8 9 Ö 3 3 Sept. 8 Communication on Applied Mathematics and Computation Vol.3 No.3 DOI.3969/j.issn.6-633.8.3.7 Õ Îµ Ï̺ Eule»²Ö µ ÝÙÚ ÛÞ ØßÜ ( Ñ É ÉÕ Ñ 444 Î ÇÄ Eule ± Æà ¼ Û Â Þ Û ¾ ³ ÇÄ Eule ± Å Å Þ Å Å ¼ 35A; 35B44; 35B65 ÒǼ O75.7 ÈÊ Ð A È 6-633(83-68- Blowup of egula solutions fo adial elativistic Eule equations with damping LIU Jianli, LUAN Liping, FANG Yaoli (College of Sciences, Shanghai Univesity, Shanghai 444, China Abstact In this pape, we mainly conside the blowup of the egula solutions of the adial elativistic Eule equations with damping. Unde the appopiate assumptions on the initial data, we obtain the singulaity fomation fo the egula solutions to the Cauchy poblem of the adial elativistic Eule equations with damping. Key wods adial elativistic Eule equations; damping; egula solutions; blowup Mathematics Subject Classification 35A; 35B44; 35B65 Chinese Libay Classification O75.7 Ó Ú¾ È Ò È ĐÔ È Ó À Ù Ã ÄÝÐ À ³ Ä ¾ À Ä À È Eule Ï Ù Ì Ã Á Æà Æà Eule Ï Û ÚÉÆ ÆÃÀ È Í Õ ¾ ß Smolle Õ Temple [] Í Æà Eule Ï Ì Ð Þ Chen Õ Zhou [] Í Ê Á Æà Eule Ð Ð Ä Đ Riemann ¹ [3] Í Ê Ä½ÂÀ 7-4-4; ¹ÂÀ 7-5-9 Í Ø²ÊÞ Ï (4367; «Ô «Þ Ï (338 Æ Ï ² E-mail: jlliu@shu.edu.cn
3 ÇÄ Eule ± Þ 69 Đß Æà Eule Ï ÚÆ Úµ Ñ Å ¹ [4-6] Í Æà Eule ½ÚÅÕ½Æ ºÁÅ Æà Eule ØÅ ÆÃÀ Å» È Pan Õ Smolle [7] Æß Æà µ» Ä Í Í µ ¾Õµ Ñ Å Guo Õ Tahvilda-Zadeh [8] Í Ú¾ Úµ ÓÆ Æà Eule Ú Geng [9] Í ³ Æà Eule Ú «Æ ½ÚÅÜ»Ó ¹ Æ Æà Eule ÄÍ Ú Á µ Ï Ý Ú ½ÉÍ Û Æà Eule ( n ( n t + v /c v /c v + nv v /c =, ( p/c +ρ ( p/c t v /c v +ρ + v /c v + p+ p/c +ρ v /c v = (p v /c 4 (p/c +ρ ( (v /c v, dn nc = dρ p + ρc, t = : ρ(, = ρ (, v(, = v (, n, ρ, p Å ÊÄ «ÊÄÕË v = (v, v, v 3 Å ÀÀ Ä c µ p Å γ Á p = ρ γ, γ >, γ µ Ù ½É³ Ë Ã ( µ [, R], ÐÅ H = { A= max A =R 3 +, A = R + R 5γ+3 8(γ v d > A, (, A 3 = R +, A 4 = R + R 45 8 { 8 8(γ < R < min, 45 5γ + 3, 3 A, A (5γ + 3 + 45A 3 + 45 A 3 + 56A 3 6, A 4 + A 4 + 6A 4(γ 4(γ ÝÆ ¾ v ÐÅ v c R (γ }, A (5γ + 3 + 56A (γ, 6(γ } Ý (ρ, v ÅÜÚ [, T Ô Á ³ ŵ { A R 3 T =max H (A R 3, 8A R 5 (γ H [8R (γ (A R 3 A R 3 (5γ + 3], 8A 3 R 5 H [8R (A 3 R 3 45A 3 R 3 ], A 4 R 5 (γ H [R (γ (A 4 R 3 A 4 R 3 ] }.
6 3 Ô µã¹² Å ½É Ë Đ Eule Ï Í ¹ [-] Í Eule Ï Úµ ÊÄ ÑÂ Í ÎÀÑ ÜÚ ¹ [3-4] ÐÜÎÒ ÏÌ Í Á Ú Ë Ã Å«µ Ã Í Ú Æ½ÉÅ ( Ý Á Đ Å Û (ω, v = (p γ γ, v = (ρ γ, v C, p (ρ < c, Ý (ρ, v Ý Ô 3 Û (ρ, v Ý Ý (ρ, v Í (ρ, v ( Cauchy» Ý ÝÚ [, T (ρ, v ܹ t = (t;» Ë «Á À ³ Ûµ d dt (t; = v(t, (t;, (; =. B = { R} B(t = {(t; B }. ( Ï d ( dt n + v /c n ( v v /c + =, (3 d dt Å «d dt = t + v. n, Å (3 Đ Äß n v /c = n v /c exp ( t ( v + (t, (t; dt. (4 ρ Õ n Å Û ρ = n ( + e c, (5 e ¹ B(t ρ(t,, Ý n(t,. ËÅ ÍÚÀ Ë B(t Ð À Ä v(t, (t; =. µï Í Á q = p/c + ρ, Ý ( ß Ï ³ µ dρ dn = q n. (6
3 ÇÄ Eule ± Þ 6 ( Ï ³Ûµ qv ρ t + vρ = c ( v /c v q t v /c v qv. (7 Å ( à ØÏ ³ Áµ ( p v /c 4 q ( v /c v + ( p v /c 4 q ( v /c v t + ( p /c qv ( v /c v qv p /c ( v /c + p ρ =, (8 Ý v + v t + p /c p v /c 4vv + ( v /c p q( p v /c 4 ρ = ( v /c v p /c ( p v /c 4. (9 Ú B(t Ð ρ(t, (t;, Ý p = γρ γ =. ÖÉ Û ½ÉÅ Á Ú (9 Å ÅÚ B ÐÔ ¾Ò µ µ ÁÁ Á Ý ω = p γ γ = ρ γ, γ >, ( Ý ρ = ω =. Ý (9 ³Ûµ p q ρ = Å ( Ú B(t Ð Đ ß γργ ρ γ /c + ρ ρ = γ /(γ ω ω γ /c + ω =, v + v t + vv =, B(t, ( v(t, = v(, e t. ( (t = + v(, t =, B, ³ (t, B(t, (t R, Ý > R ρ(t, =, v(t, =, ¹ ρ(t, R =, v(t, R =, (3
6 3 Ú Æ ½É ÐË Á Ñ ¹ Ã Í (4 Õ (5 ³ n(t,, Ý ρ(t,, ¹ (9 ³ Áµ v + v t + p /c p v /c 4 vv + ( v /c p q( p v /c 4 ρ, (4 v + v t + vv p /c ( v /c p v /c 4 vv + ( v /c p q( p v /c 4 ρ. (5 Å (5 Æ Äß ³ + vd+ v t d+ dv + 4 p ( p v /c 4 d v c (v /c p q(p v /c 4 dρ. (6 ˽ÉÆ (6 Õ ß ÌÕ (3, p ( 4 p v /c 4 d v c = [ p ( v /c ] R 4 p v /c 4 p ( v /c 4 p v /c 4 d 4 = p ( v /c 4 p v /c 4 d ( v 4 p = ρ γ, Ûµ ( v /c γρ γ γc (ρ γ /c + ρ( p v /c 4 dρ = γ Ï (7 Ï (8 4 = γc γ γc γ d ( v p c p v /c 4 c d p p v /c 4. (7 p ( v /c p v d, /c4 ( v /c p v /c 4 d ln( + ργ /c ( v /c ln( + ρ γ /c p v /c 4 d γc ( v /c ln( + ρ γ /c γ p v /c 4 d. ln( + ρ γ /c d ( v /c p v /c 4. (8
3 ÇÄ Eule ± Þ 63 (6 ³ vd + γc γ v t d + dv 4 d ( v p c p v /c 4 ln( + ρ γ /c d ( v /c p v. (9 /c4 ˽ÉÅ Ñ Ã ( Cauchy» Ý Ñ Á Ý (9 4 d ( v p c p v /c 4, γc γ ln( + ρ γ /c d ( v /c p v, /c4 Ý (9 ³ Û vd + v t d 4 v d. ( Á H = H(t = Cauchy-Schwatz ³ vd = ( vd R v d ( vd. ( d, ( Ö (, Á ( ³ vd + d dt Ñ Æ Ë ³ Ï 4H R H + dt vd 4 v d. (3 ẋ = a x + a x, v d H R 3. (4 H R 3. (5 a >, a Û ÊºÏ a x + a x = ÇÚ ẋ <, Ñ ẋ >. (5 dt H H. (6 R3
64 3 ½É³ ÊºÏ H R 3 H = µ H =, H = R 3, R 3 > µµ H = v d > R 3, ½É R < 3 A ³ ¹ H(t ÅÚ T = Á Ý (9 4 γc γ = γ γ dt H(t dt >, H > A = R 3 +. (6, H R 3 H A A R 3 A R 3 H, (7 A R 3 H A R 3 H (A R 3 t, (8 A R 3 Ý H (A R 3 d ( v p c p v /c 4 = 4 ln( + ρ γ /c d ( v /c p v /c 4 ln( + ρ γ /c ( v (p + (p /c 4 (v c ( p v /c 4 d, ( v /c v /c (p + ( + p v /c 4 + p /c ( v /c (v ( p v /c 4 d, Ý (9 ³Û vd + v t d + dv ( v /c ( p v /c 4 [ 4 + γ γ ln( + ργ /c v c ] (p d ( v /c [ (p ( ( p v /c 4 c 4 v c γ γ ln( + ργ /c ( p v ] c 4 p c (v d. (9 µ (9 Õ Ð Û Ï [5] ln( + ρ γ /c ρ γ /c = p γc, (3 f(y = y p = kc 5 f(y, q 4 + q dq, ρ = kc 3 g(y, y g(y = 3 q + q dq,
3 ÇÄ Eule ± Þ 65 k Ý ± Ý p = c y 3( + y, p = 9kcy ( + y 5 >. p c y lim ρ p = lim y 3( + y = c 3, (3 p c 3, ( p v /c 4 ( /3 = 9 4. (3 Ö (3 (3, (9 Û ( v /c [ ( p v /c 4 4 + γ ] γ ln( + ργ /c v c (p d 9 4 = ( v /c [ ( p v /c 4 4 + γ ] γ ln( + ργ /c v c (p d (p d v 3(3γ + 6(γ ( c 4 + 3(γ v (p 3(3γ + c d 6(γ v d. (33 Á v c ( v /c v /c, ÏË (ρ, v ( Cauchy» Ý ¹ (p (p = γωω c. ¼ (9 Û (v /c [ (p ( (p v /c 4 c 4 v c 9 4 5 γ 4(γ v c 9 γ γ 3γ ½É (v c ( 3 vv d γ γ ln( + ργ /c ( p v c 4 p c ] (v d v d, (34 vd + = v v c. Å (33 Õ (34 Ü (9, v t d 4 v d 5γ + 3 + 3(γ v d, (35 (3 Õ (36, ³ vd + vd + d dt v t d 4 ( R ( vd R 5γ + 3 8(γ v d. (36 5γ + 3 H 8(γ R, (37
66 3 Ý H = v R d > R 5γ+3 8(γ ( dt 5γ + 3 H H. (38 R 8(γ R, Å«³ dt >, Ñ (7 ¼ ³ ( dt 5γ + 3 H R 8(γ R H, (39 A R < A(5γ+3+ A (5γ+3 +56A (γ 6(γ ³ H(t ¹ ÅÚ T = 4 Á 3 Ý (9 Ý (9 ³Û 8H A R 5 (γ 8A R 5 (γ H [8R (γ (A R 3 A R 3 (5γ + 3]t, (4 d ( v p c p v /c 4, γc γ vd + (3 (3 8A R 5 (γ Ý H [8R (γ(a R 3 A R 3 (5γ+3] v t d + vd + dv 4 v t d 4 ( R 45 8 ln( + ρ γ /c d ( v /c p v, /c4 d ( v p c p v. (4 /c4 v d. (4 Á ( ³ ( dt R 45 H H. (43 8 R Ý H = v d > R, Å«, ³ R 45 dt >. Ñ (7 Õ (39 ¼ ³ 8 ( dt R 45 H 8 R H, (44 A 3 R < 45A3+ 45 A 3 +56A3 6 ³ ¹ ÅÚ T = 4 H(t Á 4 Ý (9 8H A 3 R 5 8A 3 R 5 H [8R (A 3 R 3 45A 3 R 3 ]t, (45 8A 3R 5 Ý H [8R (A 3R 3 45A 3R 3 ] d ( v p c p v /c 4, γc γ ln( + ρ γ /c d ( v /c p v, /c4
3 ÇÄ Eule ± Þ 67 Ý (9 ³Ûµ vd + v t d + (3 (3 Á ( ³ ݵ H = vd + v R d > dv γc γ v t d 4 ln( + ρ γ /c d ( v /c p v. (46 /c4 ( R (γ ( dt R H (γ R H. R (γ, Å«, ³ dt >, Ý ( dt R H (γ R H, A 4 v d. (47 R < A4+ A 4 +6A4(γ 4(γ ³ H(t ¹ ÅÚ T = Ð Ú H A 4 R 5 (γ A 4 R 5 (γ H [R (γ (A 4 R 3 A 4 R 3 ]t, (48 A 4R 5 (γ Ý H [R (γ(a 4R 3 A 4R 3 ] { 8 8(γ < R < min, 45 5γ + 3, 3 A, A (5γ + 3 + 45A 3 + 45 A 3 + 56A 3 6, A 4 + A 4 + 6A 4(γ 4(γ A (5γ + 3 + 56A (γ, 6(γ }, Ð v c ( Cauchy» Ý Ú { A R 3 T = max H (A R 3, 8A R 5 (γ H [8R (γ (A R 3 A R 3 (5γ + 3], Ý ± ÉË 8A 3 R 5 H [8R (A 3 R 3 45A 3 R 3 ], A 4 R 5 (γ H [R (γ (A 4 R 3 A 4 R 3 ] [] Smolle J, Temple B. Global solutions of the elativistic Eule equations [J]. Communications in Mathematical Physics, 993, 56(: 67-99. [] Chen Y, Zhou Y. Simple waves of the two dimensional compessible full Eule equations [J]. Acta Math Sci Se B Engl Ed, 5, 35(4: 855-875. }
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