328 .. W period HORN comma J period SOKO L-suppress OWSKI nnoindentInitial conditions328 nquad on u areW given . HORN as follows , J . : SOKO nL OWSKI InitialEquation: conditions open parenthesis on 1 $ 3 u closing $ are parenthesis given .. as u openfollows parenthesis : x comma y comma 0 closing parenthesis = u sub 0 open parenthesis x comma y closing parenthesis on Capital Omega comma Equation: open parenthesis 1 4 closing parenthesis .. u sub t open parenthesis x comma y comman begin 0f closinga l i g n ∗g parenthesis = u sub 1 open parenthesis x comma y closing parenthesis on Capital Omega period uWe ( will assume x , that y the initial , 0 conditions ) = for v u andf theta0 g are( compatible x , with y those ) on nOmega , n tag ∗f$ ( 1 3 ) $gnn u f t g ( x , y , 0 ) = u f 1 g ( x , y ) on nOmega . n tag ∗f$ ( 1 4 ) $g for328 u commaW . HORN i period , J . SOKO e periodLOWSKI Initial conditions on u are given as follows : nendEquation:f a l i g n ∗g open parenthesis 1 5 closing parenthesis .. v open parenthesis x comma 0 closing parenthesis = v sub 0 open parenthesis x closing parenthesis = u sub 0 open parenthesis x comma y closing parenthesis bar sub Q comma Equation: open parenthesis 1 6 closing parenthesis .. v sub nnoindent We will assume that the initial conditions for $ v $ and $ ntheta $ are compatible with those t open parenthesis x comma 0 closing parenthesisu(x; y; =0) v sub = 1u open0(x; y parenthesis) on Ω; x closing parenthesis = u sub 1 open(13) parenthesis x comma y closing for $u ,$ i.e. parenthesis bar sub Q comma Equation: openut parenthesis(x; y; 0) = 1 7u closing1(x; y) parenthesis on Ω: .. theta open parenthesis x comma(14) 0 closing parenthesis = theta sub 0 period n beginWeWe conclude willf a l assumei g n ∗g this that section the by initial formally conditions showing for thatv and theθ are energy compatible of this system with those for u; i . e . vdecreases ( x period , .. Formally 0 ) comma = v onef can0 computeg ( the x energy ) of = this u structuref 0 g as follows( x period , y ) nmid f Q g , n tag ∗f$ ( 1After 5 multiplying ) $gnn v openf parenthesist g ( 4 x closing , parenthesis 0 ) by = u sub v tf and1 integratingg ( x the result ) = over Capital u f 1 Omegag ( one x receives , : y ) nmid f Q g , n tag ∗f$ ( 1 6 ) $gnn ntheta ( x , 0 ) = ntheta f 0 g . n tag ∗f$ ( 1 7 ) $g Equation: open parenthesis 1 8 closing parenthesisv(x; 0) = .. 2 tov0( thex) = poweru0(x; of y) 1jQ dt; to the power of d integral sub Capital(15) Omega open parenthesis bar u nendf a l i g n ∗g sub t bar to the power of 2 plus bar nabla u barvt(x; to0) the = powerv1( ofx) 2 = closingu1(x; y parenthesis) jQ; dxdy = integral sub Q Row 1(16) partialdiff u Row 2 partialdiff nu . u sub t dx period θ(x; 0) = θ0: (17) WeMultiplying conclude open this parenthesis section 7 by closing formally parenthesis showing by v sub that t and the integrating energy over of Q this and adding system the result to the integral d e c r e a s e s . nquad Formally , one can compute the energy of this structure as follows . of openWe concludeparenthesis this 8 closing section parenthesis by formally over showing Q yields that the energy of this system decreases . Formally , one can 2compute to the power the energy of 1 dt of to this the structure power of as d follows integral . sub Q open parenthesis bar v sub t bar to the power of 2 plus R bar v sub xx bar to the power nnoindent After multiplying ( 4 ) by $ u f t g$ and integrating the result over $ nOmega $ one receives : of 2After plus psi multiplying theta sub ( 1 4 bar ) by vu subt and x barintegrating to the power the result of 2 over minus Ω betaone receives bar v sub : x bar to the power of 4 plus alpha bar v sub x bar to the power of 6 plus bar theta bar closing parenthesis dx Equation: open parenthesis 1 9 closing parenthesis .. = integral sub Q fv t dx plus k integral sub Q open parenthesisn begin f a l theta i g n ∗g sub 0 minus theta closing parenthesis dx period 2 ^f 1 g dt ^f d g n int f nOmegaZ g ( nmid uZ f t g nmid ^f 2 g + nmid nnabla u nmid ^f 2 g This computation can be found in1 Sprekelsd and2 Zheng open2 parenthesis@u 1 989 closing parenthesis period Next we add open parenthesis 1 8 closing ) dxdy = n int f Q g2 n ldt e f t [ n(jbeginut j +fjarray ru j )gfdxdycg= n partial utdx:u nn n partial nnu nend(18)f array gn right ] u f t g parenthesis Ω Q @ν dxand . openn tag parenthesis∗f$ ( 1 1 9 closing 8 )parenthesis $g and see that the terms integral sub Q bracketleftbig sub partialdiff nu to the power of partialdiff u bracketrightbignendMultiplyingf a l i g n ∗g u ( sub7 ) by t dxvt commaand integrating and integral over subQ and Q fv adding t dx cancel the result period to Wethe getintegral of ( 8 ) over Q yields 2 to the power of 1 dt to the power of d integral sub Capital Omega open parenthesis bar u sub t bar to the power of 2 plus bar nabla u bar to the powernnoindent of 2 closingMultiplying parenthesis ( dxdyZ 7 ) Equation: by $ v openf parenthesist g$ and 20 integrating closing parenthesis over .. plus $ Q2 to $ the and power adding of 1 dt the to the result power of to d integral the integral sub Q of ( 8 ) over $Q$1 d yields 2 2 2 4 6 open parenthesis bar v sub2 tdt bar to(j thevt j power+R j ofvxx 2j plus+ Rθ1 barj vx vj sub−β xxj vx barj + toα thej vx powerj + j ofθ j) 2dx plus psi theta sub 1 bar v sub x bar to the power of 2 minus beta bar v sub x bar to theQ power of 4 plus alpha bar v sub x bar to the power of 6 plus bar theta bar closing parenthesis dx = k integral sub n begin f a l i g n ∗g Z Z Q open parenthesis theta sub 0 minus theta closing parenthesis dx= periodfvtdx + k (θ0 − θ)dx: (19) 2The ^f left1 g handdt side ^f ofd thisg equation n int f isQ theg time( derivativenmid ofv t f tQ g nmid ^Qf 2 g + R nmid v f xx g nmid ^f 2 g + n psi ntheta f 1 g nmid v f x g nmid ^f 2 g − nbeta nmid v f x g nmid ^f 4 g + nalpha nmid v Thisf x computationg nmid ^f can6 beg found+ innmid Sprekelsn andtheta Zhengn (mid 1 989 )) . Next dx wenn add= ( 1 8n int ) and (f 1Q 9 )g andfv see that t the dx + k n int f Q g R @u R ( termsnthetaQ[@ν ]futdx;0 gand −Q fvtdx nthetacancel .) We get dx . n tag ∗f$ ( 1 9 ) $g nendf a l i g n ∗g Z 1 d 2 2 nnoindent This computation can be found in Sprekels2 dt and(j Zhengut j + j ( r 1u j 989)dxdy ) . Next we add ( 1 8 ) Ω and ( 1 9 ) and see thatZ the terms $ n int f Q g [ ^f n partial u g f n partial nnu g ] u f t g dx 1 d 2 2 2 4 6 , $ and $ n int f+2Q dtg fv(j vt j t+R dx$j vxx j + cancel θ1 j vx .Wegetj −β j vx j +α j vx j + j θ j)dx (20) Q n begin f a l i g n ∗g Z = k (θ0 − θ)dx: 2 ^f 1 g dt ^f d g n int f nOmega g ( nmid u f t g nmidQ ^f 2 g + nmid nnabla u nmid ^f 2 g ) dxdy nn + 2 ^f 1 g dt ^f d g n int f Q g ( nmid v f t g nmid ^f 2 g + R nmid v f xx g nmidThe^f left2 handg + side ofn psi this equationntheta is thef time1 g derivative nmid ofv t f x g nmid ^f 2 g − nbeta nmid v f x g nmid ^f 4 g + nalpha nmid v f x g nmid ^f 6 g + nmid ntheta nmid ) dx n tag ∗f$ ( 20 ) $gnn = k n int f Q g ( ntheta f 0 g − ntheta ) dx . nendf a l i g n ∗g nnoindent The left hand side of this equation is the time derivative of t 330 .. W period HORN comma J period SOKO L-suppress OWSKI nnoindentSimilarly we330 integratenquad byW parts . HORN over Capital, J . SOKO Omegan toL theOWSKI power of minus to get : SimilarlyLine 1 minus we integral integrate sub 0 to by the partspower of over T integral $ n subOmega Capital^f Omega − g$ minus to get Capital : Delta u phi dxdt = integral sub 0 to the power of T integral sub Capital Omega minus nabla u nabla phi dxdt minus integral sub 0 to the power of T integral sub partialdiff Capital Omega minus partialdiff to then [ n powerbegin off a partialdiff l i g n e d g sub − n ton int the power^f T ofg uf phi0 g d sigma n int dt Linef nOmega 2 = minus− integral g nDelta sub 0 to theu powernphi of T integraldxdt sub = Capitaln int Omega^f T minusg f 0 u g Capitaln int Deltaf nOmega phi dxdt− plus g integral nnabla sub 0 tou then powernabla of Tn integralphi subdxdt partialdiff− Capital n int Omega^f T g minusf 0 ug sub n partialdiffint f n npartial to the powern ofOmega partialdiff− g n partial ^f n partial g^f u g f n g nphi d nsigma −dt nn phi d330 sigmaW dt .