On a nonlinear compactness lemma in L p (0, T ; B).

Transcription

1 On a nonlnear compactness lemma n L p (, T ; B). Emmanuel Matre Laboratore de Matématques et Applcatons Unversté de Haute-Alsace 4, rue des Frères Lumère 6893 Mulouse E.Matre@ua.fr 3t February 22 Abstract We consder a nonlnear counterpart of a compactness lemma of J. Smon [], wc arses naturally n te study of doubly nonlnear equatons of ellptc-parabolc type. Our work was motvated by prevous results J. Smon [], recently sarpened by H. Amann [2], n te lnear settng, and by a nonlnear compactness argument of H.W. Alt and S. Luckaus [3]. MSC2 : Prmary 46B5, 47H3. Secondary 34G2, 35K65. Introducton Typcal applcatons were te compactness argument stated below s useful are tose n wc te followng knd of doubly nonlnear equatons arses db(u) + A(u) = f were A s ellptc and B monotone (not strctly). It s te case, for example, n porous medum, semconductor equatons,... In our applcaton, we consdered te njecton mouldng of a termoplastc, wt a mold of small tckness wt respect to ts oter dmensons. By averagng aver-stokes equatons across te tckness of te mold, and under an assumpton (of Hele-Saw) statng tat te velocty eld s proportonal to te pressure gradent, te pressure equaton can be wrtten as a doubly nonlnear equaton [6]. ote tat n ts context, te equaton can degenerate to an ellptc one. In order to get exstence of a soluton, one usually perform a tme dscretzaton, use some result on ellptc operator and pass to te lmt as te tme step goes to zero. In nonlnear problems compactness n tme and space s ten requred. Te compactness n space s easly obtaned for u from a coercveness assumpton on te ellptc part A, but we ave no estmate on u t snce B could degenerate. Teorem uses te space compactness of u and some tme regularty on B(u) to derve a compactness for B(u), wc n turn can be useful to pass to te lmt n nonlnear terms of A (provded A as a an approprate structure, e.g. B pseudomonotone [5]). 2 Man result Let us consder two Banac spaces E, E 2. Let T >, p [, + ], and B a (nonlnear) compact operator from E to E 2,.e. wc maps bounded subsets of E to relatvely compact subsets of E 2.

2 Teorem : Let U be a bounded subset of L (, T ; E ) suc tat V = B(U) s a subset of L p (, T ; E 2 ) bounded n L r (, T ; E 2 ) wt r >. Assume lm v( + ) v + L p (,T ;E 2) = unformly for v V. () Ten V s relatvely compact n L p (, T ; E 2 ) (and n C(, T ; E 2 ) f p = + ). Remarks :. One can easly ceck tat teorem olds f we assume only U bounded n L loc (, T ; E ) and V bounded n L r loc (, T ; E 2). 2. In te case were B s te canoncal njecton from E to E 2, te assumpton on B corresponds to te compactness of te embeddng of E nto E 2, and te concluson falls n te scope of prevous results of J. Smon [], teorem Te pont n ts result s tat we do not make any structural assumpton on B (e.g. strct monotony, wc would fall n te scope of results of A. Vsntn [4]) except compactness. ote tat n te case of a compact embeddng of E nto E 2, B needs only to be contnuous from E to E 2 for te E 2 topology. Idea of te proof : A sucent condton for compactness s to prove tat for eac couple (t, t 2 ), t 2 t v(t) descrbes a relatvely compact subset of E 2 as v descrbes V. Frst te u(t), u U are truncated n norm at egt M > and form a bounded subset of E wc B maps to a relatvely compact subset V M (t) of E 2. Te key pont s tat tanks to equ-ntegrablty assumpton, t v(t) can be approxmated unformly n v by Remann sums nvolvng truncated elements of te V M (t). Proof : Tanks to te equ-ntegrablty () of V and results of [], we only ave to prove tat for eac (t, t 2 ) suc tat < t < t 2 < T, te set { } K = v(t), v V t s relatvely compact n E 2. For tat purpose, we ntroduce for u U and M > te measurable subset of [, T ] dened by G M u = { t [, T ], u(t) E > M }. From our assumptons on U, tere exsts a constant C > suc tat and snce we ave tat gves Introducng te truncated functons meas(g M u ) = u U, u L (,T ;E ) C, G M u G M u u(t) E M C M lm M + meas(gm u ) =, unformly n u. (2) we ave by constructon u M (t) = u(t) f t G M u, M >, u U, t [, T ], oterwse, u M (t) E M. (3) 2

3 Lemma Under condton (), K can be unformly approxmated by Remann sums nvolvng elements of te form v M (t) = B(u M (t)), n te followng sense : gven ε >, tere exst ntegers and M suc tat for all v = B(u) V, tere exsts s,m v ], [ suc tat v(t) v M (ξ + s,m v ) < ε (4) were = t2 t and ξ = t +. Proof : = t = We rst note tat ( ) t2 v(t) v M (ξ + s,m v ) = v(t) v M (ξ + s,m,ξ t t ] (t). (5) Ten we prove te followng nequalty, were r stands for te conjuguate exponent of r : t v(t) v M (ξ + s)χ ]ξ,ξ ds = 2T p = sup v( + σ) v L p (,T σ;e + 2 ( 2) meas G M ) r u v B() L r (,T ;E. 2) (6) σ [,] Let us denote by I te left-and sde of te stated nequalty. Ten I = = ξ v(t) v M (ξ + s) ds = Usng Fubn's teorem, and settng σ = s t we get I = = ξ t wc gves tanks to a new applcaton of Fubn's teorem, I = = mn(ξ,ξ σ) = ξ ξ ξ t v(t) v M (t + σ) dσ, max(ξ,ξ σ) v(t) v M (t + σ) dσ From te denton of v M we tus ave I mn(t2,t 2 σ) max(t,t σ) v(t) v(t + σ) dσ + mn(t2,t 2 σ) mn(t2,t 2 σ) max(t,t σ) max(t,t σ) As V s a bounded subset of L r (, T ; E 2 ) one as te second term bounded by v(t) v M (s) ds. v(t) v M (t + σ) dσ. χ G M u (t + σ) v(t) B() dσ. ( ) ( mn(t2,t 2 σ) r t 2 ) χ G M u (t + σ) v(t) B() r r E 2 dσ 2(meas G M u ) r v B() Lr (,T ;E 2). max(t,t σ) t and te Hölder nequalty gves te announced estmaton (6). 3

4 Usng (), (2) and as v belongs to a bounded subset V of L r (, T ; E 2 ), we conclude from (6) tat v(t) v M (ξ + s)χ ]ξ,ξ ds, wen M and go to nnty, unformly n v. t = (7) We clam tat tere exsts at least one s = s,m v [, ] suc tat v(t) v M (ξ + s,m,ξ, (8) t = wen M, go to nnty, unformly n v. Indeed, let us set by sake of readablty f,m v (s) = v(t) v M (ξ + s)χ ]ξ,ξ so tat te unform convergence (7) reads t = f v,m (s)ds, wen M and = go to nnty, unformly n v. (9) Ten for xed v,, M tere exsts at least one s = s,m v [, ] suc tat f,m v (s,m v ) f v,m (s)ds. If not, we would ave te reverse strct nequalty for all s [, ] wc by averagng on [, ] would lead to a contradcton. Ten as f,m v s postve, te unform convergence (9) mples f v,m (s,m v ), wen M and = wc s exactly (8). A fortor, (4) olds tanks to (5) and snce ( ) t v(t) = v M (ξ + s,m,ξ ] (t) t go to nnty, unformly n v, () v(t) v M (ξ + s,m,ξ = Ts proves lemma. To conclude te proof of teorem, note tat lemma means tat were B s te unt open ball of E 2 and { } K M, = v M (ξ + s,m v ), v M = B(u M ), u U. = K εb + K M, For xed M, and from (3) we note tat u M (ξ + s,m v ) s bounded n E unformly n u U. As B s compact, K M, s tus a relatvely compact subset of E 2. Tus K s also relatvely compact n E 2. Corollary : Let U be a bounded subset of L (, T ; E ) suc tat V = B(U) s bounded n L r (, T ; E 2 ) wt r >. Assume { } V v t = t, v V s bounded n L (, T ; E 2 ). Ten V s relatvely compact n L p (, T ; E 2 ) for any p < +. 4

5 Proof : Condton () of teorem s satsed (see [], lemma 4). References [] J. Smon, Compact sets n te space L p (, T ; B), Ann. Mat. Pura Appl. 46 (987), [2] H. Amann, Compact embeddngs of vector-valued Sobolev and Besov spaces, Glasnk Matematck 35 (55) (2), [3] H.W. Alt and S. Luckaus, Quaslnear ellptc-parabolc derental equatons, Mat. Z. 83 (983), [4] A. Vsntn, Strong convergence results related to strct convexty, Comm. Partal Derental Equatons 9 (984), [5] E. Matre and P. Wtomsk, A pseudo-monotoncty adapted to doubly nonlnear ellptc-parabolc equatons, to appear n onlnear Analyss TMA. [6] E. Matre, Sur une classe d'équatons à double non lnéarté : applcaton à la smulaton de l'écoulement d'un ude vsqueux compressble, Tess, Unversty of Grenoble I, 997 5