GLOBAL L^∞ SOLUTIONS TO SYSTEM OF ISENTROPIC GAS DYNAMICS IN A DIVERGENT NOZZLE WITH FRICTION

Acta Mathematica Scientia, 2019, 39B(5): 1213-1218https:///10.1007/sl0473-019-0501-2©Wuhan Institute Physics and Mathematics,Chinese Academy of Sciences, 2019数学物理学报http: 〃a ct a GLOBAL L 00 SOLUTIONS TO SYSTEM OF ISENTROPIC GAS DYNAMICS IN A DIVERGENTNOZZLE WITH FRICTION ** Received August 31, 201& revised March 5, 2019. The first author was supported by the Zhejiang Natural Science Foundation of China (LQ13A010022). The second author was supported by the Qianjiang professorship of Zhejiang Province of China and the National Natural Science Foundation of China (11271105).t Corresponding author: Yunguang LU.Qingyou SUN (孙庆有) Yunguang LU (陆云光)tK. K. Chen Institute for Advanced Studies, Hangzhou Normal University, Hangzhou 311121, ChinaE-mail : qysun@hznu. edu. cn; ylu2005@ustc. edu. cnChristian KLINGENBERGDeptartment of Mathematics, Wuerzburg University, Wuerzburg 97070, GermanyE-mail : klingen@mathematik. uni-wuerzburg. deAbstract In this article, we study the global L°° entropy solutions for the Cauchy problem of system of isentropic gas dynamics in a divergent nozzle with a friction. Especially when the adiabatic exponent 7 = 3, we apply for the maximum principle to obtain the L°° estimates w(p s,e ,u s '£) < B(t) and z(p s,e ,u s,e ) < B(t) for the viscosity solutions (p s 'e ,u s,e ), where B(t) is a nonnegative bounded function for any finite time t. This work, in the special case 7 > 3, extends the previous works, which provided the global entropy solutions for the same Cauchy problem with the restriction w(p s,e ,u s,£) < 0 or z(p 6,e y u 6,e ) < 0.Key words global L°° solution; isentropic gas dynamics; source terms; flux approximation;compensated compactness2010 MR Subject Classification 35L45; 35L60; 46T991 IntroductionThe following system of isentropic gas dynamics in a divergent nozzle with a friction, whose physical phenomena called “choking or choked flow ”，occurs in the nozzle (see [1, 2] for the det a ils),Pt 十的"晋叫a f (x) 2{pu)t + (pu 2 + P(p))z = - apu\u\,(1-1)where p is the density of gas, u is the velocity, P = P(p) is the pressure, a(x) is a slowly variable cross section area at x in the nozzle, and a denotes a friction constant. For the polytropic gas, P takes the special form P(p) = where 7 > 1 is the adiabatic exponent.鱼Springer1214ACTA MATHEMATICA SCIENTIA Vol.39 Ser.BThe golobal entropy solutions for the Cauchy problem (1.1) with bounded initial data(p(ar, 0), u(x, 0)) = (po(x), uq (x )), po(x) > 0,(1-2)was first studied in [2] for the usual gases 1 < 7 < |, and later, by the author in [3] for any adiabatic exponent 7 > 1, provided that the initial data are bounded and satisfy the strong restriction condition on the Riemann invariant zo(po(e), "()(©)) S 0, where the Riemann invariants of (1.1) arez (P,u) = f w(p,tz) = /s (1-3)and c is a constant.It is well-known t h at, after we have a met h od to obtain the global exis t ence of solutions for the Cauchy problem of the following homogeneous systemPt + = 0,(pu)t + (pu 2 + P(p))x = 0(1.4)with the bounded initial data (1.2), the unique difficulty to treat the inhomogeneous system (1.1) is to obtain the a-priori L°° estimate of the approximation solutions of (1.1), for instance, the a-priori L°° estimate of the viscosity solutions for the Cauchy problem of the parabolic system a z (x)Pt + ^x = --^pu + ep xx ,a f (x) 2{pu)t + (pu 2 + P(p)L =----pu 2 一 apu\u\ + e{pu)xx (1.5)a(x)with the initial data (1.2).When a\x) = 0, (1.1) is the river flow equations, a shallow-water model describing the vertical depth p and mean velocity u, where apu\u\ corresponds physically to a friction term, and a is a nonnegative constant. This kind of inhomogeneous systems is simple because the source terms have, in some senses, the symmetric behavior. We may use the Riemann invariants (w, z) of system (1.4) to rewrite (1.1) as the following symmetric, coupled systemWt + 入2% = ew xx H ----Px^x — _7_t ==:(2F ，+ pP")危—石Q (s — z)|u|,P 2z t + 入1死=ez xx + —pxZ x — (2P Z + pP")危—尹(z 一 讪"|・(1.6)We can apply for the maximum principle directly to (1.6) to obtain the necessary a-priori L°° estimates on the Riemann invariants w(p £,< M and z(p £,u £) < N 、for two suitable constants M 、N (see [4] for the det a ils).When q = 0, that is, the nozzle flow without friction, system (1.1) was well studied in ([2, 5-8]). Roughly speaking, the technique, introduced in these articles, is to control the super-linear source terms pu and pw 2 by the flux functions pu and pu 2 + P(p) in (1.1), and to deduce a upper bound of w or z by a bounded nonegative function B(x), which depends on the function a(x).When 丰 0 and q 黑 0, both the above techniques do not work because the flux functions can not be used to control the super-linear friction source terms apu\u\, and the g SpringerNo.5Q.Y.Sun et al:GLOBAL L°°SOLUTIONS TO IGD SYSTEM IN DIVERGENT NOZZLE1215functions哥？pu destroyed the symmetry of the Riemann invariants(w,z).In fact,we may copy the method given in[2,3]to obtain the following process.First,to avoid the singularity of the flux function pu2near the vacuum p=0,we still use the technique of the5-flux-approximation given in[9],and introduce the sequence of systemsPt+(-26u+pu)x=A(x)(p一26)“，(17)(ptz)t+(pu2—6u2+Pi(p,5))x=A(x){p—25)162—apu\u\to approximate system(1.1),where A(x)=5>0denotes a regular perturbation con-stant and the perturbation pressurerp+_q c/(1.8)J26tSecond,we add the viscosity terms to the right-hand side of(1.7)to obtain the following parabolic systemPt+((p一26)u)x=A(x)(p一26)"+ep xx,(pu)t+(pu2—Su2+RSe))©=4(e)(p-2(5)u2一apu\u\+£(pu)xxwith initial data(p15,6^,0),^(2:,0))=(p0(x)+2d,“o(0)),(1-9) (1-10)where(po(^),^0(^))are given in(1.2).Now,we multiply(1.9)by(w p,w^)and(z p,Zm),respectively,where(w,2are given in (1.3),to obtainWt+=£w X x H-----Px^x---------(2P+pP")危+A(x)^p—26)乜必(")—P2p2y/P\p)P(1.11) andz t+入(先=ez xx H-----Px^x——z一(2P+pP")危+4(^)(“—+au\u\,P2p2y/P\p)P(1•⑵where俟沪宁存药乂专+今丙(1.13)are two eigenvalues of the approximation system(1.7).It is obvious that the terms A(x')(p-26)u^^-in(1.11)and(1.12)are not symmetric with respect to the Riemann invariants w,z.However,with the strong restriction(/?()(©),”o(e))S 0on the initial data,in[2,3],we may obtain the uniformly upper bounds of z and w by using the maximum principle.In some senses,it is similar to obtain the estimate u<0for the following scalar equationu t+/(u)x+S(u,x,i)u=eu xx(1.14) for a suitable bounded function S(iz,x,t)when the initial data uq^x)<0.In this article,we will remove the condition z()(po(龙)，"()(©))S0,and prove the uniformly upper bound of2and w by a bounded nonnegative function B(t)of t,and obtain the global existence theorem of the entropy solutions for the Cauchy problem(1.1)and(1.2)as follows:g Springer1216ACTA MATHEMATICA SCIENTIA Vol.39 Ser.BTheorem 1.1 (I) Let P(p) = 7 > 3, the function 4(r ) < 0 (or A[x) > 0 ) be bounded. Then, there exists a bounded, nonegative function B(t), such that, u s,€) <B(t), < B(t), where E(t) depends only on the bound of the initial data.(II) Under the conditions in (I), there exists a subsequence of (p s,e (Xj t), u 6,e (x, t)), which converges pointwisely to a pair of bounded functions (p(x, t), u(x, t)) as 6, £ tend to zero, and the limit is a weak entropy solution of the Cauchy problem (1.1)-(1.2).2 Proof of Theorem 1.1In this section, we shall prove Theorem 1.1.When A(x) < 0, we rewrite systems (1.11) and (1.12) as follows:eWt + + — (—A(x)(p — 26)-----^―■ + q |u |)(w — z)2 P=£w xx + ~Px^x ~ T~Z~~ (2P + pP")危P 2p 2y /P\p)and「述 di)c SZt + 入f z ’ + —q |u |(2： — w) — A(x)(p — 25)— (=£z xx + —p x z x 一 -——y==^P ，+ pP")化.P 2p 2y/P\p)Letting — w — B(t),V2 = z — B(t), we have from (2.1) and (2.2) that⑵2)St + B f (t) + 入釦“ + -(—A(x)(p — 26)-----— + a|u|)(vi — s)2 p =+ —p x vix - 7==(2^ + pP'\化P 2p 2 yP^p)(2.1)(2.3)andV2t + B f (t) + xiv 2x + ^a|u|(v 2 一 5)+ A.(x){p 一 26)"" W2 P一％)(p 一 26)芒@ I p J,e""P ⑶ ds + B(t)A{x\p — 26)"回s P(2.4)=^2xx + —PxV2x 一 2 /577-T (2-P ，+ 淀. P 2pV p (p)Choose B(t) such that > 0,0 < B(t) < /3, where /3 be a positive constant. Then, whenV ： 2)ds 2 0 or p is large as 7 > 3, we have旳）一4仗）（卩一 26）^^P P 住⑶ds + - 26) JP® > 0.s P (2.5)When V ： &)ds v 0 or p is small, we can always choose a B(t), where B z (t) is suitable large, such that (2.5) is also true. Therefore, we have the following inequalities from (2.3) and (2.4)^it + 入如is + 石(一4@)(P — 26)------------F a|u|)(vi — V2)< svixx H -----Px^ix 2 P P (2.6)g SpringerNo.5Q.Y.Sun et al:GLOBAL L°°SOLUTIONS TO IGD SYSTEM IN DIVERGENT NOZZLE1217 and^2t+入力加+:Px^2x>(2.7)If the initial data satisfy zo(po(R),"()(©))S-B(O)and Wo(po(^)?"o(©)<B(0),then,%o(“o(忑)，"o(z))<0,i=1,2,we can apply for the maximum principle given in[10]to(2.6)and(2.7)to obtain仇t),泌气①,t))<0, i=1,2,where the coefficient A(x)^p—2d)必号亘before g is only necessary to be local bounded.In fact,we rewrite inequalities(2.6)and(2.7)as follows:(2.8)Vit+入鈿+Ci(©,t)Wi—v2)<ev lxx H-----PxVlxpand(2.9)V2t+入恂2s+C2(R,t)(Q2—5)+C3(rr,t)v2<ev2xx H-----px^2x-Pwhere c\(x y t)=寺(-4(^)(/9—26)"；的+a|u|)>0, C2(x,t)=|a|u|>0,and C3(x,t)= %)(p-2d)今回.Make a transformation如=（仇+"（*护心）評,。