stochastic stochastic parabolic and wave equations with ...itn2012/files/talk/brzezniak.pdf ·...

Introduction Stochastic geometric wave equations Stochastic Landau-Lifshitz equations Proof of some of the results on SGWEs Proof of some of the results on SLLGEs

Stochastic Stochastic parabolic and waveequations with geometric constraints

Zdzisław Brzezniak,University of York, UK

based on joint research with M. Ondreját (Prague), B.Goldys & T. Jegaraj (Sydney), A. de Bouard (Paris) and A.

Prohl (Tübingen)

Iasi, 6th July, 2012


Outline

Prerequisites from Differential GeometryPrerequisites from Stochastic AnalysisStochastic geometric wave equationsStochastic geometric heat equationsStochastic Landau-Lifshitz equations with applications toferromagnetism


Prerequisites from Differential Geometry


Let M be compact Riemanian manifold of dimension m withoutboundary which we assume is isometrically embedded into aneuclidean space Rd . In particular, the scalar product in thetangent space Tm, where m ∈ M, is equal to the restriction ofthe scalar product in Rd .

m ∈ Mu, v ∈ TmMn ⊥ TmM, i.e. n ∈ Rd TmM




Assume that M ⊂ Rn:

y : R→ Rn Newton’s second lawy = f (y , y)




Assume that M ⊂ Rn:

y : R→ Rn Newton’s second lawy = f (y , y)

PTy M [y ] = PTy M [f (y , y)]

Dt y = PTy M [f (y , y)]

Covariant derivative



Prerequisites from Differential GeometryThe second fundamental form

γ : (a,b)→ MX (t) ∈ Tγ(t)M

X (t) ∈ Rn

DtX = PTγ(t)M X

Aγ(X , γ) = P(TγM)⊥X

Ap : TpM × TpM → (TpM)⊥, p ∈ M

X = DtX + Aγ(X , γ)



Prerequisites from Differential GeometryThe second fundamental form

γ : (a,b)→ MX (t) ∈ Tγ(t)M

X (t) ∈ Rn

DtX = PTγ(t)M X

Aγ(X , γ) = P(TγM)⊥X

Ap : TpM × TpM → (TpM)⊥, p ∈ M

X = DtX + Aγ(X , γ)

Special case: if X (t) = γ(t), then

γ(t) = Dt γ(t) + Aγ(t)(γ(t), γ(t)

)


Prerequisites from Stochastic Analysis

Wiener process

Assume that (Ω,F ,P) is a probability space with completefiltration F = (Ft )t≥0. An Rd -valued standard F-Wiener processis a family W = (Wt )t≥0 of Rd -valued process such that

W0 = 0,Wt −Ws is Fs independent for t > s ≥ 0,Wt is an N(0, tI) random variable, i.e. gaussian with mean0 and variance t , for t > 0. In other words, the density ofWt is the function p(t , ·), where p(t , x) is the fundamentalsolution of the heat equation

∂tu =12

∆u =12

d∑i=1

∂xi xi u



Itô integral

∫ T

0ξ(s)dW (s) :=

N∑i=1

ξ(si−1)(W (si)−W (si−1)

)for an adapted L(Rd ,Rk )-valued step processξ =

∑Ni=1 ξ(si−1)1[si−1,si ), 0 = s0 < s1 < · · · < sN = T .



Itô integral

∫ T

0ξ(s)dW (s) :=

N∑i=1

ξ(si−1)(W (si)−W (si−1)


∑Ni=1 ξ(si−1)1[si−1,si ), 0 = s0 < s1 < · · · < sN = T . Itô

isometry (with ‖ · ‖T2 being the Hilbert-Schmidt norm)

E|∫ T

0ξ(s)dW (s)|2 = E

∫ T

0‖ξ(s)‖2T2

ds



Itô integral

∫ T

0ξ(s)dW (s) :=

N∑i=1

ξ(si−1)(W (si)−W (si−1)


∑Ni=1 ξ(si−1)1[si−1,si ), 0 = s0 < s1 < · · · < sN = T . Itô

isometry (with ‖ · ‖T2 being the Hilbert-Schmidt norm)

E|∫ T

0ξ(s)dW (s)|2 = E

∫ T

0‖ξ(s)‖2T2

ds

This can be generalized two fold: (i) Replace Rd by a Hilbertspace K with ONB (ej)

∞j=1, so we get a cylindrical Wiener

process W (t) =∑∞

i=1 wj(t)ej , where (wj)∞j=1 are independent

standard R-valued Wiener process,(ii) Replace Rk by a Hilbert space (or even a Banach space ofsufficient geometrical smoothness, so called 2-smooth) E .



Itô integral

If K, resp. E , is a Hilbert, resp. 2-smooth Banach space, then∫ T

0ξ(s)dW (s) :=

N∑i=1

ξ(si−1)(W (si)−W (si−1)

)



Itô integral


0ξ(s)dW (s) :=

N∑i=1

ξ(si−1)(W (si)−W (si−1)

)Burkholder inequality (with ‖ · ‖T2 being the γ-radonifying

norm), if 1 < p <∞,

E supt∈[0,T ]

|∫ t

0ξ(s)dW (s)|p ≤ Cp(T )E

[ ∫ T

0‖ξ(s)‖2T2

ds]p/2



Itô integral


0ξ(s)dW (s) :=

N∑i=1

ξ(si−1)(W (si)−W (si−1)

)Burkholder inequality (with ‖ · ‖T2 being the γ-radonifying

norm), if 1 < p <∞,

E supt∈[0,T ]

|∫ t

0ξ(s)dW (s)|p ≤ Cp(T )E

[ ∫ T

0‖ξ(s)‖2T2

ds]p/2

Naturally extended to progressively measurable γ(K,E)-valuedprocess.



Itô and Stratonovich differential equations

Itô integral allows one to define a solution to a SDEs

dx = g(x) dw + f (x) dt (1)

as a solution to the integral equation (in which the 1st integral isthe Itô one)

x(t) = x(0) +

∫ t

0g(x(s)) dw(s) +

∫ t

0f (x(s)) ds, t ≥ 0. (2)

A solution to (2) is not stable with respect to the piecewiselinear approximations but a solution to Stratonovich problem,where∫ t

0g(x(s))dw(s) :=

∫ t

0g(x(s)) dw(s)+

12

∫ t

0tr(g′(x(s))g(x(s))

)ds, t ≥ 0,

has such a stability property (Wong-Zakai phenomenon).



Brownian Motion on a riemannian manifold

Let M be compact Riemanian manifold of dimension m withoutboundary which we assume is isometrically embedded into aneuclidean space Rd .Let a C∞0 function π : Rd → L(Rd ,Rd ) satisfy

π(m) : Rd → TmM orthogonal projection,m ∈ M

If a ∈ M, a solution x s.t. x(0) = a, to Stratonovich equation

dx = π(x) dw(t) = π(x) dw (3)

is an M-valued Brownian Motion. In particular, the density(w.r.t. the riemannian volume measure) of x(t) is equal tou(t , ·; a) where u(t , ·; a) solves the heat equation (with theLaplace-Beltrami operator) on M such that u(0, ·; a) = δa.


Wave Equations - why they important?

The fundamental equation of wave mechanicsvibrating strings, membranes, 3D elastic bodiesseismic wavesultrasonic waves (detection of flaws in materials)sound waves, water waveselectromagnetic waves

Quantum mechanicsGeneral relativityYang-Mills theoryOptics

Geometric wave equations


Wave Equation - critical points of action functional

Lagrange - Euclidean

L(u) =

∫Rd〈Du,Du〉Eucl + V (u)

〈Du,Du〉Eucl :=∣∣∇u

∣∣2+

∆u =12

V ′(u)

Lagrange - Minkowski

L(u) =

∫R1+d〈Du,Du〉Mink + V (u)

〈Du,Du〉Mink := −∣∣ut∣∣2 +

∣∣∇u∣∣2

utt −∆u = −12

V ′(u)


Deterministic Wave Equation

utt −∆u + f (u,Du) = 0 in R× Rd

u(0, x) = u0(x), ut (0, x) = v0(x)

u(t) = K (t) ∗ u0 + K (t) ∗ v0 −∫ t

0K (t − s) ∗ f (u(s),Du(s)) ds

K (t) = F−1

sin(t |y |)|y |


Stochastic Wave Equation

utt −∆u + f (u,Du) = g(u,Du)W in R+ × Rd

u(0, x) = u0(x), ut (0, x) = v0(x)

u(t) = K (t) ∗ u0 + K (t) ∗ v0 −∫ t

0K (t − s) ∗ f (u(s),Du(s)) ds

+

∫ t

0K (t − s) ∗ g(u(s),Du(s)) dW

K (t) = F−1

sin(t |y |)|y |




u(0, x) = u0(x), ut (0, x) = v0(x)

u(t) = K (t) ∗ u0 + K (t) ∗ v0 −∫ t

0K (t − s) ∗ f (u(s),Du(s)) ds

+

∫ t

0K (t − s) ∗ g(u(s),Du(s)) dW

1 EW (t , x)W (s, y) = min t , s Γ(x , y)

2 EW (t , x)W (s, y) = min t , s Γ(x − y)




u(0, x) = u0(x), ut (0, x) = v0(x)

u(t) = K (t) ∗ u0 + K (t) ∗ v0 −∫ t

0K (t − s) ∗ f (u(s),Du(s)) ds

+

∫ t

0K (t − s) ∗ g(u(s),Du(s)) dW

Finite speed of propagation (Huygen’s principle)


spatially homogenous Wiener process

Consider(Ω,F ,P), F = (Ft ), Γ ∈ S ′

µ = Γ spectral measureW is S ′(Rd )-valued F-Wiener process:

E〈W (t), ϕ〉2 = t〈Γ, ϕ ∗ ϕ(−·)〉, ϕ ∈ S (Rd ).

Then the RKHS K can be explicitly found andLaw of W (t) is translation invariant.If Γ ∈ C(Rd ) then EW (t , x)W (s, y) = t ∧ s Γ(x − y).

Examples:

Euclidean free field dµdx = (2π)−

d2 (|x |2 + m2)−1.

Space-time white noise Γ = δ0.


Geometric Wave Equations

M a compact Riemannian manifoldu0(x) ∈ M for x ∈ Rd

v0(x) ∈ Tu0(x)M for x ∈ Rd

Definition

A map u : R× Rd → M is a solution of a GWE iff

Dtut −∑d

i=1 Dxi uxi = 0 in R× Rd

u(0, x) = u0(x)ut (0, x) = v0(x)


Wave Equation - critical points again

u : R× Rd → M

Lagrange - Minkowski

L(u) =

∫R1+d

⟨Du,Du

⟩Mink dt dx⟨

Du,Du⟩

Mink = −∣∣ut∣∣2TuM +

∣∣∇u∣∣2TuM

Dtut =d∑

i=1

Dxi uxi



M is a submanifold in Rn

A is the second fundamental form of M

γ : R→ M is a curve

LemmaThe following properties holds:

1 Dt∂tγ(t) = ∂ttγ(t)− Aγ(t)(∂tγ(t), ∂tγ(t))

2 Aγ(t)(∂tγ(t), ∂tγ(t)) ⊥ ∂ttγ(t)− Aγ(t)(∂tγ(t), ∂tγ(t))



M is a submanifold in Rn

A is the second fundamental form of M

γ : R→ M is a curve

LemmaThe following properties holds:

1 Dt∂tγ(t) = ∂ttγ(t)− Aγ(t)(∂tγ(t), ∂tγ(t))

2 Aγ(t)(∂tγ(t), ∂tγ(t)) ⊥ ∂ttγ(t)− Aγ(t)(∂tγ(t), ∂tγ(t))

Corollary

| · | and 〈·, ·〉 denote the norm and the inner product in Rn

〈∂ttγ(t)−Aγ(t)(∂tγ(t), ∂tγ(t)), ∂ttγ(t)〉 = |∂ttγ(t)−Aγ(t)(∂tγ(t))|2,


Geometric Wave Equations cont.

ExampleIf M is a sphere then u is a solution of the SGWE if and only if

utt −∆u + (|ut |2 − |∇u|2)u = 0|u| = 1


Blow-up and the global existence: deterministic case

Global existencestrong solutions d = 1 (Gu 80’)weak solutions d = 1 (Zhou 99’)weak solutions d = 2 (Müller, Struwe 96’)M is a sphereM is a homogeneous riemannian manifold (Freire 96’)

Blow-upd ≥ 3 (Cazenave, Shatah, Tahvildar-Zadeh 98’)

Non-uniquenessd ≥ 3 (Cazenave, Shatah, Tahvildar-Zadeh 98’)


Stochastic wave equations with values in R

This is a very well developed subject. Let me just mentionfew names.[Carmona and Nualart 88’, Mueller 97’][Mueller 97’, Dalang and Frangos 98’, Millet and Sanz-Solé99’, Peszat and Zabczyk 00’][Peszat and Zabczyk 00’, Peszat 02’, Dalang and Nualart’04][Ondrejat 07’, 11’, Kim 06’ and many others]


The existence and uniqueness of global strongsolutions to SGWE in R1+1

We will always assume thatEW (t , x)W (s, y) = min t , sΓ(x − y)

M a compact Riemannian manifoldgp : TpM × TpM → TpM, p ∈ M is C1

b

Dtut − Dxux = g(u,ut ,ux )W (4)

Theorem (1: ZB and M. Ondreját (JFA 2007))

If Γ ∈ C2b then An intrinsic solution⇔ an extrinsic solution.


The existence and uniqueness of global strongsolutions to SGWE in R1+1

We will always assume thatEW (t , x)W (s, y) = min t , sΓ(x − y)

M a compact Riemannian manifoldgp : TpM × TpM → TpM, p ∈ M is C1

b

Dtut − Dxux = g(u,ut ,ux )W (4)

Theorem (1: ZB and M. Ondreját (JFA 2007))

If Γ ∈ C2b then and u0 ∈ H2

loc(R,M) and v0 ∈ H1loc(R,TM) are

such that v0(x) ∈ Tu0(x)M, x ∈ R, then ∃a unique global strongsolution to (4). This solution has continuous H2

loc × H1loc

trajectories.


The existence of a global weak solution for SGWE inR1+1

M a compact Riemannian manifoldg,g0, . . . ,gd continuous

Dtut − Dxux = [g(u) + g0(u)ut +d∑

i=1

gi(u)uxi ]W

Theorem (2: ZB & M.Ondrejat (Comm PDEs (2011))If Γ ∈ Cb then there exists a global weak solution to SGWEprovided u0 ∈ H1

loc(R,M) and v0 ∈ L2loc(R,TM) are such that

v0(x) ∈ Tu0(x)M, for a.a. x ∈ R.

The solution has weakly continuous H1loc × L2

loc trajectories.


Global Existence in R1+d

M a compact Riemannian homogeneous spaceg,g0, . . . ,gd continuous

Dtut − Dxux = [g(u) + g0(u)ut +d∑

i=1

gi(u)uxi ]W

Theorem (3: ZB and M. Ondreját (Ann Prob to appear))If Γ ∈ Cb then there exists a global weak solution to SGWEprovided u0 ∈ H1

loc(R,M) and v0 ∈ L2loc(R,TM) are such that

v0(x) ∈ Tu0(x)M, for a.a. x ∈ R.

The solution has weakly continuous H1loc × L2

loc trajectories.


Physical background

We consider a ferromagnetic material fillinga domain D ⊂ Rd , d ≤ 3,

u(t , x) the magnetic moment at x ∈ D at time t ,

For temperatures not too high (below Curie point)

|u(t , x)| = 1, x ∈ D


Energy functional

Landau-Lifshitz 1935, Gilbert 1955Every configuration φ : D → S2 ⊂ R3, φ ∈ H1 of magneticmoments minimizes the energy functional

E(φ) =a1

2

∫D|∇φ|2dx +

12

∫Rd|∇v |2dx −

∫D

H · φdx

exchange energy magnetostatic energy,

H- given external field.

∆v = ∇ ·(1Dφ

), on Rd


Landau-Lifshitz-Gilbert equation

H(u) = −DuE(u) = a1∆u −∇v + H

∂u∂t = λ1u ×H(u)− λ2u × (u ×H(u)) on D

∂u∂n = 0 on ∂D

|u0(x)| = 1 on D

where λ2 > 0 and from now on

λ1 = λ2 = 1.


Connection with harmonic maps problem

E(φ) =12

∫D|∇φ|2dx

∂u∂t

= −u × (u ×∆u)

butu × (u ×∆u) = (u ·∆u)u − |u|2∆u,

|u|2 = 1 on D then

u · ∇u = 0, ⇒ u ·∆u = −|∇u|2

We obtain heat flow of harmonic maps:

∂u∂t

= ∆u + |∇u|2u


Previous works

A. Visintin 1985: weak existence, d ≤ 3,Chen and Guo 1996, Ding and Guo 1998, Chen 2000,Harpes 2004: existence and uniqueness of partiallyregular solutions, d = 2C. Melcher 2005: existence of partially regular solutions,d = 3,R. V. Kohn, M. G. Reznikoff, E. Vanden-Eijnden 2007,large deviationsA. Desimone, R. V. Kohn, S. Müller, F. Otto 2002, thin filmapproximationsR. Moser 2004, thin film approximations, magnetic vortices


Thermal noise

E(φ) = · · · −∫

DH · φ

Néel 1946: H = noise.

H = hdW

h : D → R3, W Brownian Motion

important problem:¯

to study noise-induced transition betweenminima of E


Stochastic Landau-Lifshitz-Gilbert-Equation I

H(u) = −DuE(u) = ∆u −∇v + hdW

∂u∂t = u ×H(u)− u × (u ×H(u)) on D

∂u∂n = 0 on ∂D

|u0(x)| = 1 on D

F dW is a Stratonovitch integral:

F (u) dW =12

F ′(u) · F (u)dt + FdW


Stochastic Landau-Lifshitz-Gilbert-Equation II

H(u) = ∆u − Pu + hdW

∂u∂t = u ×H(u)− u × (u ×H(u)) on D

∂u∂n = 0 on ∂D

|u0(x)| = 1 on D

(5)


Numerical experiments

(joint works with L. Banas and A. Prohl)

Figure: Switching mechanism: u(t , x) for space-time white noisewithn).


Integration by parts

∆N Neumann Laplacian

D (∆N) =

u ∈ H2 :

∂u∂n

= 0, on ∂D

.

Lemma

If v ∈ H1 and u ∈ D (∆N) then∫D〈u ×∆N , v〉 dx =

∫D〈∇u, (∇v)× u〉dx .


Weak martingale solution

A system (Ω,F ,F,P,W ,u), where F = (Ft )t≥0 is a solution to(5) iff for every T > 0 and φ ∈ C∞

(D,R3),

u(·) ∈ C(

[0,T ];H−1,2), P− a.s.

E supt≤T|∇u(t)|2L2 <∞, |u(t , x)|R3 = 1, Leb ⊗ P− a.e.

〈u(t), ϕ〉 − 〈u0, ϕ〉 =

∫ t

0〈∇u, (∇ϕ)× u〉 ds +

∫ t

0〈G(u)Pu, ϕ〉ds

−∫ t

0〈∇u,∇(u × ϕ)× u〉 ds +

∫ t

0〈G(u)h, ϕ〉 dW (s).

G(u)f = u × f + u × (u × f ), 〈Pu,∇ϕ〉 = 〈u,∇ϕ〉


Notation

Given u ∈ H1,2 we define u ×∆u as a measurable L2-valuedfunction such that

〈u ×∆u, ϕ〉 = 〈∇u,u × (∇ϕ)〉


Weak existence for d = 3

Theorem (4: ZB, T Jegaraj and B Goldys (AMReX, 2012))

Let u0 ∈ H1, h ∈ L∞ ∩W1,3 and |u0(x)| = 1. Then there exists a solution(Ω,F ,F,P,W , u) to the SLLGEs such that for all T > 0

E∫ T

0|u ×∆u|2 dt <∞,

u(t) = u0 +

∫ t

0u ×∆ uds −

∫ t

0u × (u ×∆u)ds

+

∫ t

0G(u)Pu ds +

∫ t

0G(u)h dW (s),

u ∈ Cα(

[0,T ],L2), α <

12.


Non-uniqueness

Non-uniqueness for d = 3

Theorem (6: ZB and Anne de Bouard (in preparation))

There exists u0 ∈ H1 such that |u0(x)| = 1 and a non-trivial h ∈ L∞ ∩W1,3

such that there exist infinity many solution (Ω,F ,F,P,W , u) to the SLLGEssuch that for all T > 0

E∫ T

0|u ×∆u|2 dt <∞,

u(t) = u0 +

∫ t

0u ×∆ uds −

∫ t

0u × (u ×∆u)ds

+

∫ t

0G(u)h dW (s),

u ∈ Cα(

[0,T ],L2), α <

12.


Comments on the proof of Theorem 1

To avoid unnecessary difficulties stemming from thelanguage of differential geometry we assume that

M = S2 ⊂ R3.

We take a tubular neighbourhood O of M as

O := x ∈ R3 :12< |x | < 2

M = S2 ⊂ O ⊂ R3.



Define an involution map

h : O 3 x 7→ x|x |2∈ O

We extend h to the whole R3. Note that

h(x) = x , x ∈ R3 iff x ∈ M, x ∈ R3.

Define a map

Sq(x , y) =12

(d2q h)((dqh)(x), (dqh)(y)

), q, x , y ∈ R3.



For M = S2 ⊂ R3 the SGWE takes the form (∆u = uxx )

utt −∆u + (|ut |2 − |∇u|2)u = g(u,ut ,ux )W , ; |u| = 1 (1)

u(0) = u0, ut (0) = v0 (2)

Instead of SGWE (1-2) we consider SPDE with values inR3:

utt −∆u = Su(ut ,ut )− Su(ux ,ux ) + G(u,ut ,ux )W (3)

u(0) = u0, ut (0) = v0 (2)

where G is a suitable extension of g to R3 (which is insome sense h-invariant on O) and with initial data asearlier.



Problem (3-2) has a unique continuous H2loc-valued solution

u(t), t ∈ [0, τ), where τ is the exit time of u(t) from O.By the construction of S and of G, a process u(t) := h(u(t)),t ∈ [0, τ) is also a solution of (3).Because u0(x) ∈ S2 and v0(x) ∈ Tu(x)S2, by the construction ofthe involution map h, u satisfies the same initial condition as u,i.e. (2). By the uniqueness of solutions to (3-2), u = u, i.e.

h(u(t , x)) = u(t , x), x , t < τ.

Since the fixed point set of h in O is S2 we infer that

u(t , x) ∈ S2, x , t < τ. (4)



Then using (4) we can employ some energy estimates toconclude that τ =∞ and that u is a solution to (1-2).

utt −∆u + (|ut |2 − |∇u|2)u = g(u,ut ,ux )W|u| = 1

(1)

u(0) = u0, ut (0) = v0 (2)


Proof of Theorem 3: Intro

The assumptions of the Theorem imply that (Moore-SchlaflyTheorem)M2 There exists a C∞-class function F : Rn → [0,∞) such that

M = x : F (x) = 0 and F is constant outside some largeball in Rn.

M3 There exist a finite sequence (Ai)Ni=1 of skew symmetric

linear operators on Rn such that for each i ∈ 1, · · · ,N,〈∇F (x),Aix〉 = 0, for every x ∈ Rn, (6)Aip ∈ TpM, for every p ∈ M. (7)

M4 There exist a family(hij)

1≤i,j≤N of C∞-class R-valuedfunctions on M such that

ξ =N∑

i=1

N∑j=1

hij(p)〈ξ,Aip〉RnAjp, p ∈ M, ξ ∈ TpM. (8)


Proof of Theorem 3: Main idea

Method: penalization and approximation

∂ttUm = ∆Um−m∇F (Um)+f m(Um,∇(t ,x)Um)+gm(Um,∇(t ,x)Um) dW m

We get tightness of sequences the sequences Um, V m = ∂tUm

and of, for every i ∈ 1, · · · ,N, Mni := 〈V m,AiUm〉Rd . We

prove that these three sequences have limits U,V ,Mi , whichsatisfy certain integral equations. Finally, using the previouspage, we show that the process U is a solution.


Proof of Theorem 4

Uniform estimates for the Galerkin approximations un,Tightness of the family of probability laws L (un) : n ≥ 1,Identification of the limit


Proof of Theorem 4: Galerkin approximations

en∞n=1 eigenbasis of ∆N in L2 and

πn orthogonal projection onto Hn = lin e1, . . . ,en .

dun = (Gn (un) ∆un (un) + Gn (un) Pun) dt + Gn (un) h dW ,un(0) = πnu0

Gn(u)f = πn (un × f )− πn (un × (un × f ))

For every n ≥ 1 there exists a unique strong solution in Hn.


Proof of Theorem 4: tightness

Lemma (For any p ≥ 2, q ∈ [2,6) and β > 12 )

the set of laws L (un) : n ≥ 1 is tight on

Lp (0,T ;Lq) ∩ C(

0,T ;H−β)

.


Proof of tightness

For β > 12 , α < 1

2 and p > 2

supn

E |un|2Wα,p(0,T ;H−β) <∞.

Then for −β < γ < 1

Lp (0,T ;H1) ∩Wα,p (0,T ;H−β)⊂ Lp (0,T ;Hγ) ,

with compact embedding by Flandoli&Gatarek 1995 and tightness on

Lp (0,T ;Hγ) ⊂ Lp (0,T ;Lq)

follows. Again by Flandoli&Gatarek 1995

Wα,p (0,T ;H−β1)⊂ C

(0,T ;H−β

), β > β1, αp > 1,

with compact embedding.


Doss-Sussman method

Simplified stochastic Landau-Lifshitz-Gilbert equation:¯

du = [u ×∆u − u × (u ×∆u)]dt + (u × h) dW , t > 0, x ∈ D,

∂u∂n = 0, t ≥ 0, x ∈ ∂D,

u(0, x) = u0(x), x ∈ D.


Doss-Sussman method: auxiliary facts

Bx = x × a, x ∈ R3

Then etB is a group of isometries and

etB(x × y) =(

etBx)×(

etBy), x , y ∈ R3.

For h ∈ H2 putGφ = φ× h, φ ∈ L2

Then(etG) is again a group of isometries in L2 and

etGφ = φ+ (sint)Gφ+ (1− costt)G2φ


Doss-Sussman method III: transformation

Letv(t) = e−W (t)Gu(t).

Thendvdt

= v × R(t)v − v × (v × R(t)v) (9)

whereR(t)v = e−W (t)G∆eW (t)Gv


Doss-Sussman method: transformation continued.

Lemma

For φ ∈ H2

e−tG∆etGφ = ∆φ+

∫ t

0e−sGCesGφds,

with

Cφ = φ×∆h + 2∑

i

(∂φ

∂xi

)×(∂h∂xi

).

If |v |R3 = 1 then we obtaindvdt = R(t)v + v × R(t)v +

∣∣∇etBv∣∣2 v

v(0) = u0.(10)


Doss-Sussman: regularity

Theorem (5: ZB and B Goldys (in preparation))

Let h ∈ H2 and u0 ∈W1,4. Then for every ω there existsT = T (ω) > 0 such that equation (10) has a unique solution uon [0,T ) with the property

u ∈ C(

0,T ;W1,4)

and|v(t , x)|R3 = 1, t < T , x ∈ D.


Proof of Theorem 5

Equation (10) is a strongly elliptic quasi-linear systemShow that there exists a mild solution v ∈ C

(0,T ;W1,4)

Use maximal regularity and ultracontractivity of the heatsemigroup to "bootstrap" the regularity of solutions.Show that |v(t , x)| = 1.

Note that (9) can be written in the form

dvdt

= ∆v + v ×∆v + |∇v |2v + v × L(t , v) + v × (v × L(t , v))

with L linear and|L(t , v |L2 ≤ C|v |H1

where C is a finite random variable.


Theorem (6: ZB and B Goldys (in preparation))

The process u(t) = eW (t)Gv(t) is a unique solution of thestochastic Landau-Lifshitz-Gilbert equation on [0,T ) satisfyingfor every n ≥ 1 conditions

E∫ T∧n

0|∆Nv(s)|22 <∞

E supt≤T∧n

|∇v(t)|2 <∞,

Proof: takeu(t) = eW (t)Gv(t).

Use the Itô formula to obtain the estimates.


Stochastic LLG in 1D

D = [0,1]

pathwise uniquenessmaximal regularitylarge deviations

Pathwise uniqueness

Theorem (7: ZB, T Jegaraj and B Goldys (in preparation))

Let u1,u2 : [0,T ]× L2 be two progressively measurablesolutions, such that ui ∈ C

([0,T ],L2) ∩ L8 ([0,T ],H1). Then

u1(·) = u2(·) P-a.s.

stochastic stochastic parabolic and wave equations with ...itn2012/files/talk/brzezniak.pdf ·...

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