scholarly journals Spatial Sobolev regularity for stochastic Burgers equations with additive trace class noise

2021 ◽  
Vol 210 ◽  
pp. 112310
Author(s):  
Arnulf Jentzen ◽  
Felix Lindner ◽  
Primož Pušnik
Keyword(s):  
Trace Class ◽  
Sobolev Regularity ◽  
Burgers Equations ◽  
Class Noise

scholarly journals A UNIFIED EXISTENCE AND UNIQUENESS THEOREM FOR STOCHASTIC EVOLUTION EQUATIONS

2009 ◽  
Vol 81 (1) ◽  
pp. 33-46
Author(s):  
A. JENTZEN ◽  
P. E. KLOEDEN
Keyword(s):  
Diffusion Coefficient ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Trace Class ◽  
Existence And Uniqueness Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Class Noise

AbstractAn existence and uniqueness theorem for mild solutions of stochastic evolution equations is presented and proved. The diffusion coefficient is handled in a unified way which allows a unified theorem to be formulated for different cases, in particular, of multiplicative space–time white noise and trace-class noise.

Martingale solutions for stochastic active scalar equations perturbed by non-trace class noise

2014 ◽  
Vol 17 (02) ◽  
pp. 1450010 ◽  
Author(s):  
Rongchan Zhu ◽  
Xiangchan Zhu
Keyword(s):  
White Noise ◽  
Multiplicative Noise ◽  
Trace Class ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Class Noise ◽  
Active Scalar Equations ◽  
Scalar Equations ◽  
Martingale Solutions ◽  
Active Scalar

In this paper we prove the existence of martingale solutions for the 2D stochastic fractional vorticity Navier–Stokes equation driven by space-time white noise for α ∈ (½, 1] and the 2D stochastic quasi-geostrophic equation on 𝕋2 for α ∈ (0, 1] driven by non-trace class noise. We also cover the case driven by non-trace class multiplicative noise for all α ∈ (0, 1].

scholarly journals Fokker–Planck Equations for SPDE with Non-trace-class Noise

2013 ◽  
Vol 1 (3) ◽  
pp. 281-304 ◽  
Author(s):  
G. Da Prato ◽  
F. Flandoli ◽  
M. Röckner
Keyword(s):  
Trace Class ◽  
Class Noise ◽  
Fokker Planck Equations ◽  
Fokker Planck

scholarly journals Quantum Orlicz Spaces in Information Geometry

2004 ◽  
Vol 11 (04) ◽  
pp. 359-375 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Quantum Information ◽  
Tangent Space ◽  
Selfadjoint Operator ◽  
Affine Connection ◽  
Orlicz Spaces ◽  
Trace Class ◽  
Information Geometry ◽  
Young Function ◽  
Affine Structure ◽  
Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

scholarly journals On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 220
Author(s):  
Alexey Samokhin
Keyword(s):  
Periodic Boundary ◽  
Asymptotic Formulas ◽  
Burgers Equations ◽  
Spherical Waves ◽  
Constant Height ◽  
De Vries ◽  
Spatial Dimensions ◽  
Integral Characteristics ◽  
Boundary Solutions ◽  
Shock Fronts

We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found.

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