stochastic stochastic parabolic and wave equations with ...itn2012/files/talk/brzezniak.pdf ·...
Post on 29-Jun-2020
2 Views
Preview:
TRANSCRIPT
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
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
Outline
Prerequisites from Differential GeometryPrerequisites from Stochastic AnalysisStochastic geometric wave equationsStochastic geometric heat equationsStochastic Landau-Lifshitz equations with applications toferromagnetism
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
Prerequisites from Differential Geometry
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
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
Prerequisites from Differential Geometry
Prerequisites from Differential Geometry
Assume that M ⊂ Rn:
y : R→ Rn Newton’s second lawy = f (y , y)
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
Prerequisites from Differential Geometry
Prerequisites from Differential Geometry
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
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
Prerequisites from Differential Geometry
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)
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
Prerequisites from Differential Geometry
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)
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
Prerequisites from Differential Geometry
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)
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
Prerequisites from Differential Geometry
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)
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
Prerequisites from Differential Geometry
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)
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
Prerequisites from Differential Geometry
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)
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
Prerequisites from Differential Geometry
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)
)
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
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
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
Prerequisites from Stochastic Analysis
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 .
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
Prerequisites from Stochastic Analysis
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
isometry (with ‖ · ‖T2 being the Hilbert-Schmidt norm)
E|∫ T
0ξ(s)dW (s)|2 = E
∫ T
0‖ξ(s)‖2T2
ds
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
Prerequisites from Stochastic Analysis
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
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 .
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
Prerequisites from Stochastic Analysis
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)
)
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
Prerequisites from Stochastic Analysis
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)
)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
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
Prerequisites from Stochastic Analysis
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)
)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.
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
Prerequisites from Stochastic Analysis
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).
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
Prerequisites from Stochastic Analysis
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.
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
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
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
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)
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
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 |
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 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 |
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 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
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)
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 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
Finite speed of propagation (Huygen’s principle)
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
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.
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
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)
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
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
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
Geometric Wave Equations
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))
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
Geometric Wave Equations
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,
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
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
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
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’)
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 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]
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
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.
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
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.
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
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.
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
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.
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
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
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
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
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
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.
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
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
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
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
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
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
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 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
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 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)
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
Numerical experiments
(joint works with L. Banas and A. Prohl)
Figure: Switching mechanism: u(t , x) for space-time white noisewithn).
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
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 .
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
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〉
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
Notation
Given u ∈ H1,2 we define u ×∆u as a measurable L2-valuedfunction such that
〈u ×∆u, ϕ〉 = 〈∇u,u × (∇ϕ)〉
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
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.
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
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.
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
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.
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
Comments on the proof of Theorem 1
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.
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
Comments on the proof of Theorem 1
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.
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
Comments on the proof of Theorem 1
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.
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
Comments on the proof of Theorem 1
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)
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
Comments on the proof of Theorem 1
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)
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
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)
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
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.
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
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
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
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.
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
Proof of Theorem 4: uniform estimates
Lemma (Let h ∈ L∞ ∩W1,3 and u0 ∈ H1.)
Then for p ≥ 1, β > 12 and T > 0
|un(t)|L2 = |un(0)|L2 , P− a.s.
supn
E
[sup
t∈[0,T ]
|∇un(t)|2pL2
]<∞,
supn
E∫ T
0|un(t)×∆un(t)|L2 dt <∞,
supn
E
(∫ T
0|un(t)×
(un(t)×∆un(t)
)|2L3/2 dt
)p/2
<∞.
supn
E∫ T
0|πn(un(t)×
(un(t)×∆un(t)
))|2H−β dt <∞.
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
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)
.
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
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.
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
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.
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
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
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
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
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
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)
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
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.
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
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.
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
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.
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 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.
top related