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One of the main results stated in Theorem 4.4 of our article, which appears in Trans. Amer. Math. Soc. 372 (2019), no. 10, 7191-7223, is that the structure algebra K[M(X,r)], over a field K, of a finite bijective left non-degenerate solution (X,r) of the Yang-Baxter equation is a module-finite central extension of a commutative affine subalgebra. This is proven by showing that the structure monoid M(X,r) is central-by-finite. This however is not true, even in case (X,r) is a (left and right) non-degenerate involutive solution. The proof contains a subtle mistake. However, it turns out that the monoid M(X,r) is abelian-by-finite and thus the conclusion that K[M(X,r)] is a module-finite normal extension of a commutative affine subalgebra remains valid. In particular, K[M(X,r)] is Noetherian and satisfies a polynomial identity. The aim of this paper is to give a proof of this result. In doing so, we also strengthen Lemma 5.3 (and its consequences, namely Lemma 5.4 and Proposition 5.5) showing that these results on the prime spectrum of the structure monoid hold even if the assumption that the solution (X,r) is square-free is omitted. [ABSTRACT FROM AUTHOR]

We obtain almost-sure scattering for the Schrödinger equation with a defocusing cubic nonlinearity in the Euclidean space \mathbb {R}^3, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [Trans. Amer. Math. Soc. Ser. B 2 (2015), pp. 1–50]. It also extends the results of Dodson, Lührmann and Mendelson [Adv. Math. 347 (2019), pp. 619–676] on the energy-critical equation in \mathbb {R}^4, to the energy-subcritical equation in \mathbb {R}^3. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect which makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao. Then, we propose a globalization argument in which we set up the I-method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime. [ABSTRACT FROM AUTHOR]

We investigate a regularity for weak solutions of the following generalized Leray equations (−Δ)α V − 2α−1/2α V + V ⋅ ∇ V − 1/2α x ⋅ ∇V + ∇P = 0, which arises from the study of self-similar solutions to the generalized Navier-Stokes equations in R3. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions V in the weighted Hilbert space Hαω(R3). Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from Hαω(R3) to Hω1+α(R3). This regularity result, together with linear theory for the non-local Stokes system, leads to pointwise estimates of V which allow us to obtain a natural pointwise property of the self-similar solution constructed by Lai, Miao, and Zheng [Adv. Math. 352 (2019), pp. 981–1043]. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Navier-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai Comm. Math. Phys., 328 (2014), pp. 29–44. [ABSTRACT FROM AUTHOR]

We study here the Zakharov-Kuznetsov equation in dimension 2, 3 and 4 and the modified Zakharov-Kuznetsov equation in dimension 2. Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of K solitons Rk (with distinct velocities), we prove the existence and uniqueness of a multi-soliton u such that |u(t) − ∑k=1K Rk(t)|H1 → 0 as t → + ∞. The convergence takes place in Hs with an exponential rate for all s ≥ 0. The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of H1-norms of the errors (inspired by Martel [Amer. J. Math. 127 (2005), pp. 1103-1140]), and introduce a new ingredient for the control of the Hs-norm in dimension d ≥ 2, by a technique close to monotonicity. [ABSTRACT FROM AUTHOR]

Let D = H\∪N k=1 Ck be a standard slit domain where H is the upper half-plane and Ck, 1 ≤ k ≤ N, are mutually disjoint horizontal line segments in H. Given a Jordan arc γ ⊂ D starting at ∂H, let gt be the unique conformal map from D \ γ[0, t] onto a standard slit domain Dt satisfying the hydrodynamic normalization. We prove that gt satisfies an ODE with the kernel on its right-hand side being the complex Poisson kernel of the Brownian motion with darning (BMD) for Dt, generalizing the chordal Loewner equation for the simply connected domain D = H. Such a generalization has been obtained by Y. Komatu in the case of circularly slit annuli and by R. O. Bauer and R. M. Friedrich in the present chordal case, but only in the sense of the left derivative in t. We establish the differentiability of gt in t to make the equation a genuine ODE. To this end, we first derive the continuity of gt(z) in t with a certain uniformity in z from a probabilistic expression of Зgt(z) in terms of the BMD for D, which is then combined with a Lipschitz continuity of the complex Poisson kernel under the perturbation of standard slit domains to get the desired differentiability. [ABSTRACT FROM AUTHOR]

We find Feynman-Kac type representation theorems for generalized diffusions. To do this we need to establish existence, uniqueness and regularity results for equations with measure-valued coefficients. [ABSTRACT FROM AUTHOR]