exists a positive constant such that if φ ψ ζ and inf R φ, then there exists a positive constant = such that there exists a unique solution φ ψ ζ / to A priori estimate Before establishing the a priori estimate, we first estimate the value of φ on the boundary = by the boundary condition.5. Let φ = φ. Since =, the boundary condition of.5 yields 3. θ φ µ φ φ = >
9 F. HUANG, A. MATSUMURA AND X. SHI Direct computation gives 3. φ = µ φ φ = φ It follows then, that 3.3 φ = φ /µ Now we turn to the a priori estimate. We have Proposition 3. A Priori Estimates. There exist positive constants δ and ε such that, for any given θ θ = δ δ, if φ ψ ζ / ε is a solution of.5 for some positive, then φ ψ ζ satisfies the a priori estimate 3.4 φ ψ ζ { φ ψ ζ } τ δ 4/3 φ ψ ζ 4/3 where = / min{ }, = θ /. Proposition 3. is proved by a series of Lemmas. We first give the following key Lemma. 3.5 Lemma 3.. It follows that φ ψ ζ { ψ ζ τ } τ δ 4/3 φ ψ ζ 4/3 δ φ τ 3.6 Proof. Multiplying.5 by ψ, we have [ ζ ] ψ ψ ζ ψ µ [ µ ψ ] ψ ψ ψ = ψ Since φ = ψ and 3.7 we get ζ = ζ ψ φ ζψ 3.8 µ ψ µψ = ψ
10 STABILITY OF CONTACT DISCONTINUITY where = [ ζ ] [ ψ µ ψ ] ψ Let 3.9 = ln It is easy to check that = and is strictly convex around =. We compute 3. where { } 3. = ln Substituting 3. into 3.8 yields 3. ψ = = φ φ ζψ µ ψ µψ = ψ 3.3 On the other hand, we calculate [ θ ] = θ ln θ = ζ θ θ θ and 3.4 ζ γ θ θ = κ µ θ = ζψ ζ ζ θ θ κ θ κ ζ θ θ ζ ζ θ µζ θ = ζψ ζ ζ θ θ κ θ κ θ θ ζζ κ ζ φ θ κ θ ζφ θ µζ θ κ θ ζ ζ θ ζ θ
11 F. HUANG, A. MATSUMURA AND X. SHI Substituting 3.3 and 3.4 into 3. gives θ ψ 3.5 γ κ θ ζ µ ψ = ψ ζ θ where 3.6 = κ ζ θ θ and 3.7 satisfies = γ ζ θ κ θ θ θ µψ ζζ κ ζ φ θ κ θ ζφ θ µζ θ 3.8 δ δ ψ ζ δ ζ φ if ε is suitably small. Note that = = and is a strictly convex function around = and ψ δ ψ δ 4/3 Choosing δ suitably small and integrating 3.5 over give 3.9 φ ψ ζ From Lemma., it is easy to see 3. ψ ζ ζ φ 4/3 φ ψ ζ 4/3 δ 4/3 Due to the good property of 3., the estimates of the terms containing and are omitted in what follows. Since 3. ζ / ζ φ φ / φ
12 STABILITY OF CONTACT DISCONTINUITY 3 applying Lemma. and the boundary condition 3.3, we have 3. ζ φ δ φ δ φ ζ We now estimate the term from the boundary. We compute from.5 and 3.3, 3.3 φ φ ψ φ ψ /µ ψ ψ φ 4/3 Thus there exists a positive constant δ. For any δ δ, Lemma 3. is obtained by Lemma 3.3. It follows that 3.4 φ φ τ τ δ 4/3 φ ψ ζ 4/3 Proof. Following [3], we introduce a new variable = /. Then.5 can be rewritten by the new variable as 3.5 Multiplying 3.5 by /, we have 3.6 µ ψ µ ψ θ The Cauchy inequality yields that and ζ ψ θ 3.9 φ φ On the boundary = ψ ζ = ψ ψ δ δ ζ δ ψ φ δ δ ζ =, we have 3.3 ψ = ψφ 3 φ φ
13 4 F. HUANG, A. MATSUMURA AND X. SHI Note that the right hand sides of have already been investigated in Lemma 3.. Integrating 3.6 over, using Lemma 3., , the boundary condition 3.3, the fact that ψ µ ψ 4 and choosing δ is suitably small, we get the desired estimate 3.4. Lemma 3.4. It follows that 3.3 ψ ζ { ψ ζ τ } τ δ 4/3 φ ψ ζ 4/3 Proof. Multiplying.5 by ψ and.5 3 by ζ and combining the resulting equalities, we have 3.3 ψ ζ ψ ψ ζ ζ γ γ κ ζ ψ µ µ ψ ζ = ψ ζ where 3.33 =µ κζ ψ ζ µψ ζ ψ ζ κ ζ ψ Since is constant, we have 3.34 ψ δ ψ δ φ ζ δ φ ζ On the other hand, the Cauchy inequality yields 3.35 µ ζ δ ζ δ ψ 4 ψ ζ and 3.36 δ ψ ζ δ 4 φ ψ ζ φ ψ ψ ζ ζ
14 STABILITY OF CONTACT DISCONTINUITY 5 The integral of the term ψ ζ can be estimated as follows ψ ζ sup{ ψ } ψ ζ ψ 3/ ψ / ζ δ ψ ζ δ ψ The last term of 3.36 can be also treated as in It is noted that ψ = φ exponentially decays and ζ =. On the boundary =, we have 3.38 ψ φ = ψφ ψφ = ψφ φ µ ψ /µ The right hand side of 3.38 can be estimated by the same way as 3.3. Thus integrating 3.3 over, choosing δ is suitably small, and using and Lemma 3., Lemma 3.3 imply 3.3. Lemma 3.4 is proved. Proof of Proposition 3.. Proposition 3. is obtained at once from Lemma 3. Lemma 3.4. REMARK. We note that Proposition 3. is only formally obtained in the previous argument because our solution space is not enough regular. To give a rigorous justification, we use the standard mollifier to smooth the solution. Since our solution is regular with respect to the space variable, we only need to smooth the solution with respect to the time variable. Let satisfying = and ε = /ε /ε. Let be any measurable function, we define ε = ε. For any < τ < <, there exists a positive constant ε τ such that for any < ε ε, ε [τ ]; holds if [ ];. Then we mollify the equations.5 as following: φ ε ψ ε = ψ ε ε ζ ε ε ε ψ ε ε ε φ ε = µ ε ε φ ε ε ε ε 3.39 γ ζε ε ζ ε ε φ ε ε ψ ε ε ε ζ ε ε = κ ε φ ε κ µ ε ψ ε ε ε φ ε µ ε ε ε ε ψ ε = µ φε ζ ε = where ε, =, tends to zero in [ ]; norm as ε. Following the same
15 6 F. HUANG, A. MATSUMURA AND X. SHI method in Proposition 3., we get 3.4 φ ε ψ ε ζ ε { φ ε ψ ε ζ ε } τ τ δ 4/3 φ ε ψ ε ζ ε τ 4/3 ψ ε ε ε ζ ε τ ε ε τ Since φ ε ψ ε ζ ε φ ψ ζ in norm, ψ ε ζ ε ψ ζ in norm [ ]; and φ ε ψ ε ζ ε τ φ ψ ζ τ in norm, letting ε implies 3.4 Letting τ yields Proposition 3.. φ ψ ζ { φ ψ ζ } τ τ δ 4/3 φ ψ ζτ 4/3 Proof of Theorem.. Theorem. is easy from the Propositions. and Local existence Proof of Proposition.. 4. φ = φ From.5, φ has an explicit form ψ τ τ The equation.5 and.5 3 are regarded as the initial-boundary value problem for the parabolic equation of ψ and ζ respectively: ψ ψ µ = := φ ζ φ ζ φ 4. µψ = φ ψ = = ψ ζ ζ κ = := φ ζ φ ψ φ 4.3 ζ = ζ = = ζ where 4.4 ζ = φ µ φ
16 4.5 STABILITY OF CONTACT DISCONTINUITY 7 = ζ φ φ κ ψ µ φ φ To use the iteration method, we approximate φ ψ, ζ by φ ψ ζ 3 which will be determined later. We define the sequence {φ ψ ζ } for each so that 4.6 φ ψ ζ = φ ψ ζ and for a given φ ψ ζ, ψ and ζ are respectively the solutions to ψ ψ µ φ = = φ ζ φ ζ 4.7 ψ = µ φ /µ ψ = = ψ ζ ζ κ φ = = φ ζ φ ψ 4.8 ζ = ζ = = ζ and 4.9 φ = φ ψ τ τ We now construct φ ψ ζ. First we choose φ 3 such that φ φ strongly in as. Let 4. = ψ = µ φ It is easy to check that 3, =, satisfying 4. = ψ = = = µ φ Let ψ = ψ. Then we have ψ. We choose ψ 3 such that ψ ψ strongly in. Let ψ = ψ, then we have ψ ψ strongly in. Furthermore we have ψ = /µφ which guarantees the compatibility condition for the equation 4.7. In the same way, we can
17 8 F. HUANG, A. MATSUMURA AND X. SHI construct the approximation ζ 3 satisfying ζ ζ strongly in and 4. κ ζ φ = φ φ ψ which guarantees the compatibility condition for the equation 4.8. It is obvious that we can choose the above approximation φ ψ ζ satisfying φ ψ ζ 3/, inf φ /3 for any. By the linear theory, if [ ];, =,, and ψ ζ 3, there exists a unique-local solution ψ ζ to 4.7 and 4.8 satisfying 4.3 ψ ζ [ ]; 3 [ ]; ; 4 Thus, if φ ψ ζ /, then the elementary energy estimate gives 4.4 ψ 3 exp µ if = 4.5 is suitably small. Also we have ψ τ Similarly we obtain 4.6 ζ where we have used the fact that 4.7 ψ ζ ψ ψ ζ δ ζ δ ψ δ for suitably small constant δ >. On the other hand, direct computation on 4.9 and 4.6 yields 4.8 φ and inf [ ] φ / Therefore we have φ ψ ζ /. Since φ ψ ζ 3, φ ψ ζ can be shown a Cauchy sequence in ;. Thus we have a solution φ ψ ζ by letting tends to infinity. In particular, on the boundary =, we have 4.9 φ = ψ = µ φ
18 STABILITY OF CONTACT DISCONTINUITY 9 In the same way, we can show φ ψ ζ is also a Cauchy sequence in ; taking smaller than if necessary. Letting, we obtain the desired uniquelocal solution φ ψ ζ / to.5. ACKNOWLEDGEMENTS. The work of F. Huang was supported in part by the JSPS Research Fellowship for foreign researchers and Grand-in-aid No. P-69 for JSPS from the ministry of Education, Science, Sports and Culture of Japan. References [] F.V. Atkinson and L.A. Peletier: Similarity solutions of the nonlinear diffusion equation, Arch. rational Mech. Anal , [] C.T. Duyn and L.A. Peletier: A class of similarity solution of the nonlinear diffusion equation, Nonlinear Analysis, T.M.A. 977, [3] L. Hsiao and T.P. Liu: Nonlinear diffsusive phenomenia of nonliear hyperbolic systems, Chin. Ann. Math. 4B:4 993, [4] S. Kawashima and A. Matsumura: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys. 985, [5] S. Kawashima, A. Matsumura and K. Nishihara: Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad , [6] S. Kawashima and Y. Nikkuni: Stability of stationary solutions to the half-space problem for the discrete Boltzmann equation with multiple collision, Kyushu J. Math, to appear. [7] T. Liu: Shock waves for compressible Navier-Stokes equations are stable, Comm. Pure Appl. Math. XXXIX 986, [8] T. Liu and Z. Xin: Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys , [9] T.P. Liu and Z.P. Xin: Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math. 997, [] A. Matsumura and M. Mei: Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Rat. Mech. Anal ,. [] A. Matsumura and K. Nishihara: On the stability of traveling wave solutions of a onedimensional model system for compressible viscous gas, Japan J. Appl. Math. 985, 7 5. [] A. Matsumura and K. Nishihara: Asymptotics toward the rarefaction wave of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math , 3. [3] A. Matsumura and K. Nishihara: Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Commun. Math. Phys , [4] A. Matsumura and K. Nishihara: Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect, Q. Appl. Math. LVIII, [5] T. Pan, H.X. Liu and K. Nishihara: Asymptotic behavior of a one-dimensional compressible viscous gas with free boundary, SIAM J. Math. Anal, to appear. [6] J. Smoller: Shock waves and reaction-diffusion equations, Berlin, Heidelberg, New York, Springer, 98. [7] Z.P. Xin: On nonlinear stability of contact discontinuities, Proceeding of 5th international conferences on hyperbolic problems: theory, numerics and applications, edited by Glimm, etc. World Scientific, 996,
19 F. HUANG, A. MATSUMURA AND X. SHI Feimin HUANG Dept. of Pure & Appl. Math. Grad. School. of Inform. Sci. & Tech. Osaka Univ., Osaka 56-43, Japan Inst. of Appl. Math. AMSS, Academia Sinica Beijing 8, China Akitaka MATSUMURA Dept. of Pure & Appl. Math. Grad. School. of Inform. Sci. & Tech. Osaka Univ., Osaka 56-43, Japan Xiaoding SHI Dept. of Math. Grad. School of Sci. Osaka Univ., Osaka 56-43, Japan Dept. of Math. Grad. School of Sci. Beijing Univ. of Tech. & Chem. Beijing 9, China Contact address:
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