@article{12304,
  abstract     = {We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of “propagation of degeneracy” down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that “degeneracy” on a certain space-time cylinder entails “degeneracy” on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production.},
  author       = {De Nitti, Nicola and Fischer, Julian L},
  issn         = {1532-4133},
  journal      = {Communications in Partial Differential Equations},
  keywords     = {Applied Mathematics, Analysis},
  number       = {7},
  pages        = {1394--1434},
  publisher    = {Taylor & Francis},
  title        = {{Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation}},
  doi          = {10.1080/03605302.2022.2056702},
  volume       = {47},
  year         = {2022},
}

@article{1318,
  abstract     = {We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale Ck,α-regularity theory, which in the present work is developed in the form of a corresponding Ck,α-“excess decay” estimate: For a given a-harmonic function u on a ball BR, its energy distance on some ball Br to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r0. Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.},
  author       = {Fischer, Julian L and Otto, Felix},
  issn         = {1532-4133},
  journal      = {Communications in Partial Differential Equations},
  keywords     = {Ck, α regularity, higher-ordercorrectors, Liouville principle, random elliptic operator, regularity theory, stochastic homogenization},
  number       = {7},
  pages        = {1108 -- 1148},
  publisher    = {Taylor & Francis},
  title        = {{A higher-order large scale regularity theory for random elliptic operators}},
  doi          = {10.1080/03605302.2016.1179318},
  volume       = {41},
  year         = {2016},
}

@article{22047,
  abstract     = {We undertake a comprehensive study of the nonlinear Schrödinger equation (mathematical formular) where u(t, x) is a complex-valued function in spacetime R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H^1(ℝ n ) and in the pseudoconformal space Σ := {f ∈ H^1(ℝ^n); xf ∈ L^2(ℝ^n)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical formular) . The results at the endpoint p1= 4/n are conditional on a conjectured global existence and spacetime estimate for the L2/x-critical nonlinear Schrödinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint).
As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H1/x for solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n < p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985).},
  author       = {Tao, Terence and Visan, Monica and Zhang, Xiaoyi},
  issn         = {1532-4133},
  journal      = {Communications in Partial Differential Equations},
  keywords     = {Energy-critical, Mass-critical, Nonlinear Schrödinger equation, Wellposedness},
  number       = {8},
  pages        = {1281--1343},
  publisher    = {Informa UK Limited},
  title        = {{The nonlinear Schrödinger equation with combined power-type nonlinearities}},
  doi          = {10.1080/03605300701588805},
  volume       = {32},
  year         = {2007},
}