The magnetization ripple: A nonlocal stochastic PDE perspective

Ignat, Radu; Otto, Félix

doi:10.1016/j.matpur.2019.01.010

Cited by 6 publications

(52 citation statements)

References 30 publications

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“…The solution U may be further decomposed as U = q + w, where q is the even-reflection of the function defined in Definition 6 (below) and w ∈ C 2α (R 2 ) such that w ≡ 0 on R 2 − . The function q is modelled after Ṽ(•, a 0 ), the even-reflection of the function defined in (12), according to a on R 2 . We find that the solution U = u + q + w is unique in the class of functions admitting such a splitting.…”

Section: Discussion and Statement Of Our Resultsmentioning

confidence: 99%

“…Notice that the definition of q only depends on a| {x2=0} and that, thanks to the lack of a massive term in (80), ā ≥ λ when a| {x2=0} ≥ λ. We would also like to mention that the " ′ " in (78) does not indicate a derivative, but is only meant to distinguish (78) from (12).…”

Section: Discussion and Statement Of Our Resultsmentioning

confidence: 99%

“…In our situation, we cannot directly use their method of proof due to the loss of periodicity in the x 2 -direction. Instead, we adapt a more general argument from a work by Ignat and Otto [12]. We use the notation…”

Section: This Is Used In the Following Definitionmentioning

confidence: 99%

“…where Ṽ(•, a 0 ) is the even-reflection of the function defined in (12) and v(•, a 0 ) solves (10). Each of these families (indexed by a 0 , a ′ 0 ) should satisfy a C 2α−2 -commutator estimate similar to (24).…”

Section: New "Offline" Productsmentioning

confidence: 99%

“…Let α ∈ (0, 1) and a 0 , a ′ 0 ∈ [λ, 1] for λ > 0. Recall that v(•, a 0 ) ∈ C α (R 2 ) solves (10) and V(•, a 0 ) is defined in (12); we use the notation from Definition 5. We then obtain: i) For any F ∈ C α (R 2 ), the products F ∂ 2 1 Ṽ(•, a 0 ) are well-defined as distributions and this family satisfies…”

Section: New "Offline" Productsmentioning

confidence: 99%

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The Initial Value Problem for Singular SPDEs via Rough Paths

Raithel,

Sauer

2020

Preprint

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In this contribution we are interested in developing a solution theory for singular quasilinear stochastic partial differential equations subject to an initial condition. We obtain our solution theory via a perturbation of the rough path approach developed to handle the space-time periodic problem by Otto and Weber (2019). As in their work, we assume that the forcing is of class C α−2 for α ∈ ( 2 3 , 1) and space-time periodic and, additionally, that the initial condition is of class C α and periodic. We observe that, thanks to bounds for the heat semigroup, enforcing an initial condition within the framework of Otto and Weber does not require any new stochastic results.

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Section: Discussion and Statement Of Our Resultsmentioning

confidence: 99%

Section: Discussion and Statement Of Our Resultsmentioning

confidence: 99%

Section: This Is Used In the Following Definitionmentioning

confidence: 99%

Section: New "Offline" Productsmentioning

confidence: 99%

Section: New "Offline" Productsmentioning

confidence: 99%

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The Initial Value Problem for Singular SPDEs via Rough Paths

Raithel,

Sauer

2020

Preprint

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Variational methods for a singular SPDE yielding the universality of the magnetization ripple

Ignat

Otto

Ried

et al. 2023

Comm Pure Appl Math

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The magnetization ripple is a microstructure formed in thin ferromagnetic films. It can be described by minimizers of a nonconvex energy functional leading to a nonlocal and nonlinear elliptic SPDE in two dimensions driven by white noise, which is singular. We address the universal character of the magnetization ripple using variational methods based on Γ‐convergence. Due to the infinite energy of the system, the (random) energy functional has to be renormalized. Using the topology of Γ‐convergence, we give a sense to the law of the renormalized functional that is independent of the way white noise is approximated. More precisely, this universality holds in the class of (not necessarily Gaussian) approximations to white noise satisfying the spectral gap inequality, which allows us to obtain sharp stochastic estimates. As a corollary, we obtain the existence of minimizers with optimal regularity.

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A Priori Bounds for the $$\Phi ^4$$ Equation in the Full Sub-critical Regime

Chandra

Moinat

Weber

2023

Arch Rational Mech Anal

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We derive a priori bounds for the $$\Phi ^4$$ Φ 4 equation in the full sub-critical regime using Hairer’s theory of regularity structures. The equation is formally given by "Equation missing"where the term $$+\infty \phi $$ + ∞ ϕ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions $$d<4$$ d < 4 by adjusting the regularity of the noise term $$\xi $$ ξ , choosing $$\xi \in C^{-3+\delta }$$ ξ ∈ C - 3 + δ . Our main result states that if $$\phi $$ ϕ satisfies this equation on a space–time cylinder $$D= (0,1) \times \{ |x| \leqslant 1 \}$$ D = ( 0 , 1 ) × { | x | ⩽ 1 } , then away from the boundary $$\partial D$$ ∂ D the solution $$\phi $$ ϕ can be bounded in terms of a finite number of explicit polynomial expressions in $$\xi $$ ξ . The bound holds uniformly over all possible choices of boundary data for $$\phi $$ ϕ and thus relies crucially on the super-linear damping effect of the non-linear term $$-\phi ^3$$ - ϕ 3 . A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of (*), which allows us to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications: we reduce the number of objects required with respect to Hairer’s work. Instead of a model $$(\Pi _x)_x$$ ( Π x ) x and the family of translation operators $$(\Gamma _{x,y})_{x,y}$$ ( Γ x , y ) x , y we work with just a single object $$(\mathbb {X}_{x, y})$$ ( X x , y ) which acts on itself for translations, very much in the spirit of Gubinelli’s theory of branched rough paths. Furthermore, we show that in the specific context of (*) the hierarchy of continuity conditions which constitute Hairer’s definition of a modelled distribution can be reduced to the single continuity condition on the “coefficient on the constant level”.

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The magnetization ripple: A nonlocal stochastic PDE perspective

Cited by 6 publications

References 30 publications

The Initial Value Problem for Singular SPDEs via Rough Paths

The Initial Value Problem for Singular SPDEs via Rough Paths

Variational methods for a singular SPDE yielding the universality of the magnetization ripple

A Priori Bounds for the $$\Phi ^4$$ Equation in the Full Sub-critical Regime

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