ON EXISTENCE, UNIQUENESS AND L r -EXPONENTIAL STABILITY FOR STATIONARY SOLUTIONS TO THE MHD EQUATIONS IN THREE-DIMENSIONAL DOMAINS

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1 ANZIAM J , ON EXISTENCE, UNIQUENESS AND L -EXPONENTIAL STABILITY FOR STATIONARY SOLUTIONS TO THE MHD EQUATIONS IN THREE-DIMENSIONAL DOMAINS CHUNSHAN ZHAO and KAITAI LI Received 4 August, 000; evised Febuay, 003 Abstact The existence of stationay solutions to the MHD equations in thee-dimensional bounded domains will be poved. At the same time if the assumption of smallness is made on extenal foces, uniqueness of the stationay solutions can be guaanteed and it can be shown that any L > 3 global bounded non-stationay solution to the MHD equations appoaches the stationay solution unde both L and L noms exponentially as time goes to infinity.. Intoduction Let be a thee-dimensional bounded domain with smooth and Q =.0; /. Conside the MHD equations [3] inq as ¹u +.u /u.b /B +. B / + ² 5 = f.x/;.x; t/ B ½B +.u /B.B /u = 0;.x; t/ Q; u = 0; B = 0;.x; t/ Q; = 0; = 0; t.0; /; u.x; 0/ = u 0.x/; B.x; 0/ = B 0.x/; x : E:S In E:S, u =.u.x; t/; u.x; t/; u 3.x; t// and B =.B.x; t/; B.x; t/; B 3.x; t// ae an unknown velocity vecto and magnetic field espectively, f.x/ is a known extenal Depatment of Mathematics, The Univesity of Iowa, Iowa City, IA 54, USA; chuzhao@math.uiowa.edu. School of Science, Xi an Jiaotong Univesity, Shaanxi, 70049, China; ktli@mail.xjtu.edu.cn. c Austalian Mathematical Society 004, Seial-fee code /04 95

2 96 Chunshan Zhao and Kaitai Li [] foce and 5 is pessue and can be uniquely detemined by.u; B; f / up to a constant. We note that ¹, ¼, ² ae constants of kinematic viscosity, magnetic pemeability and density of Euleian flow espectively and that ½ = =¼ with electical esistivity. Clealy, the stationay solutions of E:S satisfy the following equations: ¹v+.v /v.b /b+. b /+ ² ³ = f.x/; x ; ½b +.v /b.b /v = 0; x ; S:S v = 0; b = 0; x ; = 0; = 0: In a simila manne, ³ can also be uniquely detemined by.v; b; f / up to a constant. In this pape, we mainly study the existence and uniqueness of stationay solutions, and stability elations between the global L > 3 bounded non-stationay solutions and stationay solutions. Fist, let us ecollect some elated methods and esults about stability fo stationay solutions to the well-known Navie-Stokes equations.n:s/. Temam [] studied the existence and uniqueness of stationay solutions to.n:s/ and obtained L -exponential stability fo the stationay solutions unde the assumption that extenal foces ae sufficiently small o the viscosity constant of fluids is sufficiently lage. Recently, Schonbek [9] and Guo and Zhang [5] pesented some decay popeties of solutions to the MHD equations and showed that unde some assumptions on the decay popeties of extenal foces, any non-stationay solutions to the MHD equations on the whole thee-dimensional o two-dimensional space decay to zeo algebaically as time goes to infinity. Qu and Wang studied L p stability fo stationay solutions of.n:s/ on thee-dimensional bounded domains in [7]. Motivated and inspied by the above mentioned wok, we studied existence, uniqueness and L. > 3/-exponential stability fo stationay solutions to the MHD equations on thee-dimensional bounded domains. We pove the existence of at least one stationay solution to the MHD equations in thee-dimensional bounded domains and uniqueness of the stationay solutions if extenal foces ae sufficiently small o ¹ and ½ ae sufficiently lage. The main pupose of this pape is to pove that unde the assumption of smallness of extenal foces thee exist positive constants c, k, þ independent of t, u, B such that u.t/ v + B.t/ b c u 0.x/ v + B 0.x/ b k e þt fo all t > 0, whee denotes the usual L -nom. This pape is aanged as follows. We pesent some mathematical peliminaies in Section. In Section 3, we pove the existence of stationay solutions to E:S. Moeove, if extenal foces ae sufficiently small o ¼ and ½ ae sufficiently lage, uniqueness of the stationay solution can be guaanteed. Additionally in this section, we also pove ou main esults on L -exponential stability of stationay solutions to

3 [3] Stationay solutions to the MHD equations in 3D domains 97 E:S. The main esults on L. > 3/-exponential stability ae pesented and poved in Section 4.. Mathematical peliminaies In this section we pesent some mathematical peliminaies. Fist, let us intoduce some vecto-valued functional spaces and notation as follows: L./ denotes the usual vecto-valued functional space consisting of -times integable functions on. Hee H = closue of { C 0./; = 0} in L./ and H ³; = closue of { ; C./} in L./. As descibed by Temam [], L./ = H H ³; : Hee W m;./ m 0 denotes the well-known vecto-valued Sobolev spaces. In paticula, if =, H m = W m;./. Let P : L./ H be a Helmholtz pojection opeato [] and let A = P be the well-known Stokes opeato with domain D.A / = H W ; 0./ W ;./. Let V = closue of { C./; = 0} in H 0./ and let V be the dual space of V. Fo simplicity, we omit subscipts in the notation H, A, P and. We denote by ; and. ; / the dual poduct between V and V and the inne poduct of H espectively. Applying P to both sides of the fist two equations in E:S and in S:S yields + ¹ Au + P.u /u P.B /B = B + ½AB + P.u /B P.B /u = 0. ¹ Av + P.v /v P.b /b = Pf;.3 ½Ab + P.v /b P.b /v = 0:.4 Since A : H D.A/ is a positive compact opeato, A has eigenvalues {½ j } j = ; ;::: and coesponding eigenvectos {! j } j = ; ;::: satisfying Moeove, ½ j as j. A! j = ½ j! j ; 0 <½ <½ ½ j : DEFINITION. Fo any u 0.x/, B 0.x/ H L./ > 3 and f.x/ L./,.u.x; t/; B.x; t// is called an L -bounded solution of E:S if

4 98 Chunshan Zhao and Kaitai Li [4] i both u.x; t/ and B.x; t/ ae unifomly bounded with espect to t [0; /; ii u.x; t/; B.x; t/ W ;./, fo any t.0; /; iii.u.x; t/; B.x; t// satisfies E:S in a weak o stong sense. Fo existence and uniqueness of local o global stong weak solutions of E:S, see efeences [, 6, 8]. 3. Existence, uniqueness and L -exponential stability fo stationay solutions In this section, we study the existence, uniqueness and L -exponential stability fo stationay solutions to the MHD equations. We use a Fadeo-Galekin appoximation combined with Schaefe s fixed point theoem to show the existence of stationay solutions, and the enegy estimates method to show uniqueness and L -exponential stability fo stationay solutions. THEOREM 3.. If f.x/ V, then thee exists at least one solution.v; b/ V V to.3.4. Moeove, if f.x/ H, then v;b D.A/. Additionally, if f.x/ is sufficiently small o both ¹ and ½ ae sufficiently lage, then the solution.v; b/ to.3.4 is unique. PROOF. We implement the well-known Fadeo-Galekin method [] to pove the existence of solutions.v; b/ to.3.4. Fo m N, we look fo an appoximate solution.v m ; b m / such that v m = m ¾ i= im! i, b m = m ¾ i= im! i, ¾ im ; ¾ im R,and ¹. v m ; ˆv/ + b.v m ;v m ; ˆv/ b.b m ; b m ; ˆv/ = f; ˆv ; 3. ½. b m ; ˆb/ + b.v m ; b m ; ˆb/ b.b m ;v m ; ˆb/ = 0; 3. fo evey ˆv; ˆb W m = span{! ;:::;! m },whee b.u;v;w/ = 3 i; j= Equations ae also equivalent to j w j dx: ¹ Av m + ˆP m.v m /v m ˆP m.b m /b m = ˆP m f; 3.3 ½Ab m + ˆP m.v m /b m ˆP m.b m /v m = 0 3.4

5 [5] Stationay solutions to the MHD equations in 3D domains 99 in V,whee ˆP m : H W m is a pojection opeato. Next we pove the existence of a solution.v m ; b m / to by the well-known Schaefe fixed point theoem [4]. Since W m is a finite-dimensional subspace of H, to apply the fixed point theoem, it is sufficient to show the unifom boundedness of v m and b m with espect to 0 Þ fo all possible solutions of the following equations: ¹ Av m + Þ ˆP m.v m /v m Þ ˆP m.b m /b m = ˆP m f; 3.5 ½Ab m + Þ ˆP m.v m /b m Þ ˆP m.b m /v m = 0: 3.6 In fact, taking the dual poduct of V and V with v m on both sides of 3.5 and with b m = on both sides of 3.6, then summing them togethe, yields Theefoe ¹ v m + ½ b m = ˆP m f;v m ¹ v m + ¹ f V : ¹ v m + ½ b m ¹ f V : Thus due to the Schaefe fixed point theoem acting on W m W m, the existence of.v m ; b m / is guaanteed. Next we give some a pioi estimates of v m and b m. Fist, taking ˆv = v m, ˆb = b m = in and summing them togethe, we get Theefoe ¹ v m + ½ ²¹ b m = f;v m ¹ f V + ¹ v m : ¹ v m + ½ b m ¹ f V : 3.7 So we can extact fom.v m ; b m / a subsequence.v m ; b m / which conveges weakly in V to some limit.v; b/, and since the injection of V in H is compact,.v m ; b m / conveges to.v; b/ stongly in H, thatis,b m b and v m v weakly in V and stongly in H. Passing to the limit in 3. 3., we find that.v; b/ is a solution of inv. We note that if v;b V, then accoding to [,.6] it follows that P.v /v, P.b /b, P.v /b and P.b /v belong to D.A = /. Hence v = ¹ A Pf P.v /v + P.b /b D.A 3= /; 3.8 b = ½ A.P.b /v P.v /b/ D.A 3= /: 3.9

6 00 Chunshan Zhao and Kaitai Li [6] If f.x/ L./, applying [, Lemma.] to yields P.v /v, P.b /b, P.v /b, P.b /v H. Theefoe v and b D.A/. Since v;b D.A/ H./ we can define v ; b fo any accoding to the Sobolev embedding theoem [4]. Next we give some a pioi estimates of.v; b/ in V and D.A/ espectively. If f.x/ L./, fom 3.7 and lowe-semicontinuity of the nom, it follows that ¹ v + ½ b f : ¹½ Fo the nom in D.A/, we infefom.3.4 fo = that ¹ Av f + P.v /v + P.b /b f +c v 3= Av = + b 3= Ab = f + ¹ 4 Av +c ¹ v 3 + ½ 4 Ab + c ² ¼ ½ b 3 ; whee we have used Young s inequality. Similaly, Finally, ½ Ab ¹ 4 Av +c ¹ b 3 + ½ 4 Ab +c ½ v 3 : ¹ Av +½ Ab f + c v 3 + b 3 + c ¹ ² ¼ ½ b 3 + c ½ v 3 [ f +c ¹ 3 ½ 3= ¹ + +./3= ½.½¹½ / 3= ² ¼ ½ + ] f 3 : 3.0 ¹ To pove the uniqueness esult, let us assume that.v i ; b i /, i = ;, ae two solutions of.3.4 fo =, that is, ¹. v i ; ˆv/ + b.v i ;v i ; ˆv/ b.b i ; b i ; ˆv/ = f; ˆv ; ½. b i ; ˆb/ + b.v i ; b i ; ˆb/ b.b i ;v i ; ˆb/ = 0; fo i = ; and ˆv; ˆb V. Let w = v v ; b = b b.weget ¹. w; ˆv/ + b.v ;w; ˆv/ + b.w; v ; ˆv/ b.b ; b; ˆv/ + b. b; b ; ˆv/ = 0; ½. b; ˆb/ + b.v ; b; ˆb/ + b.w; b ; b/ b.b ;w; ˆb/ + b. b;v ; ˆb/ = 0:

7 [7] Stationay solutions to the MHD equations in 3D domains 0 Taking ˆv = w and ˆb = b in the above equalities and adding them togethe yields ¹ w + ½ b = b.w; v ;w/+ b. b; b ;w/+ b. b;v ; b/ b.w; b ; b/ c w v + c c w + f ½¹½ + c b Theefoe ¹ c ¹½ = ¹½ = + ¹½ = + b w b + b v ½¹½ f : f w ½¹½ f b 0: ½¹½ + ½ c + ¹½ = If f.x/ is sufficiently small o both ¹ and ½ ae sufficiently lage so that and ¹ c + f > 0 ¹½ = ½¹½ ½ c + f > 0; ¹½ = ½¹½ it follows that v = v and b = b. Hence uniqueness of the solution is poved and the poof of Theoem 3. is completed. THEOREM 3.. Let.u.t/; B.t// be any solution of.. with initial data u 0 ; B 0 H and f.x/ H be sufficiently small o ¹ and ½ be sufficiently lage. Then thee exist constants c, þ > 0 independent of t, u and B such that, fo all t > 0, u.t/ v + B.t/ b c u 0 v + B 0 b e þt : PROOF. Let w.t/ = u.t/ v and B.t/ = B.t/ b. +¹ Aw+ P.u /w+ P.w /v P.B / B + P. B /b = 0; B + ½A B + P.u / B + P.w /b P.B /w + P. B /v = 0; 3. w.x; 0/ = u 0 v; B.x; 0/ = B 0 b:

8 0 Chunshan Zhao and Kaitai Li [8] Taking the scala poduct of 3. with w.t/, and of 3. with B.t/= in H and summing them togethe, we get d w.t/ + dt B.t/ + ¹ w + ½ B = b.w;v;w/+ b. B; b;w/+ b. B;v; B/ b.w; b; B/ : 3.3 Next we give estimates of each tem on the ight-hand side of 3.3 b.w;v;w/ c w 3 w = Av ¹ 6 w + c ¹ =3 w Av 4=3 ; b. B; b;w/ c B 3=4 B =4 w 3=4 w =4 Ab ¹ 6 w + + c ¹ =3 w Ab 4=3 ; whee we have used Young s inequality. Similaly, ½ 6 B + c./ =3 ½ =3 B Ab 4=3 b. B;v; B/ ½ 6 B + c./ =3 B Av 4=3 ; ½ =3 b.w; b; B/ c B 3=4 B =4 w 3=4 w =4 Ab ¹ 6 w + + c ¹ =3 w Ab 4=3 ; also using Young s inequality hee. Substituting the above estimates into 3.3, we get d dt w + B ½ 6 B + c./ =3 ½ =3 B Ab 4=3 + ¹ w + ½ B c w ¹ =3 Av 4=3 + Ab 4=3 ¹ =3 + c B./ =3 ½ =3 Av 4=3 +./=3 ½ =3 Ab 4=3 : Theefoe d w + dt B + þ w + B 0; 3.4

9 [9] Stationay solutions to the MHD equations in 3D domains 03 whee þ = min.¹; ½/½ { c Av 4=3 max + c Ab 4=3 ; c./ 4=3 Av 4=3 + c }./ 4=3 Ab 4=3 : ¹ =3 ¹ =3 ½ =3 ½ =3 Fom 3.0, we know and Av ¹ f +c Ab ½ f +c [ ¹ 4 ½ 3= [ ½¹ 3 ½ 3= ¹ + +./3= ½ ¹ 5=.½½ / 3= ² ¼ ½ + ] f 3 ¹ ¹ + +./3= ½ ½ 5=.¹½ / 3= ² ¼ ½ + ] f 3 : ¹ Thus it follows that if ¹ and ½ ae sufficiently lage o f is sufficiently small, then þ>0. Taking 3.4 into account, we get u.t/ v + B.t/ b c u 0 v + B 0 b e þt fo all t > 0. The poof of Theoem 3. is completed. 4. Main esults of L > 3-exponential stability In this section we will study the stability of the solutions to E:S with initial values u 0.x/; B 0.x/ H L./. The main tools we use in this section ae enegy estimates. Fom E:S and S:S, by diffeence, we ¹w +.u /w +.w /v.b / B +. B /b +. B b / + ².5 ³ / = 0;.x; t/ B ½ B +.u / B +.w /b D:E:S.B /w. B /v = 0;.x; t/ Q; w = 0; B = 0;.x; t/ Q; = 0; = 0; t.0; /; w.x; 0/ = u 0.x/ v; B.x; 0/ = B 0.x/ b; x :

10 04 Chunshan Zhao and Kaitai Li [0] Let us denote 5 =. B b /=./ +.5 ³ /=². FomD:E:S ¹w +.u /w +.w /v.b / B +. B /b + 5 = 0; B ½ B +.u / B +.w /b.b /w. B /v = 0: 4. LEMMA 4.. The following estimate of 5 holds: 5.+/= c w + v c + + w + + B + b + + B + : 4.3 PROOF. To estimate 5.+/=, we take the divegence of 4. and obtain 5 = n j w i.u j + v j / n i; j= Fom the Caldeon-Zygmund inequality [0] it follows that 5.+/= c n i; j= w i.v j + u j / + c.+/= Noticing that u = w + v and B = B + b we obtain 5.+/= c n i; j= w i.v j + w j / Then by Holde s inequality, we get 5.+/= c w + Lemma 4. is poved..+/= j. B i B j + b j / : n i; j= n i; j= B i.b j + B j /.+/= : B i.b j + B j /.+/= : v c + + w + + B + b + B + + : Next we pesent two lemmas which wee poved in []. LEMMA 4.. Let > and N.w/ = w w dx. Then fo any w W ;./ and w = 0. N.w/ c w 4=3. / w +.4=3. // ; LEMMA 4.3. Let > 3 and w W ;./. Then w + + c w N.w/ 3=.

11 [] Stationay solutions to the MHD equations in 3D domains 05 Multiplying both sides of 4. by w w and both sides of 4. by B B and integating ove, afte suitable integations by pats, we obtain d 4¹. / dt w + ¹ N.w/ + w = dx =.u /w w w dx.w /v w w dx +.B / B w w dx +. B /b w w dx 5 w w dx 4.4 and d dt B + ½N 4½. /. B/ + B = dx =.u / B B Bdx.w /b B Bdx +.B /w B Bdx+. B /v B Bdx: 4.5 Noticing that the fist integal on the ight-hand side of vanishes since u, w and B ae divegence fee, we need only give estimates of othe tems on the ight-hand sides of , that is,.w /v w w dx. / w w v dx =. /N.w/ = w v dx ¹ 3 N 8. /.w/ + w ¹ + v +: On the othe hand, w cn +.w/ 3=.+/ w. /=.+/. Thus w + v + ¹ 3 N.w/ + c w v.+/=. / + : Theefoe.w /v w w dx ¹ 6 N.w/ + c w v.+/=. / : 4.6 Similaly,. B /b w w dx. /N.w/ = ¹ 3 N.w/ + = w B v dx 8. / w + ¹ B + v : +

12 06 Chunshan Zhao and Kaitai Li [] Noticing that w we get + B cn +.w/ 3. /=.+/ w. /. /=.+/ N. B/ 6=.+/ B. /=.+/ ; w + B + v + ¹ 3 N.w/ + ½ 3 N. B/ + c w v.+/=. / + c B v.+/=. / : Hence. B /b w w dx ¹ 6 N.w/ + ½ 3 N. B/ By a simila manne, we get.w /b B Bdx ¹ 3 N.w/ + ½ 6 N. B/ + c w v.+/=. / + + c B v.+/=. / + : c w b.+/=. / + + c B b.+/=. / + ; 4.8. B /v B Bdx ½ 6 N. B/ + c B v.+/=. / 4.9 and 5 w w dx. / 5 w w dx. /N.w/ = w. /= + 5.+/= Applying Lemma 4.,we have w + 5.+/= w + v + + w + ¹ 3 N.w/ + c w + 5.+/=: c + w + B + b + B + + cn.w/ 3=.+/ w. /=.+/ v + cn +.w/ 3= w + cn.w/ 3. /=.+/ w. /. /=.+/ N. B/ 6=.+/ B. /=.+/ b + + cn.w/ 3. /=.+/ w. /. /=.+/ N. B/ =.+/ B 4. /=.+/ ¹ 3 N.w/ + ½ 3 N. B/ + c w v.+/=. / + c w. /=. 3/ + c w b.+/=. / + + c B b.+/=. / + + c B. /=. 3/ :

13 [3] Stationay solutions to the MHD equations in 3D domains 07 Hence 5 w w dx ¹ 6 N.w/ + ½ 6 N. B/ + c w v.+/=. / + c w. /=. 3/ + c w b.+/=. / + + c B b.+/=. / + c B. /=. 3/ : 4.0 Since.B / B w w dx.b / B w w dx +. B / B w w dx ;.b / B w w dx. / w w b B dx ¹ 3 N.w/ + ½ 3 N. B/ + c w b.+/=. / + c B b.+/=. / + ;. B / B w w dx. / w w B dx ¹ 3 N.w/ + c w + B 4 + and Lemma 4.3 tells us w + B 4 + cn.w/ 3. /=.+/ w. /. /=.+/ N. B/ =.+/ B 4. /=.+/ ¹ 3 N.w/ + ½ 3 N. B/ + c w. /. /=.+/. 3/ B 4. /=.+/. 3/ ¹ 3 N.w/ + ½ 3 N. B/ + c w. /=. 3/ + c B. /=. 3/ ; we get.b / B w w dx ¹ 6 N.w/ + ½ 6 N. B/ + c w b.+/=. / + c B b.+/=. / + c w. /=. 3/ + c B. /=. 3/ : 4. Similaly,.B /w B Bdx ¹ 6 N.w/ + ½ 6 N. B/ + c w b.+/=. / + c B b.+/=. / + + c w. /=. 3/ + c B. /=. 3/ : 4.

14 08 Chunshan Zhao and Kaitai Li [4] Substituting estimates into and summing them togethe, we get d dt w + B + min.¹; ½/.N.w/ + N. B// c w + B v.+/=. / + + b.+/=. / + + c w. /=. 3/ + B. /=. 3/ : By intepolation, we have w w 4=.3 / w 3. /=.3 / 3. Fom Lemma 4. it follows that N.w/ c w 4=.3. // w +.4=3. //. Using these estimates, we get d dt w + B + min.¹; ½/ w 4 =3. / w + B 4=3. / B +.4=3. // c w + B.+/=. / v + b.+/=. / + + c B. /=. 3/ + w. /=. 3/ : Fom 3.0, Theoem 3. and H./, L +./, weget d dt +.4 =3. // w + B + c w0 + B 0 k e þkt w + B +.4=3. // c w + B whee k = =3. /. Let y.t/ = w.t/ + B.t/.Fom4.3weget + c w + B. /=. 3/ ; 4.3 y.t/ + c w 0 + B k e 0 þkt y.t/ +.4=3. // cy.t/ + cy.t/. /=. 3/ ; whee k = 4=3. /. Applying Lemma 4.5, we get the following theoem. THEOREM 4.4. Suppose.u; B/ is any L -bounded solution of E:S, f.x/ L./ and u 0.x/, B 0.x/ L./ H,whee.v; b/ is the solution of S:S. IftheL -nom of f.x/ is sufficiently small o ¹ and ½ ae sufficiently lage, then thee exist constants c, þ, k > 0 independent of t, u and B such that, fo all t > 0, u.t/ v + B.t/ b c u 0 v + B 0 b k e þkt : LEMMA 4.5 [7]. Let us make the following assumptions: i Thee exists a constant M > 0 such that fo all t > 0, the function y.t/ <M.

15 [5] Stationay solutions to the MHD equations in 3D domains 09 ii Thee exists a diffeentiable function h.t/ >0 and continuous functions f.t/; :::; f m.t/ fo t > 0 such that h.t/ lim t h.t/ = l > 0; lim t f i.t/ h.t/ = 0;.i = ;:::;m/: iii Thee ae constants a > a > > a m > a 0 >. iv The function y.t/ satisfies the diffeential inequality m y.t/ + ch.t/y.t/ a0 b 0 y.t/ + f i.t/y.t/ ai ; whee c; b 0 > 0. Then the estimate y.t/ a0.b 0 =cl/h.t/ holds fo all t > 0. i= Acknowledgement The second autho was subsidised by the special funds fo majo state basic eseach unde poject G The authos want to give thei thanks to the anonymous efeees fo thei valuable comments. Refeences [] H. Beiao da Veiga, Existence and asymptotic behavio fo the stong solutions of the Navie- Stokes equations in the whole space, Indiana Univ. Math. J [] E. V. Chizhonkov, On a system of equations of magneto-hydodynamic type, Soviet Math. Dokl [3] T. G. Cowling, Magnetohydodynamics, Intescience Tacts on Physics and Astonomy 4 Intescience Publishes, New Yok, 957. [4] L. Evans, Patial diffeential equations Ameican Mathematical Society, Povidence, RI, 998. [5] B. Guo and L. Zhang, Decay of solutions to magnetohydodynamics equations in two space dimensions, Poc. Roy. Soc. London Se. A [6] G. Lassene, Ube ein Rand-anfangswet-poblem de Magnetohydodinamik, Ach. Rational Mech. Anal [7] C. Qu and P. Wang, L p exponential stability fo the equilibium solutions of the Navie-Stokes equations, J. Math. Anal. Appl [8] M. Rojas-Meda and J. Boldini, Global stong solutions of equations of magneto-hydodynamic type, J. Austal. Math. Soc. Se. B [9] M. Schonbek, The long time behavios of solutions to the MHD equations, Math. Ann [0] E. M. Stein, Singula integals and diffeentiability popeties of functions Pinceton Univesity Pess, Pinceton, NJ, 970. [] R. Temam, Navie-Stokes equations Noth-Holland, Amstedam, 979. [] R. Temam, Navie-Stokes equations and nonlinea functional analysis SIAM, Philadelphia, PA, 983.

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal