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86d67cee40a48d36eda36a99cafd9846f55c59f6 | subsection | 39 | 49 | Examples of CMC-surfaces in | All these examples are invariant under the 1-parametric group of isometries \lbrace I(\theta ) \times \tau _\theta ,\, \theta \in \mathbb {R}\rbrace of M^2(\epsilon ) \times \mathbb {R}, where \tau _\theta : \mathbb {R}\rightarrow \mathbb {R} is \tau _\theta (t) = t + \theta \sqrt{b} and I(\theta ): M^2(\epsilon ) \rig... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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1ef0fb0cf098cde6a3da1e0244752afe256a7300 | subsection | 40 | 49 | Examples of CMC-surfaces in | So, by a direct computation we getH(x, y) = \frac{\sqrt{b}}{2} \left(\frac{\sqrt{b}}{h^{\prime }(x)}\bigl ((c-h(x)) \psi _x - \psi _y\bigr ), \frac{h^{\prime }(x)}{\epsilon (a-h^2(x))} \right)From this we have that \Psi is a CMC-immersion with |H|^2 = b/4 and it is straightforward to check that the associated Abresh-Ro... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
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0cdb9b28938b131d1515554551f32c6dd8719bd2 | subsection | 41 | 49 | Examples of CMC-surfaces in | In the second case, if \Phi is given by \Phi (x,y)=(\alpha (x),\beta (y)) with \alpha and \beta curves in M^2(\epsilon ) with constant curvature k_{\alpha } and k_{\beta } and |\alpha ^{\prime }|=|\dot{\beta }|=1, following Theorem REF , the Frenet data of \Phi _1 and \Phi _2 are given by u = 0, H_1 = H_2, \nu _1 = \nu... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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2b5abc114d4748d1f7a1234a9cf34b4960313d89 | subsection | 42 | 49 | Examples of CMC-surfaces in | Following the notation, for each real number 0<H<1/2, \Psi _0=(\psi _0,\eta _0):]-\pi /2,\pi /2[\times \mathbb {R}\rightarrow \mathbb {H}^2\times \mathbb {R} given by\begin{split}
\psi _0(x,y) &= \frac{1}{\sqrt{1-4H^2}} \left(\tan x,\frac{\sinh y}{\cos x} + 2H^2e^{-y}\cos x,\frac{\cosh y}{\cos x} -2H^2e^{-y}\cos x\righ... | {
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"raw": "M.L. Leite. An elementary proof of the Abresh-Rosenberg theorem on constant mean curvature immersed surfaces in \\mathbb {S}^2\\times \\mathbb {R} and \\mathbb {H}^2\\times \\mathbb {R}. Quart.J. Ma... | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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fb3f669b5794469b7a553ee01665a8c80520f7cd | subsection | 43 | 49 | Examples of CMC-surfaces in | Furthermore, \epsilon (a-h^2(x)) > b because the minimum for the function \operatorname{dn} is \sqrt{1-\kappa ^2} and it is easy to see that a (1-\kappa ^2) > b if and only if a > b.Now the function f appearing in Proposition REF is given byf(x, y) = y + \frac{1}{\sqrt{a-b}}\arctan \left( \frac{\operatorname{cn}(\sqrt{... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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43ad0f45dec5af1a904d2ea2a626343a9df2dcf1 | subsection | 44 | 49 | Examples of CMC-surfaces in | \end{split}We consider the local isometry t\in \mathbb {R}\mapsto \frac{\sqrt{b}}{\sqrt{a-b}}e^{i\frac{\sqrt{a-b}}{\sqrt{b}}t}\in \mathbb {S}^1(\frac{\sqrt{b}}{\sqrt{a-b}}) and the CMC-immersion\hat{\Phi }_{a,b}=(\phi _{a,b},\hat{\eta }_{a,b}):\mathbb {R}^2\rightarrow \mathbb {S}^2\times \mathbb {S}^1(\frac{\sqrt{b}}{\... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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9e0ccf8fe8de92179701f885065cdac9119a8cc9 | subsection | 45 | 49 | Examples of CMC-surfaces in | In the first case, looking at the immersion we obtain that y=\hat{y}. In the other two cases, \operatorname{cn}\hat{x}=-\operatorname{cn}x and \operatorname{dn}\hat{x}=\operatorname{dn}{x}. So, looking again at the immersion we easily get that\cos (\kappa \hat{y}-\kappa y)=\frac{a\operatorname{dn}^2x-\operatorname{cn}^... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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b66cb0bfedf0b98a3295f08713052ab1d9c1e636 | subsection | 46 | 49 | Compact PMC-surfaces | In this section we are going to prove some properties of compact PMC-surfaces of M^2(\epsilon )\times M^2(\epsilon ). Let \Phi :\Sigma \rightarrow M^2(\epsilon )\times M^2(\epsilon ) be an PMC-immersion of an orientable surface \Sigma . We define two vector fields X_j,\,j=1,2, tangent to \Sigma as the tangential compon... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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759b655c7fe8e0aa51a31bd9f4f8b3f4328dd162 | subsection | 47 | 49 | Compact PMC-surfaces | If \epsilon =1, then the degrees of \phi and \psi are zero.
If K\ge 0, then either \Phi (\Sigma ) is a CMC-sphere of \mathbb {S}^2\times \mathbb {R} with 4|H|^2\ge 1 or \Phi (\Sigma ) is a torus of Example REF .
If \epsilon =1 then K cannot be negative. If \epsilon =-1, then K cannot be less than -1.
If some of the ... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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bf666f24548331a30141ecc4f32b558e40ed4461 | subsection | 48 | 49 | Compact PMC-surfaces | \Theta _1=0, then from (REF ), (REF ) and (REF ) we obtain that16|H|^2\int _{\Sigma }K\,dA=\int _{\Sigma }(4|H|^2+\epsilon (1-C_1^2))^2\,dA.In particular \int _{\Sigma }K\,dA\ge 0 and again either \Sigma is a sphere and so \Theta _2=0 or \Sigma is a torus in Example REF with \Theta _1=0, which is impossible looking at ... | {
"cite_spans": []
} | 0807.1808 | Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2 | [
"Francisco Torralbo",
"Francisco Urbano"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
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4d260c805ca5b3a90191ae14729b3f28f80f3b58 | abstract | 0 | 2 | Abstract | Scanning tunneling microscopy and spectroscopy (STM/S) measurements in the
superconducting dichalcogenide 2H-NbS2 show a peculiar superconducting density
of states with two well defined features at 0.97 meV and 0.53 meV, located
respectively above and below the value for the superconducting gap expected
from single ban... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.166407 | 0807.1809 | Superconducting density of states and vortex cores of 2H-NbS2 | [
"I. Guillamon",
"H. Suderow",
"S. Vieira",
"L. Cario",
"P. Diener",
"P. Rodiere"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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7f8e719667114b1120b9e140781952761397b18c | subsection | 1 | 2 | Acknowledgments. | We acknowledge J. Martial at IMN for her help in sample preparation, and discussions with A. Mel'nikov, A.I. Buzdin, F. Guinea, J.G. Rodrigo, V. Crespo and J.P. Brison. The Laboratorio de Bajas Temperaturas is associated to the ICMM of the CSIC. This work was supported by the Spanish MEC (Consolider Ingenio 2010, MAT a... | {
"cite_spans": []
} | 10.1103/PhysRevLett.101.166407 | 0807.1809 | Superconducting density of states and vortex cores of 2H-NbS2 | [
"I. Guillamon",
"H. Suderow",
"S. Vieira",
"L. Cario",
"P. Diener",
"P. Rodiere"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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abbfb94577d03dbc6872ba6ad00b402d9f4a102a | abstract | 0 | 75 | Abstract | We deal with a class of abstract nonlinear stochastic models, which covers
many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD
models and 2D magnetic B\'enard problem and also some shell models of
turbulence. We first prove the existence and uniqueness theorem for the class
considered. Our main r... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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... |
116f5c298e55d03447414431fc41ad23b80abe5a | subsection | 1 | 75 | Introduction | In recent years there has been a wide-spread interest in the study of
qualitative properties of stochastic models which describe
cooperative effects in fluids by taking into account macroscopic
parameters such as temperature or/and magnetic field. The
corresponding mathematical models consists in coupling the stochasti... | {
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{
"arxiv_id": "",
"doi": "",
"end": 1697,
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"raw": "G. Da Prato & J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.",
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"start": 1602
... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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ebb8317bb9ce0d7b649beb7101fc4646ddbaa634 | subsection | 2 | 75 | Introduction | However, since we deal with an
abstract hydrodynamical model with a forcing term which contains a stochastic control
under a minimal set of hypotheses,
the argument requires substantial modifications compared to that of
or .
It relies on a two-step Gronwall lemma (see Lemma REF
below and also ).Our main result (see T... | {
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{
"arxiv_id": "",
"doi": "",
"end": 225,
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"raw": "S. S. Sritharan & P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stoch. Proc. and Appl. 116 (2006), 1636–1659.",
"source_ref_id": "2aea9... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
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] | 2,008 | en | Mathematics | [
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a78f0779b9696f02b423523abe801020a6f3f831 | subsection | 3 | 75 | Introduction | Thus the elegant method in , which is
based on some compactness property of the family of solutions in
{\mathcal {C}}([0,T],V^{\prime }) obtained by means of the Ascoli theorem,
cannot be applied here, and we use a technical time
discretization.
Let us also point out that due to the bilinear term which arises in hydrod... | {
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"end": 450,
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"raw": "J. Ren & X. Zhang, Freidlin-Wentzell Large Deviations for Stochastic Evolution Equations, J. of Functional Analysis, 254 (2008), 3148–3172.",
"source_ref_id": "cf1a570a55dda6be3b6fced11afaccad... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
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41511aa4cf43cc657599fedd542fae9d0e6f4a4a | subsection | 4 | 75 | Description of the model | Let (H, |.|) denote a separable Hilbert space, A be an
(unbounded) self-adjoint positive linear operator on H. Set
V=Dom(A^{\frac{1}{2}}). For v\in V set \Vert v\Vert =
|A^{\frac{1}{2}} v|. Let V^{\prime } denote the dual of V (with respect
to the inner product (.,.) of H). Thus we have the Gelfand
triple V\subset H\su... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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e42cbbb28b2a9d0ba331d7462416290f4aeae48d | subsection | 5 | 75 | Description of the model | On the other hand,
if we put in (REF ) \eta C_1^{-1} u_3 instead of u_3,
then we recover (REF ) with C_\eta =C_1C_2 \eta ^{-1}
Thus the requirements (REF ) and(REF ) are equivalent.
If for u_3\ne 0 we put now \eta =\Vert u_1\Vert _{\mathcal {H}}\Vert u_2\Vert _{\mathcal {H}}\Vert u_3\Vert ^{-1} in
(REF ) with C_\eta =C... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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a3a79dc99bddca2dc4b6d62fa77e904c22419f9f | subsection | 6 | 75 | Motivation | The main motivation for the condition (C1) is that it covers a wide class
of 2D hydrodynamical models including the following ones.
An element of {\mathbb {R}}^2 is denoted u=(u^1,u^2). | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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19ff77d631a2cd5fb36de58d1d1c69f523bb7ff4 | subsection | 7 | 75 | 2D Navier-Stokes equation | Let D be a bounded, open and simply connected domain of {\mathbb {R}}^2.
We consider the Navier-Stokes equation with
the Dirichlet (no-slip) boundary conditions:\partial _t u - \nu \Delta u + u\nabla u + \nabla p =f , \quad \mbox{\rm div}\, u=0
~~\mbox{ in }~~D,\qquad u=0\quad \mbox{on}\quad \partial D,where u= (u^1(x,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1172,
"openalex_id": "",
"raw": "R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd Edition, SIAM, Philadelphia, 1995.",
"source_ref_id": "b753560f1f05b144715b0a15a4a22c70d34b5472",
"start": 727
... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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9794aef42ab7c317c3748ecfe4bb6a17b3af1afc | subsection | 8 | 75 | 2D Navier-Stokes equation | For this we only need to
shift the spectrum away from zero by changing A into A+Id and
introducing R=-Id. | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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0ebaf37231171bdaff52ce9d9cca3733808219be | subsection | 9 | 75 | 2D magneto-hydrodynamic equations | We consider magneto-hydrodynamic (MHD) equations
for a viscous incompressible resistive fluid in a 2D domain D,
which have the form
(see, e.g., ):\partial _tu-\nu _1\Delta u+
u \nabla u= -\nabla \left(p+\frac{s}{2} |b|^2\right)
+s b \nabla b+
f,\partial _tb-\nu _2\Delta b+
u \nabla b=
b\nabla u+
g,{\rm div\,}u=0, \quad... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 461,
"openalex_id": "",
"raw": "R. Moreau, Magnetohydrodynamics, Kluwer, Dordrecht, 1990.",
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... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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08084f3a17b64e751e54eb8c76f6218202de6897 | subsection | 10 | 75 | 2D magneto-hydrodynamic equations | We also set V=V_1\times V_2 and define
B : V\times V \rightarrow V^{\prime } by the relation\langle B(z_1,z_2), z_3\rangle =\langle B_1(u_1,u_2), u_3\rangle -\langle B_1(b_1,b_2), u_3\rangle +\langle B_1(u_1,b_2), b_3\rangle -
\langle B_1(b_1,u_2), b_3\ranglefor z_i=(u_i,b_i)\in V=V_1\times V_2, where B_1 is given by
(... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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17d758ebc8f58829b478eea3235bd4b929492f9a | subsection | 11 | 75 | 2D
Boussinesq model for the Bénard convection. | The next example is the following coupled system of Navier-Stokes and heat equations
from the Bénard convection problem
(see e.g. and the references therein).
Let D =(0, l) \times (0, 1) be a rectangular domain in the
vertical plane, (e_1, e_2) the standard basis in
{\mathbb {R}}^2 and x=(x^1,x^2) an element of {\mathb... | {
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"raw": "C. Foias, O. Manley & R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension. Nonlinear Analysis 11 (1987), 939–967.",
"source_ref_id": "dc3c8f26... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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fd7894862fc2aca8ad7a22605abdd80329f57421 | subsection | 12 | 75 | 2D
Boussinesq model for the Bénard convection. | We also denoteV_3 = & \left\lbrace u\in H_{(3)}\cap \left[H^1(D)\right]^2,\; u |_{x^2=0}=u |_{x^2=1}=0,\;
u \; \mbox{is $l$-periodic in}\; x^1 \right\rbrace , \\
V_4= & \left\lbrace \theta \in H^1(D),\; \; \theta |_{x^2=0}=\theta |_{x^2=1}=0,\;
\theta \; \mbox{is $l$-periodic in}\; x^1
\right\rbrace .Let A_3 be the Sto... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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2e929cd5652c4121a0861bc131f8e4f6cdeb2472 | subsection | 13 | 75 | 2D magnetic Bénard problem. | This is the Boussinesq model coupled with magnetic field (see ).
As above let D =(0, l) \times (0, 1) be a rectangular domain in the
vertical plane, (e_1, e_2) the standard basis in
{\mathbb {R}}^2. We consider the equations\partial _t u + u { \nabla }u-\nu _1 u̥ + \nabla \left(p+\frac{s}{2} |b|^2\right)
-s b \nabla b ... | {
"cite_spans": [
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"arxiv_id": "",
"doi": "",
"end": 64,
"openalex_id": "",
"raw": "G.P. Galdi & M. Padula, A new approach to energy theory in the stability of fluid motion, Arch. Rational Mech. Anal. 110 (1990), 187–286.",
"source_ref_id": "842866e6191a6d7dc64b63c80276b2bc863... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
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] | 2,008 | en | Mathematics | [
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59757e74d54ecc90f79e86b50389509ca2bab288 | subsection | 14 | 75 | 2D magnetic Bénard problem. | The bilinear operator B is defined by\langle B(z_1,z_2), z_3\rangle & = &\langle B_1(u_1,u_2), u_3\rangle -\langle B_1(b_1,b_2), u_3\rangle \\ & &
+\, \langle B_1(u_1,b_2), b_3\rangle -
\langle B_1(b_1,u_2), b_3\rangle + \sum _{i=1,2} \int _D u_1^i \, \partial _i \, \theta _2\; \theta _3 \, dxfor z_i=(u_i,\theta _i,b_i... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
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c563100496ff3417f25eb3ef98b0b7f625561540 | subsection | 15 | 75 | 3D Leray | The theory developed in this paper can be also applied to some 3D models.
As an example we consider 3D Leray \alpha -model
(see ; for recent development of this model we refer to , and
to the references therein).
In a bounded 3D domain D we consider the following
equations:& \partial _t u - \nu \Delta u + v\nabla u + \... | {
"cite_spans": [
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"end": 213,
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"raw": "J. Leray, Essai sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math. 63 (1934), 193–248.",
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}... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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99f2dc3c507f7713894f3ace88486e3c6efbcf9e | subsection | 16 | 75 | 3D Leray | Furthermore, Hölder's inequality
and the embedding H^1(D)\subset L^6(D) imply that for u_1, u_2,
u_3\in V,| \langle B(G_\alpha u_1,u_2)\, ,\, u_3\rangle | &
\le & C\Vert u_2\Vert \, |G_\alpha u_1|_{L^6(D)} \, | u_3|_{L^3(D)} \le C\Vert u_2\Vert \; \Vert G_\alpha u_1\Vert \; | u_3|_{L^3(D)} \\
& \le &
C\Vert u_2\Vert \;... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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79ba2c2e757e04bc78b330b44dbc2106cc8abbb2 | subsection | 17 | 75 | Shell models of turbulence | Let H be a set of all sequences u=(u_1, u_2,\ldots ) of complex numbers
such that \sum _n |u_n|^2<\infty . We consider H as a real Hilbert space
endowed with the inner product (\cdot ,\cdot ) and the norm |\cdot | of the form(u,v)={\rm Re}\,\sum _{n=1}^\infty u_n v_n^*,\quad |u|^2 =\sum _{n=1}^\infty |u_n|^2,where v_n^... | {
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"raw": "K. Ohkitani & M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence, Prog. Theor. Phys. 89 (1989), 329–341.",
"sou... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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6601f642271466a5fb6035ccf995e2e10f72d43d | subsection | 18 | 75 | Shell models of turbulence | In
both cases the equation (REF ) is an infinite sequence of ODEs.One can easily show (see for the GOY model and
for the Sabra model) that
the trilinear form\langle B(u,v), w\rangle \equiv {\rm Re}\, \sum _{n=1}^\infty [B(u,v)]_n\, w_n^*possesses the property (REF ) and also satisfies the inequality\left|\langle B(u,v... | {
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"raw": "D. Barbato, M. Barsanti, H. Bessaih, & F. Flandoli, Some rigorous results on a stochastic Goy model, Journal of Statistical Physics, 125 (2006) 677–716.",
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deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
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] | 2,008 | en | Mathematics | [
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f9fa46df8c7fbb8c5d15a26fcc406a207dbaa551 | subsection | 19 | 75 | Stochastic model | We will consider a
stochastic external random force f
of the equation in (REF ) driven by a Wiener process W
and whose intensity may depend on the solution u. More precisely,
let Q be a linear
positive operator in the Hilbert space H which belongs to the trace class,
and hence is compact. Let H_0 = Q^{\frac{1}{2}} H. T... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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41dde09b4aa5f82bd1f119fe3f5fd937e9cc86b7 | subsection | 20 | 75 | Stochastic model | For details concerning this Wiener process
we refer to , for instance.The noise intensity \sigma : [0, T]\times V \rightarrow L_Q(H_0, H) of the stochastic perturbation
which we put in (REF ) is
assumed to satisfy the following growth and Lipschitz conditions:Condition (C2): \sigma \in C\big ([0, T] \times V; L_Q(H_0, ... | {
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... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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1026d671e528e36ff5013ae235955ee842309ca9 | subsection | 21 | 75 | Stochastic model | Thus we also need to consider the corresponding shifted problem.To describe a set of admissible random shifts we introduce the class
\mathcal {A} as the set of H_0-valued
(\mathcal {F}_t)-predictable stochastic processes h such that
\int _0^T |h(s)|^2_0 ds < \infty , \; a.s.
LetS_M=\Big \lbrace h \in L^2(0, T; H_0): \i... | {
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"raw": "A. Budhiraja, P. Dupuis & V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems. Ann. Prob. 36 (2008), 1390–1420.",
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deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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d9b7cf989e58b3a0953c0cf50e6c8b3d390bec2d | subsection | 22 | 75 | Stochastic model | Define\mathcal {A}_M=\lbrace h\in \mathcal {A}: h(\omega ) \in S_M, \; a.s.\rbrace .In order to define the stochastic control equation, we introduce another intensity
coefficient \tilde{\sigma } and also nonlinear feedback forcing \tilde{R}
(instead of R)
which satisfyCondition (C3): (i) {}\;{\tilde{\sigma }}
\in C\big... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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d14d93428ba3e2794d2e8406113e3d318f8dc351 | subsection | 23 | 75 | Stochastic model | Under Conditions (C2) and (C3) we consider the nonlinear
SPDE with initial condition
u_h(0)=\xi :d u_h(t) + \big [ A u_h(t) + B\big (u_h(t) \big ) + \tilde{R} (t, u_h(t)) \big ]\, dt
= \sigma (t,u_h(t))\, dW(t) + \tilde{\sigma }(t, u_h(t)) h(t)\, dt.Fix T>0 and let X: = C\big ([0, T]; H\big ) \cap L^2\big (0, T;V\big )... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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bedac87ca5d4bcf8c0f97eb4d91bbd283bd935f3 | subsection | 24 | 75 | Stochastic model | Note that this solution is a strong one in the probabilistic meaning, that is written
in terms of stochastic integrals with respect to the given Brownian motion W.The following assertion shows that equation (REF ), as well
as (REF ), has a unique solution in X, and the X-norm of the solution u_h to (REF )
satisfies a p... | {
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"Igor Chueshov",
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0.1455078125,
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0.1... |
4324f3e090c0889951cfe918fc90a285a4469ef2 | subsection | 25 | 75 | Large deviations | We consider large deviations using a weak convergence approach
, , based on variational representations of
infinite dimensional Wiener processes.
Let \varepsilon >0 and let u^\varepsilon denote the solution to the following equationdu^\varepsilon (t) + [A u^\varepsilon (t) +B(u^\varepsilon (t)) +
\tilde{R} (t,u^\vareps... | {
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deviations | [
"Igor Chueshov",
"Annie Millet"
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1a1f928728888a9f20fee7291cf727d2be271249 | subsection | 26 | 75 | Large deviations | Define {\mathcal {G}}^0: C([0, T], H_0) \rightarrow X by
{\mathcal {G}}^0(g)=u_h for g=\int _0^. h(s)ds \in {\mathcal {C}}_0
and {\mathcal {G}}^0(g)=0 otherwise.
Since the argument below requires some information about the difference of the
solution at two different times, we need an additional assumption about the
re... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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5e991a9cf69cf983af8128362af3b68bb7dd15a2 | subsection | 27 | 75 | Large deviations | Given N>0, h\in {\mathcal {A}}_M,
and for t\in [0,T], letG_N(t)=\Big \lbrace \omega \, :\, \Big (\sup _{0\le s\le t} |u_h^\varepsilon (s)(\omega )|^2 \Big )\vee \Big ( \int _0^t \Vert u_h^\varepsilon (s)(\omega )\Vert ^2 ds \Big ) \le N\Big \rbrace .As in Proposition REF ,
we can use a relaxed form of condition (C3 (i)... | {
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"start": ... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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7e6240702d0df09fa1f7eea4913c5d83f7c65d1e | subsection | 28 | 75 | Large deviations | Let h\in {\mathcal {A}}_M, \varepsilon \ge 0;
for any s\in [0,T], Itô's formula yields|u_h^\varepsilon (\psi _n(s))-u_h^\varepsilon (s)|^2 =2\int _s^{\psi _n(s)} (u_h^\varepsilon (r)-u_h^\varepsilon (s), d u_h^\varepsilon (r))
+\varepsilon \int _s^{\psi _n(s)}|\sigma (u_h^\varepsilon (r))|^2_{L_Q}d r .Therefore
I_n(h,\... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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e670d482bcac7cf8f40c0cee612e73eefadb1258 | subsection | 29 | 75 | Large deviations | \big \langle B( u_h^\varepsilon (r))\, , \, u_h^\varepsilon (r)-u_h^\varepsilon (s)\big \rangle \, dr\Big ) , \\
I_{n,6}&=&- 2 \, {\mathbb {E}}\Big ( 1_{G_N(T)} \int _0^T \!\! ds \int _s^{\psi _n(s)} \!\!
\big (\tilde{R} (u_{h}^\varepsilon (r))\, , \, u_h^\varepsilon (r)-u_h^\varepsilon (s)\big )\, dr\Big ) .Clearly G_... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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0015b07d427366363405cf36ca1fb3976d8bab98 | subsection | 30 | 75 | Large deviations | \big ( K_0+K_1\, |u_h^\varepsilon (r)|^2 +
K_2\, \Vert u_h^\varepsilon (r)\Vert ^2\big )\, \Big ( \int _{(r-c2^{-n})\vee 0}^r ds\Big ) \, dr
\Big ]^{\frac{1}{2}} \\
&\le & C_1 2^{-\frac{n}{2}}for some constant C_1 depending only on K_i, \tilde{K}_i, i=0,1,2, L_j, \tilde{L}_j, j=1,2,
R_1, M,
\varepsilon _0, N and T.
The... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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d602f915e8a653fe79a520b9db4fc35d59250524 | subsection | 31 | 75 | Large deviations | \big (\tilde{K}_0 +\tilde{K}_1|u_h^\varepsilon (r)|^2 + \tilde{K}_2 \Vert u_h^\varepsilon (r)\Vert ^2
\big )^{\frac{1}{2}}\, |h(r)|_0 |\, u_h^\varepsilon (r)-u_h^\varepsilon (s)|\, dr\Big )
\\
& \le 4 \sqrt{N} \; {\mathbb {E}}\int _{0}^T 1_{G_N(T)}
|h(r)|_0 (\tilde{K}_0 +\tilde{K}_1 N + \tilde{K}_2 \Vert u_h^\varepsilo... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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58bfa9d786a501b76bec4020a081366adadbbc5c | subsection | 32 | 75 | Large deviations | \! ds \int _s^{\psi _n(s)} \!\!\!
dr \big [ - \Vert u_h^\varepsilon (r)\Vert ^2 + \Vert u_h^\varepsilon (r)\Vert \Vert u_h^\varepsilon (s)\Vert \Vert \big ]\Big ) \\
&\le &
\frac{1}{2}\; {\mathbb {E}}\Big ( 1_{G_N(T)} \int _0^T ds \; \Vert u^\varepsilon _h(s)\Vert ^2 \,
\int _s^{\psi _n(s)} dr \Big ) \le c \; N \; 2^{-... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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2acd8d46efacee4bb2df223e780f754951a9ab18 | subsection | 33 | 75 | Large deviations | \int _0^T \!\!\!
dr \ \Vert u^\varepsilon _h(r)\Vert ^2 \Big ) \le CN 2^{-n}.Using (REF ), we deduce that on G_N(T) we have\int _0^T \!\!\! \Vert u^\varepsilon _h(s)\Vert ^4_{\mathcal {H}}ds\le a_0^2 \sup _{s\in [0,T]}| u^\varepsilon _h(s)\Vert ^2
\int _0^T \!\!\! \Vert u^\varepsilon _h(s)\Vert ^2 ds\le a_0^2 N^2.ThusI... | {
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} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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fca5b1556cc9158c0ba662e3282b93cdf759cc96 | subsection | 34 | 75 | Large deviations | Let u_{h_\varepsilon }, or strictly speaking, u^\varepsilon _{h_\varepsilon }, be
the solution of the corresponding stochastic control equation
with initial condition u_{h_\varepsilon }(0)=\xi \in H:d u_{h_\varepsilon } + [Au_{h_\varepsilon } +B(u_{h_\varepsilon })+\tilde{R}(t,u_{h_\varepsilon })]dt
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deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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0f4281721b82e1491c792d28a0da48f9b1bab5b4 | subsection | 35 | 75 | Large deviations | That is, as \varepsilon \rightarrow 0,
{\mathcal {G}}^\varepsilon \Big (\sqrt{\varepsilon } \big ( W_. + \frac{1}{\sqrt{\varepsilon }} \int _0^. h_\varepsilon (s)ds\big ) \Big ) converges in
distribution to {\mathcal {G}}^0\big (\int _0^. h(s)ds\big ) in X.Since {\mathcal {A}}_M is a Polish space (complete
separable me... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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5ea8616a07da5fcde5ba6a61ce402bc942605236 | subsection | 36 | 75 | Large deviations | To lighten notations, we will
write (\tilde{h}_\varepsilon , \tilde{h}, \tilde{W}^\varepsilon )=(h_\varepsilon ,h,W).Let U_\varepsilon =u_{h_\varepsilon }-u_h; then U_\varepsilon (0)=0 andd U_\varepsilon + \big [AU_\varepsilon & +B(u_{h_\varepsilon })-B(u_h)+\tilde{R}(t,u_{h_\varepsilon })-\tilde{R}(t,u_h)\big ]dt
\\
&... | {
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} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
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0.... |
49df63e527dd802558e9e2fead34dd88ab194fc6 | subsection | 37 | 75 | Large deviations | \big ( {C}_{\frac{1}{2}} \, \Vert u_h(s)\Vert ^4_{\mathcal {H}}+
R_1 + \sqrt{L_1} |h_\varepsilon (s)|_0\big ) |U_\varepsilon (s)|^2 ds,whereT_1(t,\varepsilon )&=& 2\sqrt{\varepsilon }\int _0^t \big ( U_\varepsilon (s), \sigma (s,u_{h_\varepsilon }(s))\, dW(s) \big ), \\
T_2(t,\varepsilon )&= & \varepsilon \int _0^t (K_... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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4e6efbf17fb03f55f182c1b6d31580825650c420 | subsection | 38 | 75 | Large deviations | Fix N>0 and for t\in [0,T] letG_N(t)&=&\Big \lbrace \sup _{0\le s\le t} |u_h(s)|^2 \le N\Big \rbrace \cap \Big \lbrace \int _0^t \Vert u_h(s)\Vert ^2 ds \le N
\Big \rbrace , \\
G_{N,\varepsilon }(t)&=& G_N(t)\cap \Big \lbrace \sup _{0\le s\le t} |u_{h_\varepsilon }(s)|^2 \le N\Big \rbrace \cap \Big \lbrace \int _0^t \V... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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0cce69515ea02b498085d2127cc439150421bc26 | subsection | 39 | 75 | Large deviations | We also use here the fact that by (REF )\int _0^T \!\!\! \Vert u_h(s)\Vert ^4_{\mathcal {H}}ds\le a_0\sup _{s\in [0,T]}| u_h(s)|^2
\int _0^T \!\!\! \Vert u_h(s)\Vert ^2 ds\le a_0 N^2\quad \mbox{on $G_{N,\varepsilon }(T)$.}Using again (REF ) we deduce that for some constant
\tilde{C}=C(T,M,N),
one has for every \varepsi... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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78f232eb36c97412d67c94e542a528ba899f2da2 | subsection | 40 | 75 | Large deviations | Indeed, by the Burkholder-Davis-Gundy inequality, (C2) and the definition of G_{N,\varepsilon }(s),
we have{\mathbb {E}}(\lambda _\varepsilon ) & \le 6\sqrt{\varepsilon } \; {\mathbb {E}}\Big \lbrace \int _0^T 1_{G_{N,\varepsilon }(s)} \, |U_\varepsilon (s)|^2 \;
|\sigma (s, u_{h_\varepsilon }(s))|^2_{L_{Q}}
ds\Big \rb... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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ca9017c60411fac531823ae11820af4568a63826 | subsection | 41 | 75 | Large deviations | For any n,N \ge 1, if we set t_k=kT2^{-n} for 0\le k\le 2^n, we obviously have{\mathbb {E}}\Big ( 1_{G_{N,\varepsilon }(T)}\sup _{0\le t\le T} |T_3(t,\varepsilon )| \Big )
\le 2\; \sum _{i=1}^4 \tilde{T}_i(N,n, \varepsilon )+ 2 \; {\mathbb {E}}\big ( \bar{T}_5(N,n,\varepsilon )\big ),where\tilde{T}_1(N,n,\varepsilon )=... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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3d3aeb809b5dcf1cd691ece50882360304722ffe | subsection | 42 | 75 | Large deviations | Using Schwarz's inequality and (C2) we deduce for \bar{C}_4=C(N,M)
and any \varepsilon \in ]0, \varepsilon _0]\tilde{T}_4(N,n,\varepsilon )&\le {\mathbb {E}}\Big [ 1_{G_{N,\varepsilon }(T)} \sup _{1\le k\le 2^n}
\big (K_0+K_1| u_h(t_k)|^2
\big )^{\frac{1}{2}} \int _{t_{k-1}}^{t_k}\!\! |h_\varepsilon (s)-h(s)|_0
\, ds \... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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10d0c57cb028d8304ba2ceb1382df80e53b84496 | subsection | 43 | 75 | Large deviations | Furthermore,
\bar{T}_5(N,n,\varepsilon ,\omega )
\le C(K_0,K_1,N, M) and hence the dominated convergence theorem proves that for any
fixed n,N, {\mathbb {E}}(\bar{T}_5(N,n,\varepsilon ))\rightarrow 0 as \varepsilon \rightarrow 0.Thus, (REF )–(REF ) imply that for any fixed N\ge 1 and any integer n\ge 1\limsup _{\vareps... | {
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"Igor Chueshov",
"Annie Millet"
] | [
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3f6b77627e977b803b2a21c289011c3cf6937fd3 | subsection | 44 | 75 | Large deviations | Then K_M is a compact subset of X.Let \lbrace u_n\rbrace be a sequence in K_M, corresponding to solutions of
(REF ) with controls \lbrace h_n\rbrace in S_M:d u_n(t) + \big [A u_n(t) +B(u_n(t))+\tilde{R} (t,u_n(t))\big ]dt =\sigma (t,u_n(t)) h_n(t) dt, \;\;
u_n(0)=\xi .Since S_M is a
bounded closed subset in the Hilbert... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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076fd27e5de24ab3921efad1ea3a82a3be8bfa04 | subsection | 45 | 75 | Large deviations | \Vert U_n(s)\Vert ^2 ds + 2 \int _0^t |U_n(s)|^2 \big (
C_{\frac{1}{2}}\Vert u(s)\Vert ^4_{\mathcal {H}}+ R_1 +\sqrt{ L_1} \, |h_n(s)|_0\big )\, ds
\\
&\quad + 2 \int _0^t \Big ( \sigma (s,u(s))\, [h_{n}(s)-h(s)]\; ,\; U_n(s)\Big ) \, ds .The inequality (REF ) implies that there exists a finite positive constant \bar{C... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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a87782d0bf833636b12e15e41c3685667427cba5 | subsection | 46 | 75 | Large deviations | \big ( |u_n(s)-u_n(\bar{s}_N)|^2 + |u(s)-u(\bar{s}_N)|^2\big ) ds
\Big )^{\frac{1}{2}}
\\
&\le C_1 \; 2^{-\frac{N}{4}} \, ,\\
I_{n,N}^3 &\le C_0 \Big (\int _0^T |u(s)-u(\bar{s}_N)|^2 ds \Big )^{\frac{1}{2}}
\Big ( \int _0^T |h_n(s)-h(s)|_0^2\, ds\Big )^{\frac{1}{2}} \le C_3\; 2^{-\frac{N}{4}}\, ,
\\
I_{n,N}^4 &\le C_0 ... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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e3df86c853367e1f924d6f2a62ea0e3da34cda7e | subsection | 47 | 75 | Large deviations | This shows that every sequence in K_M has a convergent
subsequence. Hence K_M is a sequentially relatively compact subset of X.
Finally, let \lbrace u_n\rbrace be a sequence of elements of K_M which converges to v in X.
The above argument
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deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
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b74e2360491363783f3280095b619e34b4383a2b | subsection | 48 | 75 | Appendix : Proof of the Well posedness and apriori bounds | The aim of this section is to prove Theorem REF . We at first suppose that conditions (C1)-(C3) are satisfied.
Let
F :[0,T]\times V\rightarrow V^{\prime } be defined byF(t,u)= -A u -B(u,u) -\tilde{R}( t,u)\; , \quad \forall t\in [0,T],\; \forall u\in V.To lighten notations, we suppress the dependence of \sigma , \tilde... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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31ebfb2643cb75622c35738d57992fd8392ffa78 | subsection | 49 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Then for k=1, \, \cdots , \, n we haved(u_{n,h}(t), \varphi _k)&=&\big [ \langle F(u_{n,h}(t)),\varphi _k \rangle +(\tilde{\sigma }(u_{n,h}(t))h(t), \varphi _k) \big ]dt\\
&& + \sum _{j=1}^n q_j^{\frac{1}{2}} \big ( \sigma (u_{n,h}(t))e_j\, ,\, \varphi _k \big ) d\beta _j(t).Note that for v\in H_n the map
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"raw": "H. Kunita, Stochastic flows and stochastic differential equations, Cambridge University Press, Cambridge-New York, 1990.",
"source_ref_id": "b583b30bcf29bba6f58853c95f57fe2f179de24d",
"... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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f65f0de4079aca8f72d3e9f093839c13d33c41a1 | subsection | 50 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Assume that
I is non-decreasing and there exist non-negative constants
C, \alpha , \beta , \gamma , \delta with the following properties\int _0^T \varphi (s)\, ds \le C\; a.s.,\quad 2\beta e^C\le 1,\quad 2\delta e^C\le \alpha ,and such that
for 0\le t\le T,X(t)+ \alpha Y(t) & \le & Z + \int _0^t \varphi (r)\, X(r)\, dr... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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eca53d2e5725141ac248026a56fedc7cfc78b8e0 | subsection | 51 | 75 | Appendix : Proof of the Well posedness and apriori bounds | \tau _{n,h}=T a.s.) with a modification u_{n,h} \in C([0, T], H_n) and
satisfying\sup _n {\mathbb {E}}\,\Big (\, & \sup _{0\le t\le T}|u_{n,h}(t)|^{2p} +
\int _0^T
\Vert u_{n,h}(s)\Vert ^2 \, |u_{n,h}(s)|^{2(p-1)} ds \, \Big )
\le C \big ( {\mathbb {E}}|\xi |^{2p} +1\big )for some positive constant C (depending on
p,K_... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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6cfb42b2a68a13bf5740c54cce2f56887d2ca259 | subsection | 52 | 75 | Appendix : Proof of the Well posedness and apriori bounds | This yields for t \in [0, T], and
any integer p\ge 1 (using the convention p(p-1) x^{p-2}=0 if
p=1)|u_{n,h}(t\wedge \tau _N)|^{2p} + 2p \int _0^{t\wedge \tau _N} \!\! |u_{n,h}(r)|^{2(p-1)} \,
\Vert u_{n,h}(r)\Vert ^2 \, dr
\le \; |P_n\xi |^{2p} + \sum _{1\le j\le 5} {T}_j(t),where{T}_1(t) &= & 2p\, \int _0^{t\wedge \ta... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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edbbd49b62a35daeb2fa5169632ecc51718a2720 | subsection | 53 | 75 | Appendix : Proof of the Well posedness and apriori bounds | \Big ( \sqrt{\tilde{K}_0} +
\sqrt{\tilde{K}_1} \, |u_{n,h}(r)| + \sqrt{\tilde{K}_2} \,
\Vert u_{n,h}(r)\Vert \Big ) \,|h(r)|_0 \,|u_{n,h}(r)|^{2p-1} dr \\
& \le \; \frac{ p}{2} \, \int _0^{t\wedge \tau _N}
\!\!\Vert u_{n,h}(r)\Vert ^2
\, |u_{n,h}(r)|^{2(p-1)}\, dr
+ 2p\, \tilde{K}_2 \, \int _0^{t\wedge \tau _N} \!\! |h... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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14c7abe9eb0ba876f7cc2a0b9afc437b98215321 | subsection | 54 | 75 | Appendix : Proof of the Well posedness and apriori bounds | \left[\left(\sqrt{\tilde{K}_0}+
\sqrt{\tilde{K}_1}\right) |h(r)|_0+
\tilde{K}_2|h(r)|_0^2 \right] |u_{n,h}(r)|^{2p} dr.Using condition (C2), relation (REF ) and also the fact that\Vert \sigma (u)\Vert _{{\mathcal {L}}(H_0,H)} = \Vert \sigma ^*
(u)\Vert _{{\mathcal {L}}(H,H_0)}\le |\sigma (u)|_{L_Q},we deduce that{T}_4(... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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5fb80ba81f0d7a52b44574694c13778d2902a4ab | subsection | 55 | 75 | Appendix : Proof of the Well posedness and apriori bounds | \varphi (r)
|u_{n,h}(r)|^{2p}\, dr+I(t)for t\in [0,T],
where I(t)=\sup _{0\le s\le t}|T_2(s)| and\varphi (r)=c_p\left(R_0+ R_1 +K_0+ K_1 + \left[\sqrt{\tilde{K}_0}+
\sqrt{\tilde{K}_1}\right] |h(r)|_0+
\tilde{K}_2|h(r)|_0^2 \right)for some constant c_p>0.
The Burkholder-Davies-Gundy inequality, (C2) and Schwarz's inequa... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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87c3cf4072ae03348bca318d393f29df1c3e6cf1 | subsection | 56 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Thus there exists \bar{K}_2 such
that for 0\le K_2 \le \bar{K}_2 we have\sup _n {\mathbb {E}}\Big ( \sup _{0\le s\le \tau _N} |u_{n,h}|^{2p} + \int _0^{\tau _N} \Vert u_{n,h}(s)\Vert \,
|u_{n,h}(s)|^{2(p-1)}\, ds \Big ) \le C(p)for all n and p, where the constant C(p) is independent of n.Now we are in position to concl... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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e351193ad152c8b673f206c5884d98c19c26bca3 | subsection | 57 | 75 | Appendix : Proof of the Well posedness and apriori bounds | The inequalities (REF ) and (REF ) imply that
for K_2\in [0, \bar{K}_2] we have the following additional a priori estimate\sup _n {\mathbb {E}}\int _0^T \Vert u_{n,h}(s) \Vert ^4_{\mathcal {H}} ds
\le C_{2} (1+ {\mathbb {E}}|\xi |^4).The proof consists of several steps.Step 1:
The inequalities (REF )
and (REF ) imply ... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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a1f7728776f7df9021d19b53209cca10c78e28d2 | subsection | 58 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Since for u,v\in L^2(\Omega _T,V), {\mathbb {E}}\int _0^T \langle Au(t)\,, \, v(t)\rangle \, dt =
{\mathbb {E}}\int _0^T \langle u(t)\, , \, Av(t)\rangle \, dt,{\mathbb {E}}\int _0^T \langle A u_{n,h}(t), v(t)\rangle \, dt
\rightarrow \;
{\mathbb {E}}\int _0^T \langle A u_h(t)\, ,\, v(t)\rangle \, dt .Using (REF ) with... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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477c2b44e6d32783703bb5116c117019dc8aba87 | subsection | 59 | 75 | Appendix : Proof of the Well posedness and apriori bounds | \left(K_1 |u_{n,h}(t)|^2 + K_2 \Vert u_{n,h}(t)\Vert ^2\right) dt
< \infty .Finally, using (REF ) in (C3), Hölder's inequality, (REF )
with p=2 and (REF ),
we deduce{\mathbb {E}}\int _0^T |\tilde{\sigma }_n( & u_{n,h} (s)\, h(s) |^{\frac{4}{3}} \, ds \le {\mathbb {E}}\int _0^T \big [\sqrt{ \tilde{K}_0} + \sqrt{ \tilde{... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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ae2e735efdaccfb882d24f837e1ba0845fb9d8fe | subsection | 60 | 75 | Appendix : Proof of the Well posedness and apriori bounds | The Itô formula
implies that for any j\ge 1, and for 0 \le t \le T,\big ( u_{n,h}(T)\, ,\, g_j(T)\big ) = \big ( u_{n,h}(0)\, ,\, g_j(0)\big )
+\sum _{i=1}^4 I_{n ,j}^i,whereI_{n ,j}^1 = \int _0^T (u_{n,h}(s), \varphi _j) f^{\prime }(s) ds,&&
I_{n ,j}^2 = \int _0^T \big ( \sigma _n(u_{n,h}(s)) \Pi _n dW(s), g_j(s)\big ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1351,
"openalex_id": "",
"raw": "S. S. Sritharan & P. Sundar, Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise, Stoch. Proc. and Appl. 116 (2006), 1636–1659.",
"source_ref_id": "2aea... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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f6cd439748db7b01392c51de9ef0c8c031fc7950 | subsection | 61 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Furthermore, (vi) shows that for every G \in \mathcal {P}_T, as n\rightarrow \infty ,
\big (\sigma _n(u_{n,h}) \Pi _n, G\big )_{\mathcal {P}_T} \rightarrow (S_h, G)_{\mathcal {P}_T}
weakly in L^2(\Omega ).Finally, as n\rightarrow \infty , P_n\xi =u_{n,h}^\varepsilon (0) \rightarrow \xi in H and by (iv), (u_{n,h}(T), g_... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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2acd56d43d1a9732b78c9466964ed4fe3e13a2eb | subsection | 62 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Now
note that j is arbitrary and {\mathbb {E}}\int _0^T |S_h(s)|^2_{L_Q} ds <
\infty ; we deduce that for 0\le t \le T,u_{h} (t) = \xi + \int _0^t S_h(s)dW(s) + \int _0^t F_h(s) ds +\int _0^t \tilde{S}_h(s)ds \in H.Moreover \int _0^t F_h(s) ds\in H .
Let f=1_{(-\delta ,
T+\delta )}; using again (REF ) we obtain\tilde{u... | {
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deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
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976bcabf58bca6e928a45b512fa76b0da939b6ba | subsection | 63 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Moreover, we also have thatr\in L^1(\Omega ,L^\infty (0;T)), \; e^{-r}\in L^\infty (\Omega _T),\;
r^{\prime }\in L^1(\Omega _T),\;
r^{\prime }e^{-r}\in L^\infty (\Omega , L^1((0,T)).Weak convergence in (iv) and the property P_n\xi \rightarrow \xi in H imply that{\mathbb {E}}\big ( |u_h(T)|^2\, e^{-r(T)}\big )-{\mathbb ... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
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] | 2,008 | en | Mathematics | [
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fc8837a4cc7449246f1fa6e93effc1e989929db2 | subsection | 64 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Using (REF ), (REF ) and letting u = v + (u-v)
after simplification, from (REF ) we obtain{\mathbb {E}}& \int _0^T \!\! e^{-r(s)}\, \big [ -r^{\prime }(s)\big \lbrace \big |u_h(s)-v(s)\big |^2
+ 2\big ( u_h(s)-v(s)\, ,\, v(s)\big )\rbrace
+ 2\langle F_h(s),u_h(s)\rangle \\
&+ |S_h(s)|^2_{L_Q} + 2\big (\tilde{S}_h(s)\,... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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95c2aea28e276997499f6edb8a9bb15b579116d8 | subsection | 65 | 75 | Appendix : Proof of the Well posedness and apriori bounds | For \lambda \in {\mathbb {R}}, \tilde{v} \in L^\infty (\Omega _T,V),
set v_\lambda =u_h -\lambda \tilde{v} ; then it is clear that v_\lambda \in {\mathcal {X}}.
Applying (REF ) to v:=v_\lambda and
neglecting | \sigma (u_h(s)) - \sigma (v_\lambda (s))|^2_{L_Q}, yields{\mathbb {E}}\int _0^T e^{-r_\lambda (s)} & \Big [ -... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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ef94bf88a36b1ab9544fe5493b7dec7e1cc5095f | subsection | 66 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Hence, by the dominated convergence theorem,\lim _{\lambda \rightarrow 0} {\mathbb {E}}\int _0^T \! e^{-r_\lambda (s)}\Big (
\tilde{S}_h(s)-\tilde{\sigma }(v_\lambda (s))
h(s)\, , \, \tilde{v}(s)\Big ) ds {}\qquad {}
\\= {\mathbb {E}}\int _0^T \! e^{-r_0(s)}\Big ( \tilde{S}_h(s)-\tilde{\sigma }(u_h(s))h(s)\, ,\,
\tilde... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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0bfbcd92f7262356b32e3415b9ce033f07a79ca8 | subsection | 67 | 75 | Appendix : Proof of the Well posedness and apriori bounds | \lambda <0 )
and letting \lambda \rightarrow 0 we obtain that for every
\tilde{v} \in L^\infty (\Omega _T,V),
which is a dense subset of L^2(\Omega _T, V),{\mathbb {E}}\int _0^T e^{-r_0(s)}\Big [ \big \langle F_h(s)- F(u_h(s))\, , \, \tilde{v}(s)\big \rangle + \big ( \tilde{S}_h(s)-\tilde{\sigma }(u_h(s)) h(s)\, ,\,
\t... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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e288615680b3989024480010f3a2299a07b41d58 | subsection | 68 | 75 | Appendix : Proof of the Well posedness and apriori bounds | \tilde{\sigma }(u_h(s))\, h(s)\, ds also belongs to
C([0,T],H).
Finally, condition (C2) implies
{\mathbb {E}}\int _0^T |e^{-\delta A} \sigma (u_h)(s)|^2_{L_Q}\, ds <+\infty . Thus
\int _0^. e^{-\delta A} \sigma (u_h(s))\, dW(s) belongs to C([0,T],H) a.s. (see e.g. , Theorem 4.12).
Therefore it is sufficient to prove th... | {
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}... | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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bb7ec4841dbaed50530780ca5a68d50601d13a5a | subsection | 69 | 75 | Appendix : Proof of the Well posedness and apriori bounds | By the Burkholder-Davies-Gundy and Schwarz inequalities we have{\mathbb {E}}\sup _{0\le t\le T} |I(t)| &\le &
C{\mathbb {E}}\left( \int _0^{T}\!\! |G_\delta u_{h}(s)|^2
|G_\delta \, \sigma (u_{h}(s))|^2_{L_Q} ds\right)^{1/2} \\
&\le & \frac{1}{2}\; {\mathbb {E}}\sup _{0\le t\le T} |G_\delta u_{h}(t)|^2 + \frac{C^2}{2} ... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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b137bfa5e00a9fed749306d9bd01ed11aa35647a | subsection | 70 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Furthermore, given u\in V we have \Vert G_\delta ^2 u\Vert \rightarrow 0 as \delta \rightarrow 0
and \sup _{\delta >0} |G_\delta |_{L(V,V)}\le 2 . Hence
\big \langle B(u_h(s))+ \tilde{R} (u_{h}(s)) +
\tilde{\sigma }(u_{h}(s)) h(s), G^2_\delta u_{h}(s)\big \rangle \rightarrow 0 for almost every
(\omega ,s).
Therefore, ... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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de88939bac53952e69f10abc42a72efeaaa667a8 | subsection | 71 | 75 | Appendix : Proof of the Well posedness and apriori bounds | By Itô's formula we havee^{-a \int _0^{t\wedge \tau _N} \Vert u_h(r)\Vert ^4_{\mathcal {H}}dr}
|U(t\wedge \tau _N)|^2 \;\; = \int _0^{t\wedge \tau _N}\Psi (s) ds +\Phi (t\wedge \tau _N),where\Psi (s) &
=
e^{- a\int _0^{s}
\Vert u_h(r)\Vert ^4_{\mathcal {H}}dr} \Big [ -a \Vert u_h(s)\Vert ^4_{\mathcal {H}}|U(s)|^2
\\
&\... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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e68766763e722d576a4aa1ddc728a64a818a82c6 | subsection | 72 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Below we use the notationsX(t)=
\sup _{0\le s\le t}\left\lbrace e^{- a\int _0^{s\wedge \tau _N}
\Vert u_h(r)\Vert ^4_{\mathcal {H}}dr}
|U(s\wedge \tau _N)|^2\right\rbrace , ~~
Y(t)=
\int _0^{t\wedge \tau _N}\!\! \! e^{- a\int _0^{s}
\Vert u_h(r)\Vert ^4_{\mathcal {H}}dr}
\Vert U(s)\Vert ^2 ds.Then it follows from
(REF ... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
"Annie Millet"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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2bf02076a06363b8b0ee97eefaf6305c26079bef | subsection | 73 | 75 | Appendix : Proof of the Well posedness and apriori bounds | Therefore,
since \sup _{0\le s\le T}\left\lbrace e^{- a\int _0^{s\wedge \tau _N}
\Vert u_h(r)\Vert ^4_{\mathcal {H}}dr}
|U(s\wedge \tau _N)|^2\right\rbrace \le 2N, relation (REF )
implies that {\mathbb {E}}X(t)=0 for all t and hence,{\mathbb {E}}\; \sup _{0\le s\le T} \; \left\lbrace e^{- a\int _0^{s\wedge \tau _N}
\Ve... | {
"cite_spans": []
} | 10.1007/s00245-009-9091-z | 0807.1810 | Stochastic 2D hydrodynamical type systems: Well posedness and large
deviations | [
"Igor Chueshov",
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2f9b91ab0e58b5d808cbccc168ed2c95cb5824ce | subsection | 74 | 75 | Appendix : Proof of the Well posedness and apriori bounds | For h\ne 0 set \tilde{W}^h_t=W_t+\int _0^t h(s)\, ds and let \tilde{{\mathbb {P}}} be the probability
defined on (\Omega ,{\mathcal {F}_t}) by\frac{d\tilde{{\mathbb {P}}}}{d{\mathbb {P}}}=\exp \Big ( -\int _0^t h(s)\, dW_s
-\frac{1}{2} \int _0^t |h(s)|_0^2\, ds\Big ).The Girsanov theorem implies that \tilde{W}^h
is a \... | {
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"Igor Chueshov",
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96e66db463e3779427833be5afe2bbcfa28c377d | abstract | 0 | 40 | Abstract | If I is a nilpotent ideal in a $\mathbb{Q}$-algebra $A$, Goodwillie defined
two isomorphisms from $K_*(A,I)$ to negative cyclic homology, $HN_*(A,I)$. One
is the relative version of the absolute Chern character, and the other is
defined using rational homotopy theory. The question of whether they agree was
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7488d07a6de54439861e1e1810e6cb97c80096f6 | subsection | 1 | 40 | Introduction | When A is a unital ring, the absolute Chern character
is a group homomorphism
ch_*: K_*(A)\rightarrow HN_*(A)
, going from algebraic K-theory to negative cyclic homology (see
). There is also a relative version, defined for any ideal
I of A:ch_*: K_*(A,I)\rightarrow HN_*(A,I).Now suppose that A is a \mathbb {Q}-algebra... | {
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4c823fc5f95b8b5cc6f49a853573162eea11bea1 | subsection | 2 | 40 | Notation | If M=(M_*,b,B) is a mixed complex (), we will write
HH(M) for the chain complex (M_*,b), and HN(M) for the total complex
of Connes' left half-plane (b,B)-complex (written as \mathcal {B}M^- in
). By definition, the homology of HH(M)
is the Hochschild homology HH_*(M) of M, the homology of HN(M)
is the negative cyclic h... | {
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d27f9e7dd8f24c77189295d02f5d6a909d58588f | subsection | 3 | 40 | Cyclic homology of cocommutative Hopf algebras | If A is the group algebra of a group G, then the bar
resolution \mathrm {E}(A)=k[\mathrm {E}G] admits a cyclic G-module structure and the
bar complex \mathrm {B}(A)=k[\mathrm {B}G] also admits a cyclic k-module structure
. In this section, we show that the cyclic modules
\mathrm {E}(A) and \mathrm {B}(A) can be defined... | {
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e521ce9e423c1571b275cfa174bdea5ec7dbe3e5 | subsection | 4 | 40 | Bar resolution and bar complex of an augmented algebra | Let k be a commutative ring, and A an augmented unital
k-algebra, with augmentation \epsilon :A\rightarrow k. We write \mathrm {E}(A)
for the bar resolution of k as a left A-module
(); this is the simplicial A-module
\mathrm {E}_n(A)=A^{\otimes n+1}, whose face and degeneracy operators
are given by\mu _i:\mathrm {E}_n(... | {
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"Charles Weibel"
] | [
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bf029d5b8aef1d387dddc06ee977f8ce422c56c1 | subsection | 5 | 40 | The cyclic module of a cocommutative coalgebra | If C is a k-coalgebra, with counit \epsilon :C\rightarrow k
and coproduct \Delta , we have a
simplicial k-module R(C), with R_n(C)=C^{\otimes n+1}, and face
and degeneracy operators given by\varepsilon _i:R_n(C)\rightarrow R_{n-1}(C)\qquad (i=0,\dots ,n)\\
\varepsilon _i(c_0\otimes \dots \otimes c_n)=
\epsilon (c_i) c_... | {
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519462707344f47539891108580292035f77e54f | subsection | 6 | 40 | The case of Hopf algebras | Let \mathsf {H} be a Hopf algebra with unit \eta , counit \epsilon and
antipode S. We shall assume that S^2=1, which is the case for all
cocommutative Hopf algebras. We may view R(\mathsf {H}) as
a simplicial left \mathsf {H}-module via the diagonal action:a\cdot (h_0\otimes \dots \otimes h_n)=a^{(0)}h_0\otimes \dots \... | {
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} | 0807.1811 | Relative Chern characters for nilpotent ideals | [
"Guillermo Cortiñas",
"Charles Weibel"
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b70045b82765b022572c45b463fde374d191e1a3 | subsection | 7 | 40 | Cyclic complexes of cocommutative Hopf algebras | From now on, we shall assume that \mathsf {H} is a cocommutative Hopf
algebra. In this case the cyclic operator \lambda :R(\mathsf {H})\rightarrow R(\mathsf {H})
of REF is a homomorphism of \mathsf {H}-modules. Thus R(\mathsf {H}) is a cyclic
\mathsf {H}-module, and we can use the isomorphisms \alpha and \beta
of Lemm... | {
"cite_spans": []
} | 0807.1811 | Relative Chern characters for nilpotent ideals | [
"Guillermo Cortiñas",
"Charles Weibel"
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f44e737a6ba8d2855bb62d980454c30d54cca061 | subsection | 8 | 40 | Cyclic complexes of cocommutative Hopf algebras | The cyclic operator
t:=\beta \lambda \alpha on \mathrm {E}(\mathsf {H}) is given by the formulas
t(h)=h, t(h_0\otimes h_1)=-h_0h_1^{(0)}\otimes Sh_1^{(1)} and:t(h_0\otimes \dots \otimes h_n)=(-1)^nh_0 h_1^{(0)}\dots h_n^{(0)}\otimes S(h_1^{(1)}\dots h_n^{(1)})\otimes h_1^{(2)}\otimes \dots \otimes h_{n-1}^{(2)}.If g_0,... | {
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8b858aa32515de1366d1a9b780e2abb52cd84895 | subsection | 9 | 40 | Cyclic complexes of cocommutative Hopf algebras | Then B^{\prime }=B^{\prime \prime } on \mathrm {E}(\mathsf {H})_\mathrm {norm} because
the relations s_0t_n=(-1)^nt_{n+1}^2s_n and tN=N yield
for all x\in \mathrm {E}_n(\mathsf {H}):(B^{\prime \prime }-B^{\prime })(x)&=((-1)^{n+1}ts_nN+(-1)^n(1-t)ts_nN)(x)\\
&=(-1)^n(-ts_nN(x)+ts_nN(x)-t^2s_nN(x))\\
&=-s_0tN(x)=-s_0N(x... | {
"cite_spans": []
} | 0807.1811 | Relative Chern characters for nilpotent ideals | [
"Guillermo Cortiñas",
"Charles Weibel"
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dabfad9c6fad8b163d257b0c7479bf3f4792fb15 | subsection | 10 | 40 | Cyclic complexes of cocommutative Hopf algebras | We have-ts^{\prime \prime }t^i(h_0\otimes \dots \otimes h_n)=
(-1)^{n+1}ts_nt^i(h_0\otimes \dots \otimes h_n)\\
=(-1)^{m}t(h_0h_1^{0}\dots h_{n-i+1}^{(0)}\otimes h_{n-i+2}^{(0)}\otimes \dots \otimes h_n^{(0)}
\otimes Sh^{(1)} \otimes h_1^{(2)}\otimes \dots \otimes h_{n-i}^{(2)}\otimes 1)\\
=(-1)^{ni}h_0h_1^{(0)}\dots h... | {
"cite_spans": []
} | 0807.1811 | Relative Chern characters for nilpotent ideals | [
"Guillermo Cortiñas",
"Charles Weibel"
] | [
"math.KT",
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0a4bac78058c7dde971ef00f803109f887d19cab | subsection | 11 | 40 | Adic filtrations and completion | As usual, we can use an adic topology on a Hopf algebra to define
complete Hopf algebras, and topological versions of the
above complexes.First we recall some generalities about filtrations and
completions of k-modules, following . There is a
category of filtered k-modules and filtration-preserving maps; a
filtered mod... | {
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"Guillermo Cortiñas",
"Charles Weibel"
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1f49ebd9364c414c278f9401b5a644ec2c4ce604 | subsection | 12 | 40 | Adic filtrations and completion | The coideal condition on I means that\Delta (I)\subset \mathsf {H}\otimes I+I\otimes \mathsf {H}.This implies that \Delta :\mathsf {H}\rightarrow \mathsf {H}\otimes \mathsf {H} is filtration-preserving;
by (REF ) it induces a map
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"Guillermo Cortiñas",
"Charles Weibel"
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"math.QA"
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