@article{13135,
  abstract     = {In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are Lp(Lp)-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8], the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model.},
  author       = {Agresti, Antonio and Veraar, Mark},
  issn         = {1090-2732},
  journal      = {Journal of Differential Equations},
  number       = {9},
  pages        = {247--300},
  publisher    = {Elsevier},
  title        = {{Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity}},
  doi          = {10.1016/j.jde.2023.05.038},
  volume       = {368},
  year         = {2023},
}

@article{9240,
  abstract     = {A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension . It is a regularised and inertial version of the Dean–Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in d-dimensions when the particle-width ϵ is proportional to  for  and N is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the d-spatial dimensions, and use of the Faà di Bruno's formula.},
  author       = {Cornalba, Federico and Shardlow, Tony and Zimmer, Johannes},
  issn         = {1090-2732},
  journal      = {Journal of Differential Equations},
  number       = {5},
  pages        = {253--283},
  publisher    = {Elsevier},
  title        = {{Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions}},
  doi          = {10.1016/j.jde.2021.02.048},
  volume       = {284},
  year         = {2021},
}

@article{8691,
  abstract     = {Given l>2ν>2d≥4, we prove the persistence of a Cantor--family of KAM tori of measure O(ε1/2−ν/l) for any non--degenerate nearly integrable Hamiltonian system of class Cl(D×Td), where D⊂Rd is a bounded domain, provided that the size ε of the perturbation is sufficiently small. This extends a result by D. Salamon in \cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the regularity assumption can not be improved.},
  author       = {Koudjinan, Edmond},
  issn         = {0022-0396},
  journal      = {Journal of Differential Equations},
  keywords     = {Analysis},
  number       = {6},
  pages        = {4720--4750},
  publisher    = {Elsevier},
  title        = {{A KAM theorem for finitely differentiable Hamiltonian systems}},
  doi          = {10.1016/j.jde.2020.03.044},
  volume       = {269},
  year         = {2020},
}