@article{20155,
  abstract     = {We study time averages for the norm of solutions to kinetic Fokker–Planck equations associated with general Hamiltonians. We provide fully explicit and constructive decay estimates for systems subject to a confining potential, allowing fat-tail, subexponential and (super-)exponential local equilibria, which also include the classic Maxwellian case. The key step in our estimates is a modified Poincaré inequality, obtained via a Lions–Poincaré inequality and an averaging lemma.},
  author       = {Brigati, Giovanni and Stoltz, Gabriel},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  number       = {4},
  pages        = {3587--3622},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{How to construct explicit decay rates for kinetic Fokker–Planck equations?}},
  doi          = {10.1137/24M1700351},
  volume       = {57},
  year         = {2025},
}

@article{17372,
  abstract     = {In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the  d-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g. the Allen-Cahn equation), and dissipative systems (e.g. equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for  d∈{1,2,3,4}  and the Brusselator for  d∈{1,2,3}. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on Itô's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new  Lζ -coercivity/dissipativity conditions and the development of an  Lp(Lq) -framework for systems of reaction-diffusion equations, which are needed when treating dimensions  d∈{2,3}  in the case of cubic or higher order nonlinearities.},
  author       = {Agresti, Antonio and Veraar, Mark},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  number       = {4},
  pages        = {4870--4927},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems}},
  doi          = {10.1137/23M1562482},
  volume       = {56},
  year         = {2024},
}

@article{11739,
  abstract     = {We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.},
  author       = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Fokker--Planck equation, gradient flow, evolutionary $\Gamma$-convergence},
  number       = {4},
  pages        = {4297--4333},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}},
  doi          = {10.1137/21M1410968},
  volume       = {54},
  year         = {2022},
}

@article{12305,
  abstract     = {This paper is concerned with the sharp interface limit for the Allen--Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Ω⊂\R2. We assume that a diffuse interface already has developed and that it is in contact with the boundary ∂Ω. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant α-contact angle. For α close to 90° we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen--Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen--Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.},
  author       = {Abels, Helmut and Moser, Maximilian},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Applied Mathematics, Computational Mathematics, Analysis},
  number       = {1},
  pages        = {114--172},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°}},
  doi          = {10.1137/21m1424925},
  volume       = {54},
  year         = {2022},
}

@article{10547,
  abstract     = {We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities,
while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.},
  author       = {Fischer, Julian L and Hopf, Katharina and Kniely, Michael and Mielke, Alexander},
  issn         = {0036-1410},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Energy-Reaction-Diffusion Systems, Cross Diffusion, Global-In-Time Existence of Weak/Renormalised Solutions, Entropy Method, Onsager System, Soret/Dufour Effect},
  number       = {1},
  pages        = {220--267},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Global existence analysis of energy-reaction-diffusion systems}},
  doi          = {10.1137/20M1387237},
  volume       = {54},
  year         = {2022},
}

@article{22084,
  abstract     = {We consider the nonlinear Schrödinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip et al. [Arch. Ration. Mech. Anal., 225 (2017), pp. 469--548], who proved scattering for the whole region of the mass/energy plane where the virial quantity is guaranteed to be positive. In this paper, we prove scattering in a slightly larger region where, in particular, the virial quantity is no longer guaranteed to be sign definite.},
  author       = {Killip, Rowan and Murphy, Jason and Visan, Monica},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {NLS, scattering, viral},
  number       = {5},
  pages        = {5803--5812},
  publisher    = {Society for Industrial & Applied Mathematics},
  title        = {{Scattering for the cubic-quintic NLS: Crossing the virial threshold}},
  doi          = {10.1137/20m1381824},
  volume       = {53},
  year         = {2021},
}

@article{9039,
  abstract     = {We give a short and self-contained proof for rates of convergence of the Allen--Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen--Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.},
  author       = {Fischer, Julian L and Laux, Tim and Simon, Theresa M.},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  number       = {6},
  pages        = {6222--6233},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies}},
  doi          = {10.1137/20M1322182},
  volume       = {52},
  year         = {2020},
}

@article{71,
  abstract     = {We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in ℝd. These metrics are natural discrete counterparts to the Kantorovich metric 𝕎2, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric Wt in terms of 𝕎2, as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric.},
  author       = {Gladbach, Peter and Kopfer, Eva and Maas, Jan},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  number       = {3},
  pages        = {2759--2802},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Scaling limits of discrete optimal transport}},
  doi          = {10.1137/19M1243440},
  volume       = {52},
  year         = {2020},
}

@article{22080,
  abstract     = {We consider discrete analogues of two well-known open problems regarding invariant measures for dispersive PDE, namely, the invariance of the Gibbs measure for the continuum (classical) Heisenberg model and the invariance of white noise under focusing cubic nonlinear Schrödinger equation. These continuum models are completely integrable and connected by the Hasimoto transform; correspondingly, we focus our attention on discretizations that are also completely integrable and also connected by a discrete Hasimoto transform. We consider these models on the infinite lattice ℤ. Concretely, for a completely integrable variant of the classical Heisenberg spin chain model (introduced independently by Haldane, Ishimori, and Sklyanin) we prove the existence and uniqueness of solutions for initial data following a Gibbs law (which we show is unique) and show that the Gibbs measure is preserved under these dynamics. In the setting of the focusing Ablowitz--Ladik system, we prove invariance of a measure that we will show is the appropriate discrete analogue of white noise. We also include a thorough discussion of the Poisson geometry associated to the discrete Hasimoto transform introduced by Ishimori that connects the two models studied in this article.},
  author       = {Angelopoulos, Yannis and Killip, Rowan and Visan, Monica},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  number       = {1},
  pages        = {135--163},
  publisher    = {Society for Industrial & Applied Mathematics},
  title        = {{Invariant measures for integrable spin chains and an integrable discrete nonlinear Schrödinger equation}},
  doi          = {10.1137/19m1265314},
  volume       = {52},
  year         = {2020},
}

@article{9781,
  abstract     = {We consider the Pekar functional on a ball in ℝ3. We prove uniqueness of minimizers, and a quadratic lower bound in terms of the distance to the minimizer. The latter follows from nondegeneracy of the Hessian at the minimum.},
  author       = {Feliciangeli, Dario and Seiringer, Robert},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Applied Mathematics, Computational Mathematics, Analysis},
  number       = {1},
  pages        = {605--622},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Uniqueness and nondegeneracy of minimizers of the Pekar functional on a ball}},
  doi          = {10.1137/19m126284x},
  volume       = {52},
  year         = {2020},
}

@article{22045,
  abstract     = {We consider the initial-value problem for the cubic-quintic nonlinear Schrödinger equation (𝑖𝜕𝑡+Δ)⁢𝜓 =𝛼1⁢𝜓 −𝛼3⁢|𝜓|2⁢𝜓 +𝛼5⁢|𝜓|4⁢𝜓 in three spatial dimensions in the class of solutions with |𝜓⁡(𝑥)| →𝑐 >0 as |𝑥| →∞. Here 𝛼1, 𝛼3, 𝛼5, and 𝑐 are such that 𝜓⁡(𝑥) ≡𝑐 is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to 𝜓⁡(𝑥) →1 as |𝑥| →∞, we study the associated initial-value problem for 𝑢 =𝜓 −1 and prove a scattering result for small initial data in a weighted Sobolev space.},
  author       = {Killip, Rowan and Murphy, Jason and Visan, Monica},
  issn         = {0036-1410},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {cubic-quintic NLS, nonvanishing boundary conditions, space-time resonances, scattering},
  number       = {3},
  pages        = {2681--2739},
  publisher    = {Society for Industrial & Applied Mathematics},
  title        = {{The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions}},
  doi          = {10.1137/17m1116702},
  volume       = {50},
  year         = {2018},
}

@article{22078,
  abstract     = {Let 𝑑 ≥4 and let u be a global solution to the focusing mass-critical nonlinear Schrödinger equation 𝑖⁢𝑢𝑡 +Δ⁢𝑢 =−|𝑢|4𝑑 ⁢𝑢 with spherically symmetric 𝐻1𝑥 initial data and mass equal to that of the ground state Q. We prove that if u does not scatter, then, up to phase rotation and scaling, u is the solitary wave 𝑒𝑖⁢𝑡⁢𝑄. Combining this result with that of Merle [Duke Math. J., 69 (1993), pp. 427–453], we obtain that in dimensions 𝑑 ≥4, the only spherically symmetric minimal-mass nonscattering solutions are, up to phase rotation and scaling, the pseudoconformal ground state and the ground state solitary wave.},
  author       = {Killip, Rowan and Li, Dong and Visan, Monica and Zhang, Xiaoyi},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  number       = {1},
  pages        = {219--236},
  publisher    = {Society for Industrial & Applied Mathematics},
  title        = {{Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS}},
  doi          = {10.1137/080720358},
  volume       = {41},
  year         = {2009},
}

@article{22034,
  abstract     = {We consider the focusing mass‐critical nonlinear Schrödinger equation and prove that blowup solutions to this equation with initial data in 𝐻𝑠⁡(\R𝑑), 𝑠 >𝑠0⁡(𝑑) and 𝑑 ≥3, concentrate at least the mass of the ground state at the blowup time. This extends recent work by Colliander et al. [Math. Res. Lett., 12 (2005), pp. 357–375], Hmidi and Keraani [Remarks on the Blowup for the 𝐿2‐Critical Nonlinear Schrödinger Equations, preprint], and Tzirakis [SIAM J. Math. Anal., 37 (2006), pp. 1923–1946] on the blowup of the two‐dimensional and one‐dimensional mass‐critical focusing nonlinear Schrödinger equation below the energy space to all dimensions 𝑑 ≥3.},
  author       = {Visan, Monica and Zhang, Xiaoyi},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  number       = {1},
  pages        = {34--56},
  publisher    = {Society for Industrial & Applied Mathematics},
  title        = {{On the blowup for the L^2‐critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class}},
  doi          = {10.1137/060663969},
  volume       = {39},
  year         = {2007},
}