THE CAUCHY PROBLEM FOR ONE-DIMENSIONAL FLOW OF A COMPRESSIBLE VISCOUS FLUID: STABILIZATION OF THE SOLUTION

Transcription

1 GLASNIK MATEMATIČKI Vol. 4666, 5 3 THE CAUCHY POBLEM FO ONE-DIMENSIONAL FLOW OF A COMPESSIBLE VISCOUS FLUID: STABILIZATION OF THE SOLUTION Nermina Mujakoić and Ian Dražić Uniersity of ijeka, Croatia Abstract. We analyze the Cauchy problem for non-stationary -D flow of a compressible iscous and heat-conducting fluid, assuming that it is in the thermodynamical sense perfect and polytropic. This problem has a unique generalized solution on ],T[ for each T >. Supposing that the initial functions are small perturbations of the constants and using some a priori estimates for the solution independent of T, we proe a stabilization of the solution.. Introduction In this paper we analyze the Cauchy problem for non-stationary -D flow of a compressible iscous and heat-conducting fluid. It is assumed that the fluid is thermodynamically perfect and polytropic. The same model has been mentioned in [], where a global-in-time existence theorem for generalized solution is gien without a rigorous proof. Here, we approach to our problem as the special case of the Cauchy problem for a micropolar fluid that is considered in [5] and [6]. Therefore we know that this problem has a unique generalized solution on ],T[, for each T > and that the mass density and temperature are strictly positie. Assuming that the initial functions are small perturbations of the constants, we first derie a priori estimates for a solution independent of T and then analyze the behaior of the solution as T. We use some ideas of Ya. I. Kanel [4] applied to the case of stabilization of Hölder continuous solution for the same model. Mathematics Subject Classification. 46E35, 35B4, 35B45, 76N. Key words and phrases. Compressible iscous fluid, the Cauchy problem, stabilization. 5

2 6 N. MUJAKOVIĆ AND I. DAŽIĆ. Statement of the problem and the main result Let, and denote, respectiely, the mass density, elocity, and temperature of the fluid in the Lagrangean description. Supposing that in [5] the microrotation is equal to zero, we obtain the problem considered as follows:. t + =,..3 t = K, t = K + +D in +, where K and D are positie constants. The equations.-.3 are, respectiely, local forms of the conseration laws for the mass, momentum and energy. We take the following non-homogeneous initial conditions:.4 x, = x,.5 x, = x,.6 x, = x for x, where, and are gien functions. We assume that there exist the constants m,m +, such that.7 m x M, m x M, x. In the papers [5] and [6] it was proed that for.8,, H the problem.-.6 has, for each T +, a unique generalized solution.9 x,t,,x,t, x,t Π = ],T[, with the properties:. L,T;H H Π,., L,T;H H Π L,T;H. Using the results from [5] and [] we can easily conclude that., > in Π. Deriation of the equations.-.3 from the Eulerian description is gien in [], pp. 3-4.

3 COMPESSIBLE VISCOUS FLOW 7 We denote by B k,k N, the Banach space { }.3 B k = u C k : lim x Dn ux =, n k where D n is n-th deriatie. The norm of the space B k is defined by { }.4 u Bk = sup sup D n ux. n k x From Sobole s embedding theorem [, Chapter IV] and the theory of ector-alued distributions [3, pp ] one can conclude that from. and. follows.5 L,T;B C [,T];L,.6, L,T;B C [,T];H L,T;B and hence.7, C [,T];B, L Π. From.7 and.8 it is easy to see that there exist the constants E, E, E 3, M +, M >, such that.8 dx+k ln dx+ ln dx = E,.9 + dx = E,. dx+ ln dx E 3,. sup x < M. x <. Suppose that the quantities η and η are such that η < < η and η where E E 4.3 E 5 = K, E 4 = µe Let η eη ηdη = eη ηdη = E 5,.4 u = expη, u = expη. +M + E { } 3 K, µ = max E D,. The aim of this work is to proe the following theorem.

4 8 N. MUJAKOVIĆ AND I. DAŽIĆ Theorem.. Suppose that the initial functions satisfy.7,.8 and the following conditions:.5 E D u dx <, 6u.6 u E E u +M M 6E 4 u u + K D M + KM E 4 +E D u M < min, s lnsds, s lnsds 6u then.7 x,t, x,t, x,t, when t, uniformly with respect to all x. emark.. Conditions.5 and.6 mean that E, E and E 3 are sufficiently small. In other words the initial functions are small perturbations of the constants. In the proof of Theorem.. we apply some ideas of [4], where a stabilization of the solution that is Hölder continuous was proed for the same model. 3. A prori estimates for, and Considering stabilization problem, one has to proe some a priori estimates for the solution independent of T, which is the main difficulty. Some of our considerations are similar to those of [4]. First we derie the energy equation for the solution of problem.-.3 under the conditions indicated aboe and we estimate the function. Lemma 3.. For each t > we hae 3. dx+ ln dx+k + t where E is defined by.8. ln dx +D dxdτ = E, Proof. Multiplying.,. and.3, respectiely, by K, and, integrating by parts oer and oer ],t[ and taking into account.5 and.6, after addition of the obtained equations we easily get equality 3. independently of t.

5 COMPESSIBLE VISCOUS FLOW 9 Lemma 3.. For each t > exist the strictly positie quantities u = u t and u = u t such that 3. u x,τ u, x,τ ],t[, where 3.3 t = sup x,τ ],t[ x,τ. Proof. We multiply. by ln, integrate oer and ],t[ and use the following equalities, which follow using. and.5-.7: 3.4 ln = ln, t 3.5 t ln dxdτ = t = ln dx t = = t t ln dxdτ ln t dx t + ln dx+ ln dx+. = ln dx dxdτ, ln dx, We also take into account the following inequalities obtained by the Young s inequality , 3.9 ln = 4 +. After simple transformations we get dx+ K t 3 dxdτ 4 3. dx+ K t t dxdτ + dxdτ + ln dx+ dx

6 N. MUJAKOVIĆ AND I. DAŽIĆ for each t >. Using 3. and. from 3. we obtain 3. 4 dx + K t 3 where 3. K t = µe +t+ E 3 E and µ, E and E 3 are defined by.3,.8 and.. Now we define the increasing function ψ by 3.3 ψη = η eξ ξdξ. One can conclude the following 3.4 ln ψ ln = ψ ξdξ 3.5 where dxdτ K t, ψ ln Using 3., 3. and the Hölder s inequality we get ln ψ K t, 3.6 K t = E µ K ln dx +t+ E 3. E dx. dx We can also easily conclude that there exist the quantities η = η t < and η = η t >, such that 3.7 η eη ηdη = η eη ηdη = K t, wherek t isdefinedby3.6. Comparing3.5and3.7weconclude that 3.8 u = expη x,τ expη = u for x,τ ],t[. Now we find some estimates for the deriaties of the functions and.

7 COMPESSIBLE VISCOUS FLOW Lemma 3.3. For each t > we hae + dx u t dx K Du 4 τ + D t 8 dxdτ K 3 t, where Du 3. K 4 t =, 6u E dx dτ 3. u 6K u K 3 t = E +t t t + K u u D t +KtK t +E. Proof. Multiplying equations.and.3, respectiely, by and, integrating oer ],t[ and using the following equality t t dx 3. t dxdτ = that is satisfied for the function as well, after addition of the obtained equalities we find that 3.3 t + dx t +D dxdτ = t +K t t + t t dxdτ +K dxdτ +K dxdτ D t t dxdτ dxdτ dxdτ dxdτ.

8 N. MUJAKOVIĆ AND I. DAŽIĆ Using 3.8, 3. and the following inequality 3.4, that holds for the function as well, and applying the Young s inequality with a sufficiently small parameter on the right-hand side of 3.3 we come to the estimates as follows: 3.5 t dxdτ t dxdτ + t 4 dxdτ u t dx dx u dx dτ + t 4 dxdτ 5 t 6 dxdτ + u t 6K t u dxdτ 5 t 6 dxdτ u t + 6K t t dxdτ, u t K dxdτ t 3.6 K dxdτ + t dxdτ K u t u t K t 3.7 K t K t t dxdτ dxdτ dxdτ + 4 t dxdτ + 4 t dxdτ t dxdτ, dxdτ,

9 3.8 t K K D t K u 3 t Du COMPESSIBLE VISCOUS FLOW 3 dxdτ dxdτ + D 4 t t dxdτ + D 4 dxdτ t dxdτ, 3.9 t dxdτ t 4 dxdτ + D D 4 u t dx3 Du + D t 4 dxdτ 8u t D u dx + D t 4 dxdτ, t 3 dτ + 8 dxdτ dx dτ t dxdτ 3.3 t D D t Du dxdτ dxdτ + D 4 t dx u + D t 4 dxdτ 8DK t tu t + 3D 8 t u dxdτ. t dxdτ dx dxdτ dxdτ

10 4 N. MUJAKOVIĆ AND I. DAŽIĆ Taking into account , 3., 3. and.9 from 3.3 follows [ ] + dx+ t 6 dxdτ D t 8 dxdτ 8u t 3 D u dx dτ +K 3 t, wherek 3 t isdefinedby3.. Wealsogetanotherimportantinequality by estimating the integral dx. We hae dx = dx 3.3 Consequently, 3.33 u E u dx dx E u dx dx. dx. Inserting 3.33 into 3.3 we obtain [ ] + dx u t D u dx Du 6u E + D t 8 dxdτ K 3 t and 3.8 is satisfied. dx dτ Similarly as in [4], in the continuation we use the aboe results, as well as the conditions of Theorem.. We derie the estimates for the solution,, of problem.-.8 defined by.9-. in the domain Π = ],T[, for arbitrary T >.

11 COMPESSIBLE VISCOUS FLOW 5 Taking into account assumption. and the fact that C Π see.7 we hae the following alternaties: either 3.35 sup x,t = T M, x,t Π or there exists t, < t < T, such that 3.36 t < M for t < t, t = M. Nowweassumethat3.36issatisfiedandwewillshowlater,thatbecause of the choice of the constants E, E, E 3 and M the conditions of Theorem., 3.36 is impossible. Because K t, defined by 3.6, increases with increasing t we can easily conclude that 3.37 K t < K M for t < t and K M = E 5. Therefore we hae 3.38 u < u t, u > u t where u, u and u t, u t are defined by.-.3 and , respectiely. The quantity K 4 t, defined by 3., decreases with increasing t and for t = M it becomes Du 3.39 K 4 M =. 6uE Taking into account the assumption.5 of Theorem. and the following inclusion See C [,T];L we hae again two alternaties: either 3.4 x,tdx K 4 M for t [,t ], or there exists t, < t < t, such that 3.4 x,tdx < K 4 M for t < t, and 3.43 x,t dx = K 4 M for t < t. Now, we assume that are satisfied. Then we hae 3.44 t < M, K 4 M < K 4 t for t [,t ].

12 6 N. MUJAKOVIĆ AND I. DAŽIĆ Taking into account 3.44, from 3.9, for t = t, we obtain 3.45 x,tdx K 3 t, t t. Since K 3 t, defined by 3., increases with the increase of t, it holds 3.46 K 3 t K3 M, t [,t ]. Using condition.6 we get 3.47 K 3 t < K4 M, t [,t ], and conclude that 3.48 x,t dx < K 4 M. This inequality contradicts Consequently, the only case possible is when 3.49 t = t and then 3.4 is satisfied. Using 3.36 and 3.4 from 3.9 we can easily obtain that 3.5 x,tdx < K 3 M for < t t. Now, we introduce the function Ψ by 3.5 Ψx,t = x,t s lnsds. From.7 follows that x,t as x and hence 3.5 Ψx,t as x. Consequently, 3.53 x,t d ψx,t ψx,t = ds ψsds x = x,t x,t lnx,t dx x,t lnx,tdx x,tdx. Taking into account 3.36, 3.5 and 3. from 3.53 we get 3.54 max x,t M ψx,t = ψ t = ψm K 3 M E,

13 COMPESSIBLE VISCOUS FLOW 7 or 3.55 M s lnsds K3 M E. Since this inequality contradicts.6, it remains to assume that t = T. Hence we hae Lemma 3.4. For each T > we hae 3.56 x,t M, x,t Π, x,tdx K 4 M, t T, x,tdx K 3 M, t T. Proof. These conclusions follow from 3.36, 3.4 and 3.5 directly. Lemma 3.5. The following inequalities hold true: 3.59 < u u, x,t Π, x,t 3.6 sup x,t 8E K 4 M, x,t Π 3.6 x,t h >, x,t Π where u and u are defined by.-.4 and a constant h depends only on the data of problem.-.8. Proof. Because the quantity u t in Lemma 3. decreases with increasing t, while u t increases, it follows from 3. and 3.56 that 3.59 is satisfied. Using the inequality x 3.6 = dx dx dx and estimations 3. and 3.57 we get immediately 3.6. From 3.53, 3.56, 3.58 and 3. we hae for x,t that the following holds 3.63 x,t s lnsds K3 M E < s lnsds because of.6. Hence we conclude that there exists the constant h > such that x,t h.

14 8 N. MUJAKOVIĆ AND I. DAŽIĆ Lemma 3.6. For each T > we hae T 3.64 dxdτ K 5, T T dxdτ K 6, dxdτ K 7, dx K 8, t [,T], T T dxdτ K 9, dxdτ K, where the constants K 5, K 6, K 7, K 8, K 9, K + are independent of T. Proof. Taking into account 3.59 and 3.6 from 3., 3., 3.9 and 3.3 we get easily Proof of Theorem. In the following we use the results of Section 3. It is important to remark that all the estimates obtained aboe are presered in the domain Π = ],T[ for each T >. The conclusions of Theorem. are immediate consequences of the following lemmas. Lemma 4.. The following relations hold true: 4. x,tdx, x,tdx, when t. Proof. Let ε > be arbitrary. With the help of 3.64, 3.65 and 3.66 we conclude that there exists t > such that 4. t t t dxdτ < ε, t t dxdτ < ε, t dxdτ < ε,

15 COMPESSIBLE VISCOUS FLOW 9 for t > t, and 4.3 x,t dx < ε, x,t dx < ε. Similarly to 3.9, we hae [ ] + dx + 8u t dx [K Du 4 τ t + D t 8 t dxdτ 4.4 [ t + ]x,t dx+k t + u t 6K t u dxdτ t t +K t dxdτ + K t D t + 8DK t u t u dxdτ. t dx] dτ dxdτ dxdτ Taking into account 3.56, 3.59 and from 4.4 we obtain [ ] dx K ε for t > t, where K depends only on the data of our problem and does not depend on t. Hence relations 4. hold. Lemma 4.. We hae 4.6 x,t, x,t when t, uniformly with respect to all x. Proof. We hae see 3.6 and x,t x,tdx 4.8 ψx,t x,t lnx,tdx x,tdx, dx.

16 3 N. MUJAKOVIĆ AND I. DAŽIĆ Taking into account 3. from 4.7 and 4.8 we get 4.9 x,t E x,tdx, 4. ψx,t E dx. Using 4. and property 3.5 of the function ψ we can easily obtain that 4.6 holds. Lemma 4.3. We hae 4. when t. x,tdx Proof. From 3.66 and 3.69 we conclude that for ε > exists t > such that 4. t t t dxdτ < ε, t dxdτ < ε, x,t dx < ε for t > t. Now, from. we get the equality 4.3 = t. Multiplying 4.3 by and integrating oer and ]t,t[ we obtain 4.4 t 4 dx = dxdτ + 4 x,t dx. t Using the Young s inequality and 3.59 from 4.4 we find out u 4 t t dx u dxdτ +u t 4.5 t dxdτ + u4 x,t dx. With the help of 4. from 4.5 we get 4.. Lemma 4.4. We hae 4.6 x,t when t, uniformly with respect to x.

17 COMPESSIBLE VISCOUS FLOW 3 Proof. Similarly as for the function ψ, we hae ψ = s lnsds 4.7 ln dx dx. Taking into account 3. from 4.7 follows 4.8 ψ KE u dx and with the help of 4. we conclude 4.6. eferences [] S. N. Antontse, A. V. Kazhykho and V. N. Monakho, Boundary alue problems in mechanics of nonhomogeneous fluids, North-Holland, Amsterdam, 99. []. Dautray and J. L. Lions, Mathematical analysis and numerical methods for science and techonology. Vol., Springer-Verlag, Berlin, 988. [3]. Dautray and J. L. Lions, Mathematical analysis and numerical methods for science and techonology. Vol. 5, Springer-Verlag, Berlin, 99. [4] Ya. I. Kanel, Cauchy problem for equations of gas dynamics with iscosity, Sibirsk. Mat. Zh. 979, [5] N. Mujakoić, One-dimensional flow of a compressible iscous micropolar fluid: The Cauchy problem, Math. Commun. 5, 4. [6] N. Mujakoić, Uniqueness of a solution of the Cauchy problem for one-dimensional compressible iscous micropolar fluid model, Appl. Math. E-Notes 6 6, 3 8. N. Mujakoić Department of Mathematics Uniersity of ijeka Omladinska 4, 5 ijeka Croatia mujakoic@inet.hr I. Dražić Faculty of Engineering Uniersity of ijeka Vukoarska 58, 5 ijeka Croatia idrazic@riteh.hr eceied:..9. eised: 6...