Neustupa, J.

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Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7Jiří NeustupaThe global solvability to the equations of motion of viscous gas with anartificial viscosityIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 289--292.Persistent URL: http://dml.cz/dmlcz/702367Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czTHE GLOBAL SOLVABILITY TO THE EQUATIONS OF MOTION OF VISCOUS GAS WITH AN ARTIFICIAL VISCOSITY NEUSTUPA J., PRAGUE, Czechoslovakia It i s known t h a t weak so lu t ions to the Nav ier -S tokes equat ions for in compr e s s ib l e l iqu id e x i s t g loba l ly in t ime (due to the work of E.Hopf from the year 1951). In sp i t e of a lo t of a t tempts , i t rema ins an open quest ion w h e a t h e r t h e same may be said about the Nav ier -Stokes equa-t ions for compres s ib le f lu id . Especia l ly the var iab i l i ty of dens i ty i s a source of t roub les : e v e n t u a l "very sma l l" as we l l as "very large" va l ues of d e n s i t y toge ther w i t h insu f f i c i ent informat ions about i t s regu lar i ty make impo s s ib le to follow the approach of E.Hopf. Fai lures lead often as far as to doubts about the s y s t e m of the Nav ier -S tokes equat ions as about a convenient model of mo t ion of a v iscous compress ible f lu id . . We s h a l l dea l w i t h the following sy s t em of equat ions: ( 1 ) ( QUt)t + iQUj ut\j = - piQ)j + V-C uUJ + ^ u ; J / ] + C fiQ\j Ujlj-(/ = 1 , 2 , 3 ) ( 2 ) Q t + ( QUJ\J = f(Q\jj in QT = ft x ( 0 , D . We consider boundary cond i t ions ( 3 ) u,\ s o (/ = 1 , 2 , 3 ) , - | ~ fiQ)\ s o Uax(o.r) 3v bflxto.n (where v is an ou ter norma l vector) and i n i t i a l cond i t ions (4 ) e | ^ = e 0 . < e " , > | f . 0 = e 0 " 0 , ( ' = 1 . 2 , 3 ) . The t erm on the r i g h t hand s ide of (2) represents a so ca l led artificial viscosity. We as sume t h a t there e x i s t 8 > 0, Q1 > 0, Q2> Qv Q3> Q2* c\ > 0, .. . . , C6 >0 so t h a t p and f are twice continuous ly d i f ferent iab le and non-decreas ing funct ions on <0, +») such that piQ) * C1 Q for Q c <0, Q1 >, MO o n < 0 , Q 2 > a n d Q p"iQ) * C2 p'iQ) t piQ) * C 3 fiQ)^~*t fiQ)^CAQ2t Q f'(Q) £ -J C5 H Q ) , H Q ) * C6 Q5 for Q 2. Q . These cond i t ions are sa t i s f i ed for e x a m -ple if piQ) = const. Qa (where 1 s a < 3) for p - 0 and /"(Q) = const, (Q - Q 2 )3 for Q >. Q2. ft i s supposed to be a bounded domain in R3 wi th a su f f ic ient ly s m o o t h b o u n d a r y . T i s a given pos i t ive number. For each t e (0,7~). the a r t i f i c i a l v i s c o s i t y i s act ing only on a set ft't = ( x € ft; Q(X, t) > Q.). Sine* "19 Kurzweil, Eauadiff 7 2 8 9 the ammoun t of mass in Q. i s c o n s t a n t (=A/f), we can derive: meas(Sl^) z i M/Q2. Thus the a r t i f i c i a l v i s c o s i t y can act at most on a set as smal l as we wan t if we choose Q2 large enough . The sy s tem (1),(2) c o i n c i d e s w i t h the sy s t em of the N a v i e r - S t o k e s e q u a t i o n s for barotrop ic f lu id and the c o n t i n u i t y e q u a t i o n in the range of "reasonable values" of d e n s i t y ( i . e . the va lues i Q2) . The problem ( l ) - (4 ) e n a b l e s to derive var iou s apriori e s t i m a t e s . The most impor tan t one ( w h i c h may be ca l l ed an energy inequality in accor-dance w i th the " incompress ib le case") h a s the form (5) Sj*uiui\ d x + J Q ( Q ) | d x % J J n C u u i i u + - 3 - ( ^ y )2 3 d x d t • T + J J ai2\j a{^]j d x dt * J 7 Q0u0iu0i d x + j Q ( e 0) d x + const. 0 fl XI A (for a . a . x € ( 0 , 7 ) ) , where Q(Q) = max { P(Q); 0 ) , P(Q) = Q Si p(o)/o2 60 (for e > 0 ) , P(0) = 0 , g(Q) = J* [ p'( a) • f (<x)/a ] 1 / 2 d<j . We can obta in t h e i n -e q u a l i t y (5) if we m u l t i p l y t h e / - t h e q u a t i o n in (1) bv u;t i n t e g r a t e bv par t s and use the e q u a t i o n (2) a n d the r e l a t i o n Q P'(Q) - P(Q) = - P(Q) in an appropr iate way . Moreover, by means of a s i m i l a r t e c h n i c s , im-bedd ing t h e o r e m s , Gronwall i n e q u a l i t y , e t c . , i t c a n be s h o w n , t h a t (6 ) U Q || -S const., (7) | Q U || * const., (8 ) | f (Q) || S const., where 7, » 7 (0 , 7 ; *«>, 2 ; L c(«), H"1(Q)), ? 2 - J ( O J ; 2 , 1 ; [2 ( . l ) 3 , H " 3 ( f l ) 3 ) , 7$ = • 7 (0 , 7 ; 2,1; H 1 ^ ) , H~3(Q))t U » iuitu2,u3) and the c o n s t a n t s on t h e r i g h t hand s i d e s of (6) - (8) depend on ly on 7, ft, QQ and U0 * iu0itu0Z,u03). It X0 and X1 are Banach spaces t h e n 7 (0 , 7 ; ct0, QLA ; X0, Xy) d e n o t e s the space of func t ion s v e L0 ( = /La0(0,7; XQ)) such t h a t v' € /., ( • Lai (0 , 7; X,)) , w i th the norm equal to | |v| | . + llv'H, * LG(Cl) i s the Orlitz space corresponding -0 Lt Q to the Young f unc t ion G(Q) = max ( Q(Q); I f(a) d a ) . We wan t to prove the global e x i s t e n c e of a weak so lu t ion to the problem ( 1 ) - ( 4 ) . It w i l l be a coup le of f unc t ion s 9 c Lco(0,T\ LG(&)), U • « (i/t t u2 , u3) c Z.2(O,7;H0(f2)3) such t h a t e -. 0, f(^) € L2(0 ,7; AV^ft)), <9> J J -ew / f / | t + Qujuifij + f ( e) " , , /? , , / + fiQiuifuj + p(e> *>,,/ -" *UU*U " 3 ^ / . J * / . / 3 * * d í = " / ©O^OI- (^#| r .0? d * 0 Û л .3 for al l <p -- ( j ) , , * ^ . ^ ) € C (QT)* such t h a t «P,|aflx(0 n » 0 , «P,|fm0s 0 ( / - 1 , 2 ,3 ) and T do) J J c e * i t • e";*./ - ne)y t j i d x d t = - J e o "( * L 0) d x 0 & for al l 4> c C^tQ--) such t h a t i|>| « 0 290 It i s n e c e s s a r y to cons truct a sequence of approx imat ion s at f irst . That i s why we f o r m u l a t e a problem wh i ch approx imates ( l ) - ( 4 ) in a c o n v e n i e n t way: (11) (9U. ) ( , + (QUJU^J = -p(Q)fi + »£uin + \ujJfl * Lf(Q)jUl • ! QjU, ] (/ = 1 . 2 , 3 ; i n QT ) , (12) e i t+ ^ e - V . y = fi*\jj + 7i e,;y (in QTn), (13) ti,| . = 0 (/ = 1 , 2 , 3 ) , J L [ f ( 9 ) + l g - | s 0 , ( U ) Q L - o = Q ° ( i n Q » } ' iQU']\t.o* Qouoi (> = L 2 , 3 ; i n fl), w h e r e Qn =- { x € R3; dist( x , Q) < 1 , Q-, fl =- Gn x (0 ,7) and - deno tes the re-g u l a r i z a t i o n def ined in a fol lowing way: C(x,t) = / w ( n ( x - / ) ) r.(/,r) d / An (for ~ def ined a . e . in QTn and such that "( . ,r) e Ly(Qn) for a. a. r e ( 0 , 7 ) ; w i s a f ixed funct ion from C°°(R3) such that supp Q = { x e R 3 ; | x | i 1 } , tne i n t e g r a l of o> over R3 i s equa l to 1 and c)(x) s 0 for a l l x e R3. ) It may be shown t h a t the p rob lem (11)-(14) has a weak so lu t ion (wh i ch w i l l . b e deno ted by Un= (uf, ug, u3n), Qn in the fo l lowing) . It i s rather compl i ca -ted from the t e c h n i c a l po in t of view, the used apparatus involves the method of a d i s c r e t i z a t i o n in t i m e , theory of n o n l i n e a r e l l ip t i c and pa-rabol ic e q u a t i o n s , f ixed po in t t h e o r e m s , e tc . The de ta i l s may be found in [ 2 ] . The same e s t i m a t e s as ( 5 ) - ( 8 > may be derived for the aoDroxi-m a t i o n s Un, Qn. Thus , we s h a l l refer to ( 5 ) - ( 8 ) as to e s t i m a t e s re lated to these a p p r o x i m a t i o n s . It fol lows from ( 5 ) - ( 8 ) and imbedd ings 7, c c L2(0,7; H"1(fl)), ?.<;<; c c L2(0,T; H'UQ)3), 7 3 C Q L2(QT) t h a t there e x i s t subsequnces of {Un)t {Q") (deno ted by {Un), {Q") again ) and func t ions U, Q, V, f so that I7n---U in /.2(0,7;H01(Q)3), Qn *~ Q i n J 1 ( Qn - Q in L2( 0,7; H'Hsk)), QnUn-*V in L2(0,T; H"1(^)3), f(Qn)-~ f in L2(QT). It i s poss ib le to show t h a t V --= 9 (J, e i 0 and f=f(Q). Moreover, we have: Un— U in L2( 0,7; H01(Q)3). The f u n c t i o n s L/n, Qn sat i s fy the in t egra l r e lat ion s ana logous to (9), (10) . The c o n v e r g e n c e s m e n t i o n e d above enab le to pass to l i m i t s (con -t a i n i n g U, Q i n s t e a d of Un, Qn, i n t e g r a l s over Q i n s t e a d of over fl„ and h a v i n g the regu lar i za t ion ~ van i shed ) in a l l t e rm s in these i n t e g r a l r e l a t i o n s e x c e p t the term T ( 1 5 ) / / P(Qn) f, , dx d r . It i s po s s ib le to pass to the l im i t a lso in th i s term if the cond i t ion 291 (16) p(g) = C7'Q (for an appropriate c o n s t a n t C7>0 and Qe < 0 , Q 1 > ) i s s a t i s f i e d . Thus , we can see t h a t U, Q r e p r e s e n t t h e weak so lu t ion to the p rob lem ( i ) - ( 4 ) and we can s t a t e : T h e o r e m . Let the condition (16) be fulfilled and let Q e L^Cl), QZ •> 0 a. e. in Q, U0 • (^QI » u 02 ' u 03^ € l-2(-2)3. Then there exists a weak solution U a a (L/1 , uz, u3), Q to the problem (1) - ( 4 ) . It is. poss ib le to show t h a t t h e weak s o l u t i o n U, Q from the las t theorem s a t i s f i e s t h e energy i n e q u a l i t y (5 ) . Withou t t h e c o n d i t i o n (16), we are ab le to prove the so lvab i l i t y of the p rob l em ( l ) - ( 4 ) i n the fo l lowing s e n s e : There e x i s t U B [ui, uz> u~) e c L2(0,/"; HQ(SI)3) and a measurab le func t ion - * . t 3 - * v [ x f ] wh i ch a s s i g n s to a. e. Cx, r 3 € Qr a n o n n e g a t i v e Radon measure v [ x t ] on <0, + <x>) so t h a t t h i s measure has a so ca l led "un i t mass" and if we put + 00 + CO e ( x , r ) = / X v [ x f ] ( d X ) , i 7 ( x , r ) = / p( X) v [ x f ] ( d X ) 0 0 then Qe Lco(OtTiLG{Sl))t f(g) £ L2(0,T; HhQ)) and t h e re lat ions (9) , (10) a r e . s a t i s f i e d ( w i t h i7 i n s t e a d of P (Q) in (9 ) ) . We can obta in t h i s r e -s u l t if we use the ideas of L .Tartar C 33 and Di Perna [13 . The ment ioned r e s u l t s may be g e n e r a l i z e d for t h e case of a n o n z e -ro body force a n d n o n h o m o g e n e o u s b o u n d a r y c o n d i t i o n s . R e f e r e n c e s : [13 Di P e r n a R . J . : Convergence of approximate solutions to conservation laws, A r c h . R a t . Mech . A n a l . 82 (1983) 2 7 - 7 0 . [23 N e u s t u p a J . : The global existence of solutions to the eouations of motion of viscous gas with an artificial viscosity, To appear i n M a t h . M e t h . i n A p p l . Sc i . [33 T a r t a r L . : Compensated compactness and applications to partial diffe-rential equations, i n Nonlinear Analysis and Mechanics, Heriot- Watt Symposium, V o l . I V , P i t m a n Research No tes i n M a t h e m a t i c s , 1979, p p . 1 3 6 - 1 9 2 . 292 

The global solvability to the equations of motion of viscous gas with an artificial viscosity

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The global solvability to the equations of motion of viscous gas with an artificial viscosity

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Institute of Mathematics AS CR, v. v. i.