Generalized Stochastic Burgers' Equation with Non-Lipschitz Diffusion Coefficient

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1 Communcatons on Stochastc Analyss Volume 2 Number 3 Artcle 6-29 Generalzed Stochastc Burgers' Equaton wth Non-Lpschtz Dffuson Coeffcent Vvek Kumar Indan Insttute of Technology Roorkee, Roorkee, Uttarakhand, Inda, vvekmsc8@gmal.com Ankk Kumar Gr Indan Insttute of Technology Roorkee, Roorkee, Uttarakhand, Inda, ankkgr.fma@tr.ac.n Follow ths and addtonal works at: Part of the Analyss Commons, and the Other athematcs Commons Recommended Ctaton Kumar, Vvek and Gr, Ankk Kumar 29 "Generalzed Stochastc Burgers' Equaton wth Non-Lpschtz Dffuson Coeffcent," Communcatons on Stochastc Analyss: Vol. 2 : No. 3, Artcle 6. DOI:.339/cosa Avalable at:

2 Communcatons on Stochastc Analyss Vol. 2, No Serals Publcatons GENERALIZED STOCHASTIC BURGERS EQUATION WITH NON-LIPSCHITZ DIFFUSION COEFFICIENT VIVEK KUAR AND ANKIK KUAR GIRI* Abstract. In ths artcle, we study the exstence of weak solutons to the one-dmensonal generalzed stochastc Burgers equaton wth polynomal nonlnearty perturbed by space-tme whte nose wth Drchlet boundary condtons and α-hölder contnuous coeffcent n nose term, where α [/2,. The exstence of weak solutons s shown by solvng an equvalent martngale problem. ft, x t. Introducton The generalzed stochastc Burgers equaton s gven as = 2 ft, x x 2 ht, x, ft, x wth followng boundary condtons and ntal data where 2 W t,x t x gt, x, ft, x x σt, x, ft, x 2 W t, x t x. ft, = ft, =, t,.2 f, x = f x, x,.3 s a whte nose wth respect to space and tme both as n [33] and h = ht, x, r, g = gt, x, r, and σ = σt, x, r are Borel-measurable functons on R [, ] R. For g =,. becomes stochastc reacton-dffuson equaton e.g. [4, 6, 28, 33]. When h = σ = and gt, x, r = r2 2, equaton. gves the classcal Burgers equaton, whch bascally shows the Newton s second law and descrbes the relatonshp between the changng momentum and force on fluds elements and t s used as a smple verson of the Naver-Stokes equaton whch represents the hydrodynamcal turbulent model, see [7, 8, 4, 2] and references there n. Next, for h =, σ, gt, x, r = r2 2, equaton. s known as stochastc Burger s equaton, whch s preferred as a better model over the determnstc Burgers equaton because t also descrbes the chaotc phenomena n the flud, see, e.g., [9,, 2, 23]. One dmensonal form of the classcal stochastc Burgers Receved 28--8; Communcated by S. S. Srtharan. 2 athematcs Subject Classfcaton. Prmary 6H5, 6H4; Secondary 35Q35, 35R6. Key words and phrases. Stochastc Burgers equaton, non-lpschtz coeffcent, weak soluton, tghtness, Skorohod repersentaton theorem. * Correspondng author. 329

3 33 VIVEK KUAR AND ANKIK KUAR GIRI equaton has been ntensvely studed n the lterature [, 2, 6, 2, 3, 7, 26], where exstence and unqueness of solutons were dscussed under the varous assumpton on the dffuson coeffcents σ. Here [2, 2, 3, 7] deal wth σ havng Lpschtz contnuty whle [5, 6, 26] have used non-lpschtz dffuson coeffcent σ. But f we nclude the polynomal nonlnearty n the equaton, then ths becomes more complcated ssue to study the exstence and unqueness of solutons. To the best of our knowledge, there are very few results [8, 9, 25], whch dscuss the stochastc Burgers equaton wth polynomal nonlnearty. In 999, Gyöngy [8] has shown the exstence and unqueness of solutons for the class of a quas lnear stochastc partal dfferental equaton wth polynomal nonlnearty havng whte nose wth respect to tme only. Further, n 26, Km [25] has dscussed about the Cauchy problem for the stochastc Burger equaton by consderng the nonlnear term wth the polynomal growth on whole real lne havng whte nose wth respect to tme. In these two works, t s assumed that the dffuson coeffcent σ s ether a constant or a Lpschtz functon wth lnear growth condtons. Later n 23, the exstence and unqueness of the global soluton for Stochastc Burgers equaton wth polynomal nonlnearty drven by addtve Lévy process a stochastc process wth jumps s obtaned by Hausenblas and Gr [9]. However, n the present work we also deal wth stochastc Burgers equaton wth polynomal type nonlnearty but wth the non-lpschtz σ. The equatons.-.3, wth quadratc nonlnearty n g under the condton that σ s 2-Hölder contnuous non-lpschtz functon, was studed by Kolkovska [26] n 22, where she has establshed the exstence of weak solutons by showng the tghtness for a sequence of polygonal approxmaton for the equaton and then solvng an equvvalent martngale problem. Further, n 24, Boulanba and ellouk [6], extended the work of Kolkovska [26] n d-dmenson d 2 wth more general nose. The novelty of the present work s that t generalzes the work of [26] by extendng the quadratc nonlnearty to a class of polynomal nonlnearty and usng more general dffuson coeffcent σ. Ths work also dffers from the work of [8, 25], where they have shown exstence of weak solutons of stochastc Burgers equaton wth polynomal nonlnearty havng whte nose wth respect to tme and Lpschtz contnuty n the dffuson coeffcent σ, whereas the present work consders the α-hölder contnuty n the dffuson coeffcent σ, where α [/2, along wth a space-tme whte nose. In ths artcle, we study the one dmensonal stochastc Burger equaton wth polynomal nonlnearty perturbed by a space-tme whte nose.e. f t t, y = 2 p f t, y λ f y2 y t, y σft, W y 2 t, y.4 t y ft, = ft, =, t [, T ],.5 f, y = f y, y,.6 where p 2 s a fxed nteger and T >. Here, the exstence of weak solutons to equatons.4.6 s establshed. The proof s manly motvated by the technque used n Funak [6].

4 STOCHASTIC BURGERS EQUATION WITH NON-LIPSCHITZ COEFFICIENT 33 The structure of the paper s the followng: In the comng secton, we gve the rgorous formulaton of the problem. In Secton 3, the dscretzed form of.4.6 s obtaned, whch gves a system of stochastc dfferental equatons n fnte dmenson. Further, the exstence of unque strong soluton to ths system of stochastc dfferental equaton s establshed by showng the exstence and pathwse unqueness of weak soluton for the system, whch s motvated by [6]. Next, n Secton 4, we have shown the tghtness property of the famly of approxmatng solutons by satsfyng the mult-dmensonal Totok Kolmogorov crteron. At last, n Secton 5, the exstence of weak solutons of the orgnal problem.4.6 s shown by solvng an equvalent martngale problem. Wth no loss of generalty, we suppose that λ =. 2. Formulaton of the Problem Defnton 2.. Let Ω, F, F t, P be a stochastc bass wth fltraton F t = {F t, t [, T ]}. Then the Brownan sheet W t, x = {W t, x : t [, T ], x R} s defned as a contnuous, F t adapted and centered Gaussan random feld wth covarance n the sense of Walsh [33]. E W s, xw t, y = s tx y Remark 2.2. By the propertes of W, t can be proved that whte nose wth respect to the fltraton F t s a martngale measure over [, T ] B[, ], where B[, ] s bounded Borel subset of [, ], [24, 33]. Now, the equaton.4 can be nterpreted n the weak sense by the followng equaton 2.. Defnton 2.3. A contnuous stochastc process {ft, x; t [, T ], x [, ]}, whch s F t -adapted, s sad to be soluton of equaton.4 n a weak sense, f for every ϕ C 2 [, ], such that ϕ = ϕ =, and a.s. for each t [, T ], and x [, ], we have ft, yϕydy = λ f, yϕydy f p s, yϕ ydyds fs, yϕ ydyds σfs, yϕyw ds, dx. 2. Further, we assume followng condtons on σ. Frst condton s that the dffuson coeffcent σ satsfes Hölder s contnuty of order α [/2, on the nterval [, ].e. there exst a constant c such that and second one s σr σr 2 c r r 2 α r, r 2 [, ], 2.2 Example 2.4. Let σ : R R such that { r r f r. σr := otherwse. σ = σ =

5 332 VIVEK KUAR AND ANKIK KUAR GIRI Then, σ satsfes condtons 2.2 and 2.3. Note: Other examples of such functons can be seen n [6]. 3. The Dscretzaton Processes Let be a fxed nteger and defne the set { ; =,,, }. On ths set, consder the descretzed form of.4.6 as dy t, x = Y t, x Y p t, x dt σy t, x db t, 3. Y t, x = Y t, x =, 3.2 Y, x = f x, 3.3 for every =, 2,, and t. Here x := { } and {B t : =, 2, } s the system of Brownan motons, derved from the Brownan sheet W x, t and t s defned as EB t := =, 2,, { f j, and EB tb j s := mn{t, s} f j =, whle and denote the approxmaton of the frst and second order dervatve respectvely wth respect to the varable x n the dscrete sense and defned as and Y t, x Y t, x := 2Y t, x Y t, x 2 Qt, x Qt, x := Qt, x, for all =, 2,,. By settng Y t, x ± := y ±t, we can wrte 3. as dy t = 2 [y t 2y t y t] [y t p y t p ] dt σy tdb t 3.4 Re-wrtng 3.4 n more compact form as dy t = α j y j t β j y j t p dt σy tdb t 3.5 wth j= y t = y t = t [, T ], 3.6

6 STOCHASTIC BURGERS EQUATION WITH NON-LIPSCHITZ COEFFICIENT 333 and y = f /,, j, 3.7 where 2 f j =, α j := 2 2 f j = 3.8 otherwse and f j = β j := f j = 3.9 otherwse. It s notced that the drft and dffuson coeffcents n the 3.5 do not satsfy the Lpschtz contnuty, whch restrct us to apply the classcal results on the exstence and unqueness for the soluton of 3.5. The followng theorem s the man result of ths secton. Theorem 3.. Let Y = y, y,, y [, ], be some gven ntal random condton and 2.2 and 2.3 hold. Then for each T > and any nteger, the system dy t = α j y j t β j y j t dt p σy tdb t 3. j= y t = y t = t [, T ], 3. y = y, 3.2 where =, 2,, admts a unque strong soluton Y t = y t, y t,, y t C[, T ], [, ]. 3.3 Proof. We consder the followng modfed form of stochastc dfferental equatons as dy t = α j y j t β j gy j t dt Ky tdb t 3.4 j= y t = y t = t [, T ], 3.5 y = y, 3.6 where =, 2,,, g : R R s defned as gx = x p { x } and K : R R s defned as Kx = σx { x }. Snce, the coeffcents of are contnuous and satsfes the lnear growth condtons, by the [5, Theorem 3., Chapter 5], there exsts a weak soluton Y t to Next, t s shown that for every weak soluton Y t = y t, y t,, y t of the , y t [, ] for every =,, and t [, T ]. In order to prove ths the followng lemma [29] s requred.

7 334 VIVEK KUAR AND ANKIK KUAR GIRI Lemma 3.2. Let Z={Zt, t } be a real valued sem-martngale. Suppose that there exst a functon ρ : [, [, such that ϵ du = for all ϵ, {Zs>} ρz s and L t Z, s dentcally zero for all t a.s.. ρu d Z s < for all t a.s. Then the local tme of Z at zero,.e. Let us apply the above Lemma 3.2 for the sem-martngale y and take ρy = y, then ϵ ρu du = for all ϵ, and t {y s>} ρy s d y s = {y s>} σ 2 y sds <. y s Therefore local tme L t y s zero. Agan we use Lemma 3.2 and Tanaka s formula [3, Theorem.2 Chapter IV] for y t := max [, y t] and summng over ndces =,, we get y t = = = = [ [ = [ = = {ys } {y s } [ {ys } [ = α j y j s β j gy j sds j= j= [ {y s } = j= j= [ ] α j y j s ds {y s } j= α j y j s ]ds { ys } = ] β j gy j s ds ] β j y j s ds j= α j y j s ]ds { y s } ] y j s ds j= y s ]ds. 3.7 = Fnally, Gronwall s nequalty gves y t =, =.e. y t { {,2, }} s always non-negatve for each t [, T ]. Agan, solvng the equaton 3.7 for y t, we can obtan y t for every t [, T ] and.

8 STOCHASTIC BURGERS EQUATION WITH NON-LIPSCHITZ COEFFICIENT 335 Snce, for y t [, ], the system 3.4 concde wth system 3., therefore, the system 3. has a weak soluton wth trajectores les n C [, T ], [, ]. 3.. Pathwse unqueness of the soluton for the descretzed equatons. In order to show the pathwse unqueness of solutons to 3., suppose that Y = y,, y and Y 2 = y 2,, y2 are two dfferent weak solutons of , wth the same Brownan moton and same ntal data. Set v := y y 2, =, 2,,, and t [, T ]. Then, we have v t = j= α j v j sds β j j= σy The quadratc varaton V t of v t, satsfes V t = [ σy = y s σy 2 s p y 2 s p ds [ σy s σy 2 s] 2ds s σy 2 y s y 2 s σy s σy 2 y s y 2 s ] 2 s s db s =, 2,,. 2 s {y s y 2 s>} ds {y s y 2 s>} ds. 3.8 Snce y, y 2 [, ], mples that y y 2 [, ]. Further, usng condton 2.2 and then smplfyng, we have [ σy s σy 2 y ] 2 s {y s y 2 s>} ds s y 2 s 2α y s y 2 s {y ds < 2T <. 3.9 s y 2 s>} Therefore, applyng Lemma 3.2 to v t = y y 2 wth ρv =v, we obtan that local tme L t y y 2 = for all =, 2,,. Usng Tanaka s formula

9 336 VIVEK KUAR AND ANKIK KUAR GIRI for the contnuous sem-martngale v t, we get v t = j= α j sgnv sv sds β j sgnv sy j s p y 2 s p ds j= [ sgnv s σy j s σy 2 ] s db s where =, 2,,. Usng the fact that α j and β j are bounded, summng over all =, 2,,, and takng the expectaton, we obtan E = v t = E E,j=,j= α j v sds β j sgnv sy s p y 2 s p ds :=I I For I, we have I E 4 2 E = v s = j= α j v s ds. 3.2 Snce the values of solutons le n nterval [, ], therefore, for I 2, we estmate I 2 E 2 E v s = = j= β j v s ds Insertng 3.2 and 3.22 n to 3.2 and applyng Gronwall s nequalty, we have E = v t =,.e., the weak solutons are pathwse unque. Fnally, by a standard theorem of Yamada and Watanabe [34]or see [, pages 8-9], the exstence of a unque strong soluton s obtaned.

10 STOCHASTIC BURGERS EQUATION WITH NON-LIPSCHITZ COEFFICIENT Tghtness of the Approxmatng Processes In ths secton, we demonstrate the tghtness of the famly of the strong solutons for system of stochastc dfferental equatons Let us denote the polygon approxmaton of y t by f t, y whch s defned as f t, y := Y t, [y] y [y] Y t, [y] [y] y, 4. where t [, T ], y [, ] and [y] = k for y <, so that we have Y t, = y t = f t,, for every t [, T ] and. Suppose q t,, j, t [, T ],, j s the fundamental soluton of the dscrete heat equaton such that t q t,, j = q t, wth boundary condtons q, q t,,, j for all t [, T ], j. Then can be re-wrtten as y t = j= q t, j= j= t,, j. 4.2, j = δ j, 4.3 j j = q t,, = 4.4, j y q t s, q t s,, j β j y s p, j σy sdb s,, where the last ntegral on the rght hand sde represents the sum of Itô stochastc ntegrals. Let us defne the re-scaled formulaton of the Green functon G to the heat kernel q n [, ] by G t, y, j = q t, [y], j q y [y] t, [y], j [y] y. 4.5

11 338 VIVEK KUAR AND ANKIK KUAR GIRI Therefore the lnear nterpolaton of the f t, y, for y [, s f t, y = j= G t, j= j=, j y [ q t s,, j q t s, G t s,, j, j β j y s p y [y] ] β j y s p [y] y ds σy sdb s,, :=f t, y f 2 t, y f 3 t, y, 4.6 where {f l }, for l =, 2, 3, denote the frst, second and thrd summaton on the rght hand sde respectvely. Proposton 4.. For every, the sequence {f t, y : t [, T ]} s tght n the space C [, T ], A, where A = C [, ], [, ]. Proof. By the hypothess 2.2, we have σf t, y c mnf t, y α, f t, y α, α [/2,. 4.7 It can easly be seen from 4. that f [, ] and consequently t mples through condton 4.7 that σ s also bounded by some postve constant. Further, by usng the same technque as used n the proof of Lemma 2.2 and Proposton 2. n [6], for every < T < and µ N, we obtan that there exsts K := Kµ, T such that E ft 3, x f 3 t 2, y 2µ K t t 2 µ/2 x y µ/2 4.8 for every x, y [, ] and t, t 2 [, T ], and µ N and lm sup ft, y ft, y =. 4.9 t,y [,T ] [,] Here, f represents the fundamental soluton of f 2 f y = 2 f y Also, f 2 t, = f 2 t, = and t, k = [q t s, k, k y k s p q t s, k, k ] y k s p ds. Snce mply that q s the fundamental soluton of heat kernel assocated to, we have t f 2 t, k = f 2 t, k f t, k p f t, k p.

12 STOCHASTIC BURGERS EQUATION WITH NON-LIPSCHITZ COEFFICIENT 339 From Theorem 4.2 n [22], we obtan max k f 2 t, k t e pt max k f s, k f 3 s, k ds. 4. Hence, from , and the polygonal form of f 2, we conclude that for any fnte T > and µ N, there exsts K = KT, µ n such a way that E ft 2, x f 2 t 2, y 2µ K t t 2 µ/2 x y µ/2 4.2 for every t, t 2 [, T ], and x, y, and N. Substtutng estmates 4.8, 4.9 and 4.2 nto 4.6 and usng the multdmensonal Totok- Kolmogorov crteron [3, 32] on tghtness we conclude that for every T >, f t, x C[, T ], A and the sequence {f t, x, N} s tght. 5. The Weak Soluton In Secton 3, t s shown that the sequence f = {f t, y, } s tght n C[, T ], A and hence by Prokhorov s Theorem [3, page 59 Chapter ], f s relatvely compact n C[, T ], A, As a consequence there exst a convergent subsequence f k = {f k t, y,, k } of f n C[, T ], A, whch converges weakly to a stochastc process f n C[, T ], A. Applyng the well known Skorohod s representaton Theorem, we get another probablty space Ω, F, F t, P and a sequence of processes f t, y and ft, y adapted to the fltraton { F t } {t [,T ]}, D D n such a way that f f t, y, f f and { f t, y} converges to f almost surely on compact subset of C[, T ], A for any T > as. Also f satsfes the gven boundary condtons n.5. Next, by solvng an equvalent martngale problem to.4.6, we show that ft, y s the requred weak soluton to.4.6. Proposton 5.. For every ϕ C 2 [, ] such that ϕ = ϕ =, we have ϕ t = ft, yϕydy fs, yϕ ydyds s a martngale wth the quadratc varaton ϕ t = f, yϕydy f p s, yϕ ydyds 5. σ 2 ft, yϕ 2 ydyds. 5.2 Proof. Usng the Skorohod representaton theorem after multplyng both the sde by ϕ k n 3. and summng over all k =, 2,,, we get, for fxed C[, T ] [, ], [, ] and C[, T ], A both have equal topologes see [6, page 45]

13 34 VIVEK KUAR AND ANKIK KUAR GIRI, ϕ t = k= k= D k= = k= k ϕ k= f t, k k ϕ f t, k k ϕ f t, k k ϕ k= f, k k ϕ k= f s, k k ϕ k= σ f s, k k= f, k k= f p f p t, k k ϕ k ϕ s, k k ϕ db k s. 5.3 Snce the rght-hand sde on 5.3, each ntegral n the summaton, s an Itô ntegral and hence these are martngale also. Therefore, ϕ t s also a martngale. oreover, ϕ 2 s also an ntegrable functon, therefore k t ϕ t 2 = ϕ σ f s, k 2 db k s E ϕ t 2 = k= T k k ϕ 2 ϕ2 k σ 2 f s, k ds < cϕ, T, 5.4 where cϕ, t s a fnte constant free from and depends only on ϕ and T. Therefore, ϕ t ϕt as, where ϕ t s gven by 5.. Now, snce the quadratc varaton of ϕ t s gven by k t ϕ t = ϕ σ f s, k db k s k t t = σ2 f s, k k ϕ 2 ds. 5.5 therefore, we have lm ϕ t = k σ 2 fs, yϕ 2 ydyds = ϕ t. 5.6

14 STOCHASTIC BURGERS EQUATION WITH NON-LIPSCHITZ COEFFICIENT 34 Now, the man result of the persent work, s as follows: Theorem 5.2. Let f : [, ] [, ] be a contnuous functon and σ satsfes the condtons Then ft,x s a weak soluton of.4.6. Proof. From Chapter 2 n Walsh [33], for the quadratc varaton ϕ t, we can fnd a martngale measure ds, dx wth quadratc varaton νdx, dt = σft, xdtds. Now, as n Kono and Sga [27], we can establsh a space-tme whte nose W t ndependent of dx, ds such that W t ϕ = σfs, x {fs,x {,}}ϕxds, dx {fs,x={,}} ϕx W ds, dx 5.7 where W t corresponds to the space-tme whte nose W ds, dx such that t ϕ = σfs, xϕxw ds, dx 5.8 Therefore, from Proposton 5. and Defnton 2.3, t s proved that f s the weak soluton to.4.6. Ths complete the proof of Theorem 5.2. References. Bertn, L., Cancrn, N., and Jona-Lasno G.: The stochastc Burger equaton, Commun. ath. Phys , Bller, P., Funak, T., and Woyczynsk, W. A.: Fractal Burgers equatons, J. Dffr. Eqn , Bllngsley, P.: Weak Convergence of easures: Applcatons n Probablty, SIA, Phladelfa, Pennsylvanya, Blount, D.: Densty dependent lmts for a nonlnear reacton dffuson model, Ann. Probab , Bonnet, G. and Adler, R. J.: The Burgers superprocess, Stochastc processes and ther applcatons, 727, Boulanba, L. and ellouk,.: On a hgh dmensonal nonlnear stochastc partal dfferental equaton, Stochastc 8624, Burgers, J..: A mathematcal model llustratng the theory of turbulence, Adv. Appl. ath 948, Burgers, J..: The Nonlnear Dffuson Equaton. Assymptotc Soluton and Statstcal Problems, Sprnger, Netherlands Chambers, D. H., Adran, R., on, J. P., Stewart, D. S., and Sung, H. J.: Karhumen-Loeve expanson of Burgers model of turbulence, Phys. Fluds 3998, Cherny, A. S. and Engelbert, H. J.: Sngular Stochastc Dfferental Equatons, Sprnger, DOI:.7/b487, 25.. Cho, H., Temam, R., on, P., and Km, J.: Feedback control for unsteady flow and ts applcatons to Burgers equaton, J. flud ech , Da Prato, G., Debussche, A. and Temam, R.: Stochastc Burgers equaton, Nonlnear Dfferental equatons and applcatons, Basel, Brkhäuser 994, Da Prato, G. and Gatarek, D.: Stochastc Burgers equaton wth correlated nose, Stoch. Stoch. Rep , Dlotko, T.: The one dmensonal Burgers equaton; exstence, unqueness and stablty, Zeszyty Nauk. Unw. Jagelloń. Prace at ,

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