February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde Infinite Dimenional Analyi, Quantum Probability and Related Topic c World Scientific Publihing Company STOCHASTIC QUASI-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTION B. P. W. FERNANDO Department of Mathematic and Information Technology, Univerity of Leoben, Franz Joef-Strae 18, 87 Leoben, Autria. panif71@gmail.com S. S. SRITHARAN Center for Deciion, Rik, Control and Signal Intelligence (DRCSI, Naval Potgraduate School Monterey, California 93943, USA. ritha@np.edu Communicated by (H. H. Kuo In thi paper we conider a tochatic counterpart of Toio Kato quai-linear partial differential equation and prove exitence and uniquene of mild olution. Keyword: Stochatic quai-linear partial differential equation; hyperbolic partial differential equation; cylindrical Wiener noie; mild olution. AMS Subject Claification:6H15, 35Q35, 6H3, 35R15 1. Introduction In Ref. 9, Toio Kato preented a unified treatment a large cla of important quailinear partial differential equation including ymmetric hyperbolic ytem of firt order, 15 quai-linear wave equation, Korteweg-de Vrie equation, 14 Navier-Stoke and Euler equation, 16 equation of compreible invicid fluid, 11 ideal magnetohydrodynamic equation, Eintein field equation of general relativity, 4 coupled Maxwell and Dirac equation of quantum electrodynamic, 5 etc. In thi paper we tudy a tochatic counterpart of Toio Kato evolution ubject to Gauian cylindrical Wiener noie.. Local olution for tochatic quai-linear equation Let u conider the Cauchy problem for the tochatic quai-linear equation with additive cylindrical Wiener noie, du(t + A(t, u(tu(tdt = S α dw(t, t T, u( = u, (.1 Correponding author: +1-831-656-66 1
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde in a eparable Hilbert pace. Here A(t, u(tu(t i a quai-linear operator in the ene of Kato paper, 7, 8 and 9. S i an iomorphim from a eparable Hilbert pace to. More detail on the operator S, S α with α > and Hilbert pace, are given below. The baic idea for etablihing olvability reult for local olution for (.1 i a follow. Firt we conider a linear equation with certain valued tochatic proce v(t, du(t + A(t, v(tu(tdt = S α dw(t, t T, u( = u. (. We firt prove that (. ha a mild olution u(t and then we define a map v u = Λ(v and ue the contraction mapping theorem to obtain a fixed point for Λ(. which will be the olution of (.1. Let u begin with a et of aumption a in Ref. 9: (1 i a eparable Hilbert pace. There i another eparable Hilbert pace which i continuouly and denely embedded in. There exit an iomorphim S from onto. The norm of i obtained uch that S become an iometry. ( A i a linear operator from to and it dependence on t and v(t i characterized a a map from [, T ] B into G(, 1, β (ee preliminary ection and P.8, Ref. 9. Here B i an open ball in and β i a real number. G(, 1, β i a cla of linear emigroup generator uch that, for all t [, T ], [, and v B, e A(t,v L(, e β (.3 Kato call uch operator quai-m-accretive and point out that both m- accretive 1 and table family of operator 1 are pecial cae of quai-m-accretive operator. (3 For all t [, T ] and v B, SA(t, vs 1 = A(t, v + B(t, v, (.4 with B(t, v L(, (pace of linear bounded operator from to. And alo, B(t, v L(, C 1 (.5 Moreover the equation (.4 hould be atified in trict ene. That i x D(A(t, v iff S 1 x D(A(t, v and A(t, vs 1 x. (4 For all t [, T ] and v B, we have A(t, v L(, (in the ene that D(A(t, v and retriction of A(t, v to i in L(, with A(t, v L(, C. (a For all v B, A(t, v i continuou in the L(, -norm. (b For all t [, T ], A(t, v i Lipchitz-continuou, that i, A(t, v 1 A(t, v L(, C v 1 v,
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde 3 where C i a contant which depend on the radiu of the ball B. (5 Let a be the center of the ball B. For all t [, T ] and v B, A(t, va with A(t, va C 3. (6 The operator S (α : i a trace cla operator with α > (that i Tr(S (α <. Remark.1. Notice that, we can eaily deduce A(t, v 1 A(t, v L(, C v 1 v C (r 1 δ v 1 v δ, (.6 for all < δ < 1, t [, T ] and v 1, v B. Remark.. It i eay to obtain the etimate and U v L(, e β T (.7 U v L(, e (β+c 1 T for ome value T T, (.8 by uing the reult 3.3, in Ref. 9 together with aumption 3 and S L(, = S 1 L(, = 1..1. Preliminary Reult In thi ubection, we preent ome preliminary reult for carrying out the contruction of local mild olution of tochatic quai-linear equation with cylindrical Wiener noie. Lemma.1. Let U 1 (t, and U (t, be evolution operator on t T, aociated with operator A 1 (t and A (t on t T thoe are atifying aumption, 3 and 4(A1 repectively. Aume that D (A 1 (t D (A (t, t T. Then, U 1 (t, y U (t, y = Proof. See ection 1 in Ref. 8. U 1 (t, r [A 1 (r A (r] U (r, ydr, y. (.9 Theorem.1. Aume that there exit a unique evolution operator U(t, on t T aociated with the operator A(t on t T, which atifie aumption, 3 and 4(A1. Then G(t, = SU(t, S 1 i alo an evolution operator on t T that atifie following two propertie: (1 G(t, L(, i trongly continuou in t T, with G(, = I. ( G(t, = G(t, rg(r,, r t. Where S i an iomorphim from to. Proof. See ection 5 in Ref. 8.
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde 4 Lemma.. Suppoe that the two Hilbert pace, atify the aumption 1. Now aume that a ubet of i convex, cloed and bounded. Then it i alo cloed in. Proof. See Lemma 7.3 in Ref. 9... Contruction of Local Mild Solution Now we preent the main reult of thi paper which i a local olvability theorem for tochatic Kato evolution. Theorem.. Suppoe that the aumption (1-(6 are atified. Let u B a.. and E u <. Then (.1 ha a unique local mild olution u L (Ω; C(, T ; for ome time T > with T T. Proof. Let B be an open ball in containing initial data u with the radiu r and the center a. Denote S be the et of all trongly meaurable function v : Ω [, T ], uch that v(t B a.., for all t [, T ] and t v(t i continuou a a -valued function over [, T ] a... Here < T T i a ome time that will be determined later. Denote A v (t = A(t, v(t, for all t [, T ] and v S. The following two lemma are direct conequence of correponding determinitic reult (ee P. 41-4 in Ref. 9 will be ueful to contruct the family of evolution operator {U v (t, } on t T aociated with {A v (t}. Lemma.3. A v ( : [, T ] L(, i continuou for ome value T T, a.. Lemma.4. B v ( : Ω [, T ] L(, i trongly meaurable, where B v (t = SA v (ts 1 A v (t with B v (t = B(t, v(t and T T. Recall that A v (t G(, 1, β (That i A v (t i a quai-m-accretive operator for t [, T ] due to aumption 1. Therefore {A v (t} i a table family with tability index 1 and β. The above reult together with the Lemma.6 and.7 fulfill the aumption of Theorem 1 in Ref. 9 for the family {A v (t}. Thi immediately implie that there exit a unique family of evolution operator {U v (t, } defined on t T. The tochatic mild olution for the linear problem (. in term of the evolution operator {U v (t, } on t T i repreented a follow: u(t = U v (t, u + U v (t, S α dw(. (.1 Since U v (t, i depend upon the tochatic proce v(t, there i a problem aociate with the mild formulation (.1. The problem i random evolution proce U v (t, i not right adapted to the filtration driven by cylindrical Wiener proce.
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde 5 Therefore the tochatic integral in (.1 i not well defined in Itô ene. By applying integration by part formula, we can avoid thi difficulty a follow. ( u(t = U v (t, u U v (t, S α dw(θ U v (t, A v t ( S α dw(θ d. Now conider E up u(t t T E up U v (t, u t T + E up t Uv (t, S α dw(θ t T + E up U v (t, A v ( S α dw(θd t T (.11 Now apply reult (.8 to firt term in right hand ide of (.11 to get E up U v (t, u e(β+c 1 T E u <. (.1 t T Conider the econd term in right hand ide of (.11 E up Uv (t, S α dw(θ = E up SUv (t, S 1 S (α 1 dw(θ t T t T = E up t T Gv (t, e β T Tr(S (α 1 <. S (α 1 dw(θ (.13 The above reult can be obtained by uing theorem.1, aumption (6 with the reult (.7. Let u etimate lat term in the inequality (.11 E up U v (t, A v ( S α dw(θd = E up U v (t, A v (S 1( S (α 1 dw(θ d t T t T = E up U v (t, ( S 1 A v ( + S 1 B v ( ( S (α 1 dw(θ d t T = E up G v (t, ( A v ( + B v ( ( S (α 1 dw(θ d t T E up t G v (t, t T L(, Av ( t L(, S (α dw(θ d + E up t G v (t, t T L(, Bv ( t L(, S (α 1 dw(θ d T Ce β T t E up S (α dw(θ d + T C1e β T t E up S (α 1 dw(θ d t T t T
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde 6 T C e β T E up t T + T C 1e β T E up t T ( S (α dw(θ ( S (α 1 dw(θ + + 4 T e β T ( C Tr ( S (α + C 1Tr ( S (α 1 <. S (α dw(θ S (α 1 dw(θ d d (.14 In the reult (.14 econd and third equalitie hold by aumption (3 and theorem.1 repectively. By applying aumption (3, (4 together with remark., one can obtain econd inequality of the reult (.14. Finally by applying Burkholder-Gundy inequality, etimate can be completed. By combining (.1, (.13 and (.14, we get, E up u(t t T e(β+c 1 T E u + eβ T Tr(S (α 1 + 4 T e β T ( C Tr ( S (α + C 1Tr ( S (α 1 <. Hence u(t L (Ω; C(, T ; with ufficiently mall time T T. Now we contruct the mild local olution for tochatic quai-linear equation (.1 uing Banach fixed point theorem. Set ũ(t = u(t a and notice that dũ(t + A v (tũ(tdt = S α dw(t A v (ta, ũ( = u a. (.15 Then u(t a = U v (t, (u a U v (t, U v (t, A v ( S α dw(θ S α dw(θd U v (t, A v (ad. (.16 By following imilar tep in etimate (.1, (.13 and (.14 together with the aumption (5, E up u(t a t T E up U v (t, (u a t T + E up t Uv (t, S α dw(θ t T ( + E up U v (t, A v t ( S α dw(θ d t T + E up U v (t, A v (ad t T [ e (β+c1 T u a + C T ] 3 + e β T Tr(S (α 1
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde 7 + 4 T e β T ( C Tr ( S (α + C 1Tr ( S (α 1 <. (.17 Right hand ide can be made le than radiu r by chooing T T ufficiently mall. Then there exit a continuou modification of u(t which belong to S a... Thi implie v u = Λ(v mapping from S to S for uch a particular choice of T. Let u define a ditance function on S a follow, d(v, w = E up v(t w(t. (.18 t T Under the ditance function d(.,., S i a complete metric pace, ince a cloed ball in i a cloed ubet of (ee Lemma 7.3 in Ref. 9. Now we will how that Λ(. i trict contraction map from S to S if T T i ufficiently mall. Conider, d(λ(v 1, Λ(v = E up Λ(v 1 (t Λ(v (t t T E up (U v 1 (t, U v (t, (u a t T + E up t T + E up t T + E up t T (Uv 1 (t, U v (t, S α dw(θ (U v1 (t, A v1 ( U v (t, A v ( ad [ (U v 1 (t, A v 1 ( U v (t, A v ( ( ] S α dw(θ d (.19 Applying Lemma.1 together with Cauchy-Schwartz inequality and aumption 4, (A to the firt term on the right hand ide of (.19, E up (U v1 (t, U v (t, (u a t T = E up t T E up t T T E T E T T [ U v 1 (t, r (A v 1 (r A v (r U v (r, (u adr ] U v1 (t, r (A v1 (r A v (r U v (r, (u a dr U v 1 (t, r (A v 1 (r A v (r U v (r, (u a dr U v1 (t, r L(, (Av1 (r A v (r L(, Uv (r, L(, (u a dr T C e (β+c1 T u a E up v 1 (t v (t t T (..
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde 8 By following imilar argument a in (. accompanying with Burkholder-Gundy inequality and Remark.1 with δ = 1 we can etimate the econd term a follow. E up (Uv 1 (t, U v (t, S α dw(θ t T = E up t T [ [ U v 1 (t, r (A v1 (r A v (r U v (r, ( ] S α dw(θ dr E up (t, r (A v 1 (r A v (r U Uv1 v (r, S α dw(θ t T T T t E (t, r (A Uv1 v1 (r A v (r U v (r, S α dw(θ dr T dr T E U v 1 (t, r L(, (r A (Av1 v (r L(, Uv (r, t L(, S (α 1 dw(θ T e (β+c 1 T E up (A v 1 (r A v (r 4 t T L(, E up t 4 S (α 1 dw(θ dr t T ] dr C (r 1 T ( E e (β+c 1 T Tr(S (α 1 up v 1 (t v (t t T (.1 Now conider the third term appear in the right hand ide of (.19, E up (U v 1 (t, A v 1 ( U v (t, A v ( ad t T E up U v1 (t, (A v1 ( A v ( ad t T + E up (U v 1 (t, U v (t, A v (ad t T (. We can eaily how that the firt term appear in right hand ide of (. i bounded by E up U v1 (t, (A v1 ( A v ( ad T C e β T a t T E up v 1 (t v (t t T. (.3 Now conider lat term in (., E up (U v 1 (t, U v (t, A v (ad t T = E up t T U v 1 (t, r (A v 1 (r A v (r U v (r, A v (adrd
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde E up t T T E T E [ T r T r ] U v 1 (t, r (A v 1 (r A v (r U v (r, A v (a ddr U v1 (t, r (A v1 (r A v (r U v (r, A v (a ddr U v 1 (t, r L(, (Av 1 (r A v (r L(, C C 3 T 3 e (β+c1 T E up v 1 (t v (t t T. r U v (r, L(, Av (a ddr 9 (.4 Firt equality in (.4 hold ince Lemma.1. Then by applying Fubini Theorem and Cauchy-Schwartz inequality one can obtain econd inequality in (.4. Finally by uing the fact that U v1 L(,, A v 1, A v L(,, U v L(,, and A v (a a.. together with aumption 4, 5 one can complete the etimate. Finally, according to the tep aociated with third term we can etimate lat term in (.19 a follow. E up t T E up t T + E up t T [ (U v1 (t, A v1 ( U v (t, A v ( [ U v 1 (t, (A v 1 ( A v ( [ (U v 1 (t, U v (t, A v ( ( ( ( ] S α dw(θ d ] S α dw(θ d ] S α dw(θ d (.5 Ue aumption (4, (6, Remark. and Burkholder-Gundy inequality to etimate firt term in (.5, [ ( E up U v 1 (t, (A v 1 ( A v t ] ( S α dw(θ d t T (.6 C T e β T Tr(S (α 1 E up t T v 1 (t v (t. Now conider lat term of (.5, [ E up (U v1 (t, U v (t, A v ( t T [ = E up t T [ r E up t T T r T E ( ] S α dw(θ d ( U v 1 (t, r (A v 1 (r A v (r U v (r, A v t ( Uv 1 (t, r (A v1 (r A v (r U v (r, A v ( Uv 1 (t, r (A v 1 (r A v (r U v (r, A v ( ] S α dw(θ drd S α dw(θ ddr S α dw(θ ddr ]
February 5, 14 1:53 WSPC/INSTRUCTION FILE Mild olution for quailinear pde 1 T T E T U v 1 (t, r L(, (Av 1 (r A v (r L(, T E U v 1 (t, r L(, (Av 1 (r A v (r L(, r U v (r, t L(, SAv (S 1 S (α 1 dw(θ T r U v (r, L(, Av ( ddr T E U v 1 (t, r L(, (Av 1 (r A v (r L(, r U v (r, ( L(, A v ( + B v ( S (α 1 dw(θ T T E U v 1 (t, r L(, (Av 1 (r A v (r L(, r ( U v (r, L(, A v ( t L(, S (α 1 dw(θ 4C C T ( 1 3 e (β+c 1 T E v 1 (t v (t up t T ( up S (α dw(θ + up S (α 1 dw(θ t T t T 8C C T ( 1 3 e (β+c 1 T ( TrS (α ( + TrS (α 1 ( E ddr + B v ( L(, up t T. v 1 (t v (t <. (.7 Firt equality in (.7 hold ince Lemma.1. Then by applying Fubini Theorem and Cauchy-Schwartz inequality one can obtain econd inequality in (.4. By uing the fact that U v 1 L(,, A v1, A v L(, and U v L(,, one can obtain the third inequality. The aumption (3, A v1, A v L(, and A v 1, B v L(, lead to the ixth inequality. Finally applying aumption (6, Remark.1 with δ = 1 and Burkholder-Gundy inequality, etimate can be completed. By combining all the reult (.,(.1, (., (.3, (.4, (.5 and (.7 we immediately ee that Λ i a contraction map from S to S if T T i ufficiently mall. Therefore from the contraction mapping theorem, it follow that Λ ha a unique fixed point and which i the olution of (.1 with the propertie tated in theorem.5. Acknowledgment Thi reearch work ha been performed while the firt author held a National Reearch Council Potdoctoral fellowhip in Naval Potgraduate School, California, USA. The econd author work ha been upported by the Army Reearch Probability and Statitic Program. Firt author alo would like to thank Autrian Science Fund (FWF: P3591 for the partial upport. S α dw(θ S (α 1 dw(θ ddr ddr
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