本论文的内容分为两部分. 第一部分主要讨论拟线性抛物方程的Cauchy问题 \begin{equation}\left\{\begin{array}{ll} u_t={\rmdiv}(|\nablau|^{p-2}\nablau)+u^q，&(x，t)\inR^N\times(0，T)，\u(x，0)=u_0(x)\geq0，&x\inR^N，\end{array}\right.\label{1} \end{equation} 解的性质，得到的主要结果如下: 1.对Cauchy问题(0.0.1)证明了解具有局部化性质，即:如果$q\geqp-1$，并且给定的初始函数$u_{...This thesis is divided into two parts. The former part concerns the following quasi-linear equation \begin{equation*}\left\{ \begin{array}{ll} u_t={\rm div} (|\nabla u|^{p-2}\nabla u)+u^q, & (x,t)\in R^N\times (0,T),\u(x,0)=u_0(x)\geq0,& x\in R^N, \end{array}\right. \hspace{3.5cm} (0.0.1) \end{equation*} and we obtained the following results: 1. For the Cauchy problem $(0.0.1)$,...学位：理学博士院系专业：数学科学学院数学与应用数学系_基础数学学号：1902006015315

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ÆS10384 +K5 U> 519020060153152 UDC q    Q p-Laplacian <o=26J0E^vCfWMa Navier-Stokes =2rUH4l^Localization for the p-Laplacian EquationandGlobal Solution to the Compressible N-S Systemin 1D Case          !  	 |   }  Æ 2009  5 ~{Æ 2009  Æ 2009  9+	\  e	2009 [ 6 厦门大学博硕士论文摘要库,%#..*$+0"-(&,B#Rrd!"n%"&M~#E4d!R.)H#^z-d;=|#E4G!)W)pWd(7!R'm#aL7"flWd_X[  g厦门大学博硕士论文摘要库,%#..*$+12')/(&d~bOCT /PoR#2%T aP	3?*0T;^%%<.xBR#&(7$-aAR*LY##kN,' ÆRDi}u1aAR#Zhi/wFIDyaAR#{7#:V#R!CV8s2%Rv1V !   [CV8stau2V `!jJ6Z ” ”.$_X      g℄  [  g"n_X      g℄  [  g厦门大学博硕士论文摘要库    iR PwYnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ijYnd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii9gp KT?[_k7 p-Laplacian>3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1.1 iDxeGB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1.2 h\y`8iO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5§1.3 xeGB8tQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109;p gXNb7 Navier-Stokes>3sVI75m_ . . . . . . . . . . 33§2.1 iDxeGB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33§2.2 xeGBtQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361LYZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46zqm:/SF7Az . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50u ℄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51厦门大学博硕士论文摘要库    iiContentsAbstract (in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract (in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiChapter I p-Laplacian Equation with Strongly NonlinearSource . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§1.2 Some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5§1.3 Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Chapter II The Global Solution of Compressible N-SSystem in 1 D Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33§2.1 Introduction and the main result . . . . . . . . . . . . . . . . . . . . . . . 33§2.2 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46The Author’s Work During Doctor Study. . . . . . . . . . . . . . . . . . . . . . .50Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51厦门大学博硕士论文摘要库[ 2 R C i/ "'9X<bx3(x\}(xLy'>N`Bt9X Cauchy ; {ut = div(|∇u|p−2∇u) + uq, (x, t) ∈ RN × (0, T ),u(x, 0) = u0(x) ≥ 0, x ∈ RN ,(0.0.1)\X`CVTXLy[E1. i Cauchy ; (0.0.1) 84\pm(/`C= q ≥ p − 1, "ObX=x" u0 X9;|_X:+; (0.0.1) X\ u(x, t) #H#sH[0, T ) <}A_Xb9;"Oi9;h_KV>2|}tB>SX[&s#kh{|*X}; ; X;LyWwN`XH#{=w\0o Blow-up JUi}w2IyXk5\ut = ∆um + uq, 4SEXm(/; )}f# [1] \3v; ><T2004 ?) C. Gui and X. Kang[2] >PDX84<1X[|#E8Ah> p-Laplacian o+t9\m(/`CX84E,#i=x u0 [}Wl,XLB<1084\ u #>TB (|x| → ∞) X}AK`2. ' (0.0.1)\X Blow-up; > Blow-up;)XxR 1 < q < p−1s Blow-up ;)|5~HR q = p− 1 s Blow-up ;).f|6_X3. #}b$MLBE'\X Blow-up %9\n(xy#}4{p Navier-Stokes t9 (QK!) X Cauchy ; '5!\XH#`; # Lagrangian ℄E z3vt = ux,ut +R(θv)x = (µuxv)x,Cvθt +R(θv)ux = (κθxv)x + (µuuxv)x,(v, u, θ)(x, 0) = (v0, u0, θ0) → (v±, u±, θ±), x→ ±∞,(0.0.2)O=xX v0, θ0 (Y6XhE_$M0 < m ≤ v0, θ0 ≤M <∞. Cauchy; 5!\XH#`[9L|%RX?wl,#ELb$Mu− ≤ u+, θ− = θ+. #X|-YX_2LB (0.0.2) 5!\H#`X8厦门大学博硕士论文摘要库[ 2 R C ii4,w2;<1B/Ke2LB$M2Æ<1o./= |θ− − θ+| (X~ZKH#sH T (9L<1a#iE[d^b).u~f: p-Laplacian t9MwN`m(/ Blow-up; Navier-Stokes t9厦门大学博硕士论文摘要库J 2 R C iiiAbstractThis thesis is divided into two parts.The former part concerns the following quasi-linear equationut = div(|∇u|p−2∇u) + uq, (x, t) ∈ RN × (0, T ),u(x, 0) = u0(x) ≥ 0, x ∈ RN ,(0.0.1)and we obtained the following results:1. For the Cauchy problem (0.0.1), we proved the Localization for the solutionu. i.e. under the assumption q ≥ p− 1, the solution u(x, t) to the (0.0.1) is strictlylocalized for all exist time t ∈ [0, T ), provided the initial u0 is compactly supported.This problem is very interesting in mathematics and has important implications inphysics. The main difficulties rise from the nonlinear reaction term, which maycause Blow-up. As far as the porous medium type equation, ut = ∆um + uq, isconcerned, such a result is obtained in [1] for the one dimensional case, but left asopen for high diemnsional cases, untill C. Gui and X. Kang presented the detailedproof in [2] in 2004. In light of some ideas in [2], we obtained the similar resultfor the evolution p-Laplacian equation. Additionally, we prove the uniform decayestimate at spacial infinite.2. We classify the Blow-up set: in case of 1 < q < p−1, the Blow-up set shouldbe the whole space, while for q = p− 1, the Blow-up set is positive and bounded.3. Finally, we discuss the Blow-up rate, and obtain some results.The latter one is about the existence of global solution to the compressible fullflow system, which can be governed in the Lagrangian coordinates by the following厦门大学博硕士论文摘要库J 2 R C ivequations (restricted to the ideal gas)vt = ux,ut + Px = (µuxv)x,(e+ 12u2)t + (Pu)x = (κθxv)x + (µuuxv)x,(v, u, θ)(x, 0) = (v0, u0, θ0) → (v±, u±, θ±), x→ ±∞,(0.0.2)and the initial data v0, θ0 are bounded from up and below, i.e.0 < m ≤ v0, θ0 ≤M <∞.The global existance of solution to (0.0.2) is unclear up to the present, unless theassumption u− ≤ u+ and θ− = θ+ are made. Here we want to find that how thedifference |θ− − θ+| relies on the most time T .Keywords: p-Laplacian Equation; Localization; Nonlinear Source; Blow-up; Navier-Stokes Equation.厦门大学博硕士论文摘要库~`;AOS p-Laplacian _G 1k%0 p,i p-Laplace oe§1.1 ) w9#v,Ly'EX Cauchy ; {ut = div(|∇u|p−2∇u) + uq, (x, t) ∈ RN × (0, T ),u(x, 0) = u0(x) ≥ 0, x ∈ RN ,(1.1.1)HE N ≥ 1, p > 2, q > 0, ZKH#sH 0 < T ≤ ∞, =xw" u0(x) |m({9xX; (1.1.1)3	X|w Newton!! (>`!!) #t  S&`X!kaCEPT!Xk5\s{JP43w Newton n!t9R p > 2st9i3)gEs\ u {=>J)/JU (R 1 < p < 2 si3 g). E (1.1.1) }%H#^\<1%2?b#9xEXe\l' 1.1.1. 'Mr Cloc(RN × (0, T ))∩Lp(0, T ;W 1,ploc (RN)) S`bi( u :=u(x, t) =E03+ (1.1.1) Sxh6SlaT$∫ T0∫RN(uφt − |∇u|p−2∇u∇φ+ uqφ)dxdt+∫RNu0(x)φ(x, 0)dx = 0,[*LS φ ∈ C∞0 (RN × [0, T )) XF-+2XwN`S uq, ; (1.1.1) 43SX p-Laplacian o+t9Es)/t9 (1.1.1) X}Iy`C|\XLC#`==x" u0X9;_:+\X9; supp u(·, t) UsH t |P`%KX"Oi_bX t ∈ (0,∞), supp u(·, t) |_XMwN`S uq, %2U\℄SEst9 (1.1.1) {=ooG (Blow-up) JUl' 1.1.2. 4) (1.1.1) Sx℄ Blow-up: hLP!r T < ∞, "R[*LS t ∈ (0, T ), x u(·, t) %LySPlimt→T−max u(·, t) = ∞.厦门大学博硕士论文摘要库~`;AOS p-Laplacian _G 23<1:X[|R 0 < q ≤ 1, (1.1.1) H#5!\R 1 < q ≤ p− 1 + pNs (1.1.1) X\oo Blow-up JUR p− 1 + pN< q s (1.1.1) X\5!H#71oo Blow-up JU2Wr=x" u0 XKV,y#MwN`X; (1.1.1), sk Blow-up \ u XC#m(`'}WHX; ,09K [1], >E2Xbl' 1.1.3. E3+ (1.1.1) [S Blow-up x u ~L r8, hmkS(u) = {x ∈ RN | lim supt→T−u(x, t) > 0}%LyS\~L *8, hmkB(u) = {x ∈ R | ∃ tn → T−, xn → x s.t. u(xn, tn) → ∞(n→ ∞)}%LyST B(u) o_x u S Blow-up mkJ#e	<12},XLy[EH| Cauchy ; (1.1.1) \Xm(/`Cl 1.1.4. r8 q ≥ p − 1, I#i( u0(x) ~LzXmT < ∞ % Cauchy 3+ (1.1.1) S^MLP!rx u PLP!r t ∈ [0, T )~LEZSzXmnLPÆ R∗ > 0, "Rsupp u(x, t) ⊂ B(0, R∗),[ R∗ {FM p, q, N, T j u0.i\X9;h_d^VTEXl 1.1.5. 5x+ty q > 2p − 1, LPAFM K, T SD(C1, C2, "RW 1.1.4 [S R∗ 	℄AT$R∗ ≤ C1K + C2Tm, (1.1.2)T m = q−p+1p(q−1) , D( K > 0 "R supp u0 ⊂ B(0, K), ^MLP!r T <∞.厦门大学博硕士论文摘要库~`;AOS p-Laplacian _G 3l 1.1.6. $6}ty q > p−1, I#i( u(x, 0) = u0(|x|) %}=AQS?P |x| → ∞ !	℄ u0(|x|) < Cs|x|− pq−p+1 , LPEHaMS R∗1 > 0,Q |x| ≥ 2R∗1 !6AT$u(x, t) ≤ Cs|x|−pq−p+1 , (1.1.3)[*LS t ∈ [0, T ) EZFWE' Blow-up ;)l 1.1.7. Æ{ Blow-up.  1 < q < p− 1,  Cauchy 3+ (1.1.1) S*L`^x u X℄ Blow-up 8>? Blow-up mk%UrnB(u) = RN .l 1.1.8. *+ Blow-up.  q = p− 1, h~LzXmSI# u0(x) S&QM Cauchy 3+ (1.1.1) Sx u ℄ Blow-up, ? Blow-up mkCZ%EVLySn0 < measB(u) <∞.Z+<1[\X Blow-up %; l 1.1.9.  q > p− 1, T < ∞ %3+ (1.1.1) x u S Blow-up !rpu0(x) = u0(|x|) }=OVAQ[*LS t ∈ [0, T ) jH?S δ > 0, 	℄‖u(·, t)‖∞ ≤ C(δ)(T − t)−1q−1 , x ∈ RN \B(0, δ), (1.1.4)TD( C(δ) %FM δ SD( B(x0, ρ) = {x ∈ RN ; |x− x0| < ρ}.; 1.1.9. h p− 1 < q < p− 1 + pN, [*LS t ∈ (0, T ),‖u(·, t)‖∞ ≤ C(t−1q−1 + (T − t)− 1q−1 ).<5SW GB [3],[4] j [5].厦门大学博硕士论文摘要库~`;AOS p-Laplacian _G 4l 1.1.10.  q > p− 1, T <∞ %3+ (1.1.1) x u S Blow-up !rLPÆD( C > 0, "R[*LS 0 < t < T, x0 ∈ RN , L∫B(x0,1)u(x, t)dx ≤ C((T − t)− 1q−1 ; (1.1.5)∫ Tt∫B(x0,1)uqdxdt ≤ C(T − t)− 1q−1 . (1.1.6)2},|I℄sk)/t9 (1.1.1)\Xm(/`CHaj2; Xskh'v<1>>}IyX)/gt9ut = △um + uq (1.1.7)#2}; skhXd+ (&xFb u0 9;_): i}4XS℄ V.A.GalaktionovY^# 1995? (L [1])R_2Tb(intersectoin comparisontechnique) it9 (1.1.7) 84\Xrm(/`C"O	4XS# [1] E\3v; ># 2004 ?3z'xV# [2] Ei4S℄>62XPD4I+ [2]EX(xQ<1#,84 (1.1.1)X\&xpm(`C=Rq ≥ p− 1 s (1.1.1) X=x= u0 X9;_:+ Blow-up \ u #H#sHt ∈ (0, T )<XC#|}A_X"Oi9;XC#V\>R 1 < q < p−1s<184\X Blow-up ;)|Y~H RN . E,0y# Blow-up \X}AK \X Blow-up ;)< Blow-up %Y; "VT}W[<1#84 9ETXLy;|>p%> (1.1.1)pb9;X\ikaCt9XS℄2H\#84E|w2IyX (L [2]), 3E<1 De-Giorgi aN?NGmv>p%\MX;84 9E05TX};|4"X>pwÆt9 (1.1.1) X℄Æ Laplace VVTX4"#2Æ%=o3_(\3|}2W;<1%V%it9\X`C[DAXxA厦门大学博硕士论文摘要库~`;AOS p-Laplacian _G 5§1.2 &q:j)VZ>+284yX}W?N`1'M	Hit9 (1.1.1) X\ u [EXWO1v(ξ, τ) = (T − t) 1q−1u(x, t), (1.2.1)HEξ =x(T − t)m ∈ RN, τ = − ln T − tT∈ [0,∞), m = q − p+ 1p(q − 1) > 0, q > p− 1.I\[ξ = T−meτmx and T − t = Te−τ .JP:" v = v(ξ, τ) (YE2Xvτ = div(|∇v|p−2∇v) −mξ · ∇v − 1q−1v + vq, (ξ, τ) ∈ RN × (0,∞),v(ξ, 0) = T1q−1u0(Tmξ), ξ ∈ RN .(1.2.2)E,NT v(ξ, τ)  τ >= v(ξ, τ) = θ(ξ), :+ θ(ξ) (YOX*t9div(|∇θ(ξ)|p−2∇θ(ξ)) −mξ · ∇θ(ξ) − 1q − 1θ(ξ) + θq(ξ) = 0. (1.2.3)i}4S℄EXt9 (1.2.3), <1E2X( 1.2.1. (s29 [6])  q > p − 1, r1( N = 1, LPEVi(θs = θs(ξ) "R(|θ′s(ξ)|p−2θ′s(ξ))′ −mξθ′s(ξ) − 1q−1θs(ξ) + θqs(ξ) = 0,θs(0) = θ0 > θH :=(1q−1)1q−1, θ′s(0) = 0, limξ→∞θs(ξ) = 0.K.L Cs = C(p, q) > 0, "Rθs(ξ) = Cs|ξ|−pq−p+1 (1 + ρ(ξ)),T ρ(ξ) → 0 (|ξ| → ∞).厦门大学博硕士论文摘要库~`;AOS p-Laplacian _G 6#=x" u0 b9;XSE)/t9 (1.1.1) jT%(X`C( 1.2.2. hLPÆ K > 0, "R supp u0 ⊂ B(0, K), [*LSr > 0, Lsupx∈∂B(0, r+2K)u(x, t) ≤ infx∈B(0, r)u(x, t), ∀ t ∈ [0, T ), (1.2.4)wk$ (1.2.1), $TqMsupξ∈∂B(0, (r+2K)T−meτm)v(ξ, τ) ≤ infξ∈B(0, rT−meτm)v(ξ, τ), ∀ τ ≥ 0. (1.2.5)( 1.2.2 h3	"W [7].%uFb suppu0(x) ⊂ B(0, K) ⊂ {x ∈ RN | xn > 0}.  ũ(x′, xn; t) =u(x′,−xn; t), :+ũ(x′, 0; t) = u(x′, 0; t) t ∈ (0, T ), x′ ∈ RN−1,2Æ u  z (1.1.1) X\i xn > 0 muũ(x′, xn; 0) = u(x′,−xn; 0) ≡ 0 ≤ u(x′, xn; 0).NT" ũ {| (1.1.1)X\#V RN−1 ×R+ × [0, T ) ETVTu(x′,−xn; t) = ũ(x′, xn; t) ≤ u(x′, xn; t).E2ibX x0 ∈ B(0, r) ' x1 ∈ ∂B(0, r + 2K), bΠ = {x ∈ RN ; 〈x− 12(x0 + x1), x0 − x1〉 = 0},2Æ 〈, 〉  z RN EX<9:+ dist{Π, x0} = dist{x1, Π}. 2|3dist{Π, {0}} = 12x21 − x20| x1 − x0 |≥ (r + 2K)2 − r2| 4(r +K) | ≥ K,℄ x0 ' suppu0 d#F2 Π X&}-<℄ x0 ' x1 | Π Xi4X%;VT u(x0) ≥ u(x1).1~H℄tT\+I h2X84={VT (1.2.4) . 2厦门大学博硕士论文摘要库~`;AOS p-Laplacian _G 7( 1.2.3. LPVD( L <∞ "R6S3+(|ϑ′|p−2ϑ′)′ − 1p−2ϑ+ ϑp−1 = 0, |x1| ≤ L,ϑ(0) = ( p2(p−2))1p−2 , ϑ′(0) = 0,ϑ(±L) = 0, ϑ′(±L) = 0,(1.2.6)LP/ES[Ex ϑ := ϑ(x1), x1 ∈ R.( 1.2.3 h3	<1?#VH [0, L] h%6\ ϑ := ϑ(s), (Yϑ(s) ≥ 0; ϑ′(L) = 0, ϑ′(s) ≤ 0 if, s ∈ (0, L). (1.2.7)#t9 (1.2.6)1 g7h" ϑ′ , \+9xVp− 1p[(−ϑ′)p](x) +H(ϑ(x)) = H(ϑ(0)), (1.2.8)HE x ∈ {x| ϑ(x) ≥ 0, ϑ′(x) ≤ 0}, H(s) = − 12(p−2)s2 + 1psp.Æ ϑ(0) = ( p2(p−2))1p−2 , t9 (1.2.8) YGp− 1p[(−ϑ′)p](x) +H(ϑ(x)) = H(ϑ(0)) = 0, (1.2.9)=(−ϑ′)(x) =(− pp− 1H(ϑ(x)))1p, (1.2.10)ÆE{Vx = −∫ x0(− pp− 1H(ϑ(y)))− 1pdϑ(y) =∫ ϑ(0)ϑ(x)(− pp− 1H(s))− 1pds. (1.2.11)}bH# C1(0, L) X" ϑ(x) '2 L, wV ϑ(L) = 0. E,# (1.2.10) E x = L , <1{wT ϑ′(L) = 0. I\[ ϑ(x) {}b(Y (1.2.6) ' (1.2.7) EXyU 2厦门大学博硕士论文摘要库Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”. Fulltexts are available in the following ways: 1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submitrequests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to etd@xmu.edu.cn for delivery details.厦门大学博硕士论文摘要库

Localization for the p-Laplacian Equation and Global Solution to the Compressible N-S System in 1D Case

http://dspace.xmu.edu.cn/bitstream/handle/2288/47662/p-Laplacian%e5%8f%91%e5%b1%95%e6%96%b9%e7%a8%8b%e7%9a%84%e5%b1%80%e9%83%a8%e5%8c%96%e6%80%a7%e8%b4%a8%20%e5%92%8c%e4%b8%80%e7%bb%b4%e5%8f%af%e5%8e%8bNavier-Stokes%e6%96%b9%e7%a8%8b%e6%95%b4%e4%bd%93%e8%a7%a3%e5%ad%98%e5%9c%a8%e6%80%a7.pdf?sequence=1&isAllowed=y

Localization for the p-Laplacian Equation and Global Solution to the Compressible N-S System in 1D Case

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