[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

Stochastic equations in infinite dimensions

Stochastic differential equations and diffusion processes

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

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Stochastic Differential Equations And Applications

Stochastically Forced Compressible Fluid Flows

We consider functions $u\in L^\infty(L^2)\cap L^p(W^{1,p})$ with $1<p<\infty$ on a time space domain. Solutions to non-linear evolutionary PDE's typically belong to these spaces. Many applications require a Lipschitz approximation $u_\lambda$ of $u$ which coincides with $u$ on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids in [DRW10]. Since ${\rm div} u_\lambda=0$, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of [BDF12].

Solenoidal Lipschitz truncation for parabolic PDE's

We consider functions u ∈ L∞(L2)∩Lp(W1, p) with 1 < p < ∞ on a time–space domain. Solutions to nonlinear evolutionary PDEs typically belong to these spaces. Many applications require a Lipschitz approximation uλ of u which coincides with u on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids of Diening, Ruzicka and Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010) 1–46. Since divuλ = 0, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of Breit, Diening and Fuchs, Solenoidal Lipschitz truncation and applications in fluid mechanics, J. Differential Equations253 (2012) 1910–1942.

SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs

We study the Navier-Stokes equations governing the motion of an isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density. We establish existence of a so-called finite energy weak martingale solution under the condition that the adiabatic exponent satisfies gamma > 3/2. The proof is based on a four-layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure.

/pdf/stochastic-navier-stokes-equations-for-compressible-fluids-4keunughj7.pdf

Stochastic Navier-Stokes equations for compressible fluids

We show that Korn’s inequality in Orlicz spaces holds if and only if the Orlicz function satisfies the Δ2- and the ${\nabla_2}$ -condition. This result applies to several types of Korn’s inequality. In particular we show that Korn’s inequality is false in L1, in L log L, in Exp and in L∞.

https://www.math.uni-sb.de/service/preprints/preprint290.pdf

Dominic Breit

Papers

Stochastically Forced Compressible Fluid Flows

Solenoidal Lipschitz truncation for parabolic PDE's

SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs

Stochastic Navier-Stokes equations for compressible fluids

Sharp Conditions for Korn Inequalities in Orlicz Spaces