Title. Author(s)Cho, Yonggeun; Jin, Bum Ja. CitationJournal of Mathematical Analysis and Applications, 3. Issue Date Doc URL.

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1 Title Blow-up of viscous heat-couctig compressible flow Author(s)Cho, Yoggeu; Ji, Bum Ja CitatioJoural of Mathematical Aalysis a Applicatios, 3 Issue Date Doc URL Type article (author versio) File Iformatio JMathAalAppl_v32p819.pf Istructios for use Hokkaio Uiversity Collectio of Scholarly a Aca

2 Blow-up of viscous heat-couctig compressible flows Yoggeu Cho Departmet of Mathematics, Hokkaio Uiversity, Sapporo, 6-81, Japa Bum Ja Ji Departmet of Mathematics, Seoul Natioal Uiversity, Seoul , Korea Abstract We show the blow-up of strog solutio of viscous heat-couctig flow whe the iitial esity is compactly supporte. This is a extesio of Z. Xi s result[5] to the case of positive heat couctio coefficiet but we o ot ee ay iformatio for the time ecay of total pressure or the lower bou of the etropy. We cotrol the lower bou of seco momet by total eergy a obtai the exact relatioship betwee the size of support of iitial esity a the existece time. We also provie a sufficiet coitio for the blow-up i case that the iitial esity is positive but has a ecay at ifiity. Key wors: blow-up, smooth solutio, viscous heat-couctig compressible flow 2 MSC: 35Q3, 76N1 correspoig author aresses: ygcho@math.sci.hokuai.ac.jp (Yoggeu Cho), bumjaji@hamail.et (Bum Ja Ji). 1 The first author was supporte by Japa Society for the Promotio of Sciece uer JSPS Postoctoral Fellowship For Foreig Researchers. Preprit submitte to Elsevier Sciece 1 July 26

3 1 Itrouctio I this paper, we cosier the followig equatios for a compressible flui i R R + ( 1): where ρ t + iv (ρu) =, (1) (ρu) t + iv(ρu u) + Lu + p =, (2) (ρe) t + iv(ρeu) κ θ + p ivu = Q( u), (3) Lu = µ u (λ + µ) iv u, a Q( u) = µ 2 u + t u 2 + λ(iv u) 2. Here ρ = ρ(x, t), u = (u 1,, u ), θ, p a e eote the esity, velocity, absolute temperature, pressure a specific iteral eergy per uit mass, respectively. t u is the traspose of u. If we eote the total eergy per uit mass E by E = 1 2 u 2 + e, the the eergy equatio (3) ca be rewritte by (ρe) t + iv (ρe + up) = Lu u + Q( u). (4) The viscosity coefficiet µ a λ are assume to be costat satisfyig µ, λ + 2 µ from the physical poit of view. We also eote by κ the coefficiet of heat couctio. If µ = λ = κ =, the we call the equatios as compressible Euler equatios for gas, O the other ha, if µ > a λ+ 2 µ, the we call the equatios as compressible Navier-Stokes equatios. I particular, we call the equatios as heat-couctig compressible Navier-Stokes equatios if µ >, λ + 2 µ a κ >. A polytropic gas is a gas satisfyig the followig state of equatios: p/ρ = Rθ, e = c ν θ a p/ρ = A exp(s/c ν ) ρ γ 1, (5) where R > is the gas costat, A a positive costat of absolute value, γ > 1 the ratio of specific heats, c ν = R the specific heat at costat volume a γ 1 S the etropy. The blow-up of smooth solutios of compressible Euler equatios has bee stuie by several mathematicias. I 1985[4], T. C. Sieris showe that the life spa T of the C 1 solutio of the compressible Euler equatios is fiite whe the iitial ata is costat outsie a boue set a the iitial flow velocity is sufficietly large (super-soic) i some regio. I 1998[5], Z. Xi showe, i a ifferet way from [4], the blow-up result for the compressible Euler equatios, whe the iitial esity a iitial velocity have compact 2

4 supports. I the paper, he also showe the blow-up of smooth solutio for the compressible Navier-Stokes equatios for polytropic gas with zero heat couctio (that is, κ = ), whe the iitial esity has compact support. His theorem was erive iepeetly of the size of ata, but his poit of view caot be applie for κ >, sice i his argumet the estimatio for the lower bou of etropy or time ecay of total pressure is strogly ecessary, which seems har to be obtaie for the case κ >. As for the positive result, oe may refer to [1]. I the paper [1], the authors showe the local existece of the uique strog solutios of the compressible Navier-Stokes equatios (1)-(3) with = 3, κ a oegative esity. I particular, they obtaie for κ > that there exists a fiite time T > such that for some 3 < q 6 ρ C([, T ]; H 1 W 1, q ), ρ t C([, T ]; L 2 L q ), (u, e) C([, T ]; H 1 H 2 ), (u t, e t ) L 2 (, T ; H). 1 (6) I this paper, we exte the Xi s blow-up result to the heat-couctig compressible Navier-Stokes equatios, that is, for the case κ > i view of the regularity (6) a hece we provie a sufficiet coitio for the local result of [1]. Before statig our mai theorem, we itrouce some otatios. We eote by B R = B R () the ball i R of raius R cetere at the origi. We will use several physical quatities: m(t) = ρ(x, t)x R (total mass), M(t) = ρ(x, t) x 2 x R (seco momet), A(t) = ρ(x, t)u(x, t) xx R (raial compoet of mometum), E(t) = ρ(x, t)e(x, t) x R (total eergy) P (t) = p(x, t) x R (total pressure). We always assume that m(), M(), A(), E() < a m() >, E() >. For the proof of blow-up, we have oly to prove the followig theorem. Theorem 1 We assume µ >, λ + 2 µ >, 1 3 a κ. Let γ > 1 a T >. Suppose that (ρ, u, e) is a solutio to the cauchy problem (1), (2) a (3) with iitial ata (ρ, u, e ) such that for some q > max(2, ) ρ C([, T ]; H 1 W 1, q ), ρ t C([, T ]; L 2 L q ), (u, e) C([, T ]; H 1 H 2 ), (u t, e t ) L 2. (7) 3

5 Furthermore, assume that the iitial esity ρ is compactly supporte i a ball B R. The we have R 2 M() m() + 2 A() E() T + mi(2, (γ 1)) m() m() T 2. (8) The restrictio of imesio ca be remove by a appropriate choice of Sobolev spaces guarateeig the cotiuity of the solutio. For example, we ca take C 1 ([, T ]; H k ) for k > 2 + [ ] as i [5]. 2 Let T be the life spa of the solutio (ρ, u, e). The sice m() a E() are strictly positive, the theorem above implies that T shoul be fiite for γ > 1. It also shows the exact relatioship betwee the size of support a the life spa. For example, the rage of life spa ca be extee as the iitial support of esity become larger. Hece, from the relatio, oe ca expect the global existece of smooth solutio of compressible Navier-Stokes equatios i case that the iitial esity is positive but has a ecay at ifiity i the sese of M() <. However eve i this case, we show that there is o global solutio with u havig a little bit fast ecay as time goes o as follows: Theorem 2 Suppose that (ρ, u, e) is the solutio of (1), (2),(3) satisfyig (7), a iitial esity ρ is ot compactly supporte but its seco mometum is fiite (M() < ). The there is o global solutio of regularity (7) with T = such that lim sup t t u(x, t) x < 1. (9) 1 + x 2 L I view of the parabolic scalig αu(αx, α 2 t), it is expecte for the global solutio with the esity away from zero that lim sup t (1 + t) u(x, t) c, 1 + x L where the costat c ca be chose to be strictly smaller tha 1 uer a smalless assumptio for iitial ata, rewritig the equatio (1)-(3) with (ũ, ẽ) = ( 1+t 1+t u, e) a usig the usual eergy estimate for (ũ, ẽ) a 1+ x 1+ x elliptic regularity as i [3]. However Theorem 2 shows that eve though the bou (9) seems to be reasoable for a esity away from zero, the global existece satisfyig (9) is impossible for the iitial esity havig a ecay at ifiity i the sese of M() <, o matter how small the ata is. Our proof is base o more elemetary argumet like itegratio by parts, eergy estimate a Growall s iequality tha i [5]. The key iea is to cotrol the lower bou of the seco momet of solutio by the evolutio of 4

6 total eergy E via the total raial compoet of mometum A. The cotrol of seco momet by total eergy eables us ot to rely o the lower bou of etropy or o the time ecay of the total pressure P. The argumet ca easily give aother proof for the compressible Euler equatios a also for the Korteweg type compressible flui of o-isothermal case if the iitial esity is compactly supporte (see [2] for the later). We leave the etails of proof for the later two cases to the reaers. 2 Proof of Theorem 1 Sice we cosier the case of compactly supporte iitial esity, we ca assume that there is a positive costat R so that suppρ B R. We let (ρ, u, e) be a solutio to the Cauchy problem (1), (2) a (3) satisfyig the regularity (7). We eote by X(α, t) the particle trajectory startig at α whe t =, that is, X(α, t) = u(x(α, t), t) a X(α, ) = α. t Sice by Sobolev embeig, u C(R [, T ]) for 1 3, X is uique a ifferetiable. We set Ω() = suppρ a Ω(t) = {x = X(α, t) α Ω()}. From the trasport equatio (1), oe ca easily show that suppρ(x, t) = Ω(t) a hece from the equatio of state (5) that p(x, t) = θ(x, t) = if x Ω(t) c. Therefore, from the equatio (2) a (3), we observe that Lu = a Q( u) = a.e. i Ω(t) c. The followig lemma is a revisit of Z. Xi s i [5]. Lemma 3 We assume µ >, λ + 2 µ > a κ. Suppose that (ρ, u, e) satisfyig the regularity (7), is the solutio of (1), (2) a (3). The u(x, t) i x B c R(t) for some B R(t) cotaiig Ω(t). Moreover, we ca take R(t) = R for all < t < T. Proof. We observe that 5

7 Q( u) = 2µ ( i u i ) 2 + λ(ivu) 2 + µ i u j ) i=1 i j( 2 + 2µ ( i u j )( j u i ). i>j Assume λ. The Q( u) (2µ + λ) ( i u i ) 2 + µ i u j ) i=1 i j( 2 + 2µ ( i u j )( j u i ) i>j = (2µ + λ) ( i u i ) 2 + µ ( i u j + j u i ) 2. i=1 i>j Assume λ >. The Q( u) 2µ ( i u i ) 2 + µ i u j ) i=1 i j( 2 + 2µ ( i u j )( j u i ) i>j = 2µ ( i u i ) 2 + µ ( i u j + j u i ) 2. i=1 i>j Therefore, both of the cases imply that i u i (x, t) = i u j (x, t) + j u i (x, t) = a.e. i Bc R(t) for all i, j = 1,,. This agai implies 2 u(x, t) = a.e. i BR(t) c. Thus u is costat except for a measure zero set. But sice u H 1 a cotiuous i R (this comes from the regularity (7)), we coclue that u i BR(t) c. That is, u(x(α, t), t) = if α BR c. Thus we observe that X(α, t) = α + t u(x(α, s), s)s = α, if α B c R. This implies that we ca choose R(t) = R for t T. From ow o, we assume that Ω(t) = suppρ(, t) is cotaie i a ball B R(t). Multiplyig x 2 to (1) a itegratig it over R, we get the ietity M(t) = 2A(t). (1) t If we take ier prouct by x to (2) a itegrate it over R, the we also get the ietity t A(t) = ρ u 2 x + P (t). (11) R 6

8 Itegratig (1) a (4) over R, we fially get the ietity m(t) =, t E(t) =. (12) t The itegratio by parts applie for erivig the above ietities ca be justifie by the regularity (7). Itegratig (1), (11) a (12) over [, t], respectively, we obtai the followig ietities: t M(t) = M() + 2 A(s) s, (13) A(s) = A() + s R ρ u 2 (x, τ) xτ + s P (τ) τ, (14) m(t) = m(), E(t) = E(). (15) Usig the efiitio of E, we have from (14) a (15) ( s A(s) = A() + 2 E()τ + 2 ) s P (τ) τ. (16) γ 1 Now we first assume ( 2 ). The by (16) we obtai γ 1 Substitutig (17) ito (13), we get A(s) A() + 2E()s. (17) M(t) M() + 2A()t + 2E()t 2. (18) Secoly, we cosier the case γ (1, ). By the equatio of state p = (γ 1)ρe a the ietity (14), we have A(s) = A() + 2E()s (2 (γ 1)) s ρe xτ. (19) It follows from (15) a the efiitio of E that ρe x E(). Substitutig this ito (19), we have a hece from (13) a (2), we have A(s) A() + (γ 1)E()s. (2) M(t) M() + 2A()t + (γ 1)E()t 2. (21) 7

9 O the other ha, sice Ω(t) B R(t), from the mass coservatio (15) we ca estimate the upper bou of the seco momet as follows: M(t) = ρ(x, t) x 2 x = ρ(x, t) x 2 x Ω(t) x R(t) (R(t)) 2 m(t) = (R(t)) 2 m(). (22) Thus from (18) a (22), we coclue that m()r(t) 2 M() + 2A()t + 2E()t 2 for γ 1 + 2, a from (21) a (22) that for 1 < γ < m()r(t) 2 M() + 2A()t + (γ 1)E()t 2 Sice the solutio has strog regularity (7) i the time iterval [, T ], otig from Lemma 3 that R(t) = R for t [, T ], we get the iequality (8). 3 Proof of Theorem 2 Suppose that there is a global solutio (ρ, u, e) satisfyig (9). The there exist costats t > a c < 1 such that for all t t, u(x, t) x 1 + x 2 c L t. (23) Let M(t) = ρ(1 + x 2 ) x. The it follows from (1) a (23) that t M(t) 2 M(t) u(x, t) x 1 + x 2 for all t t. Itegratig this over [t, t], we have M(t) M(t t ) + 2c t By Growall s iequality, we fially have M(t) 2c L t M(s) s. s M(t) M(t ) exp (2c log(t/t )) = M(t ) t 2c t 2c = m() + M(t ) t 2c. (24) t 2c 8

10 From (18), (2) a (24), it follows that M() + 2A()t + (γ 1)E()t 2 m() + M(t ) t 2c t 2c for all t t. Thus the last iequality yiels the cotraictio to the hypothesis c < 1. This completes the proof of theorem. Refereces [1] Y. Cho, H. Kim, Existece results for viscous polytropic fluis with vacuum, to appear i J. Differetial Equatios, Hokkaio Uiversity preprit series i Mathematics 675. [2] R. Dachi, B. Desjaris, Existece of solutios for compressible flui moels of Korteweg type, A. Ist. Heri Poicaré Aal. oliear 18 (21) [3] A. Matsumura, T. Nishia, The iitial value problem for the equatios of motio of viscous a heat-couctive gases, J. Math. Kyoto Uiv. 2 (198) [4] T. C. Sieris, Formatio of sigularity i three imesioal compressible fluis, Comm. Math. Phys. 11 (1985) [5] Z, Xi, Blow up of smooth solutios to the compressible Navier-Stokes equatios with compact esity, Comm. Pure Appl. Math. 51 (1998)