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1. Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

   https://doi.org/10.1007/s00220-022-04522-7  

   Laurens, Thierry ( November 2022 , Communications in Mathematical Physics)

   Given a suitable solutionV(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$$u(0,x)\in V(0,x)+H^{-1}\left(R\right)$. Our conditions onVdo include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$$V(0,x)\in H^5(R/Z)$satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022.https://doi.org/10.1088/1361-6544/ac37f5) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019.https://doi.org/10.4007/annals.2019.190.1.4) where$$V\equiv 0$$$V\equiv 0$. In that setting, it is known that$$H^{-1}(\mathbb {R})$$$H^{-1}\left(R\right)$is sharp in the class of$$H^s(\mathbb {R})$$$H^s\left(R\right)$spaces.

    

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2. Local Well-Posedness of the Skew Mean Curvature Flow for Small Data in $$d\geqq 2$$ Dimensions

   https://doi.org/10.1007/s00205-023-01952-y  

   Huang, Jiaxi ; Tataru, Daniel ( January 2024 , Archive for Rational Mechanics and Analysis)

   The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${\mathbb {R}}^{d+2}$$$R^{d+2}$(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions$$d \geqq 4$$$d\geqq 4$. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\geqq 2$$$d\geqq 2$. This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.

    

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3. Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions

   https://doi.org/10.1007/s00220-021-04303-8  

   Huang, Jiaxi ; Tataru, Daniel ( January 2022 , Communications in Mathematical Physics)

   The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${{\mathbb {R}}}^{d+2}$$$R^{d+2}$(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\ge 4$$$d\ge 4$.

    

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4. An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

   https://doi.org/10.1007/s10240-024-00144-y  

   De Lellis, Camillo ; Nardulli, Stefano ; Steinbrüchel, Simone ( February 2024 , Publications mathématiques de l'IHÉS)

   We consider integral area-minimizing 2-dimensional currents$T$$T$in$U\subset \mathbf {R}^{2+n}$$U\subset R^{2+n}$with$\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$$\partial T=Q〚\Gamma 〛$, where$Q\in \mathbf {N} \setminus \{0\}$$Q\in N\setminus \left\{0\right\}$and$\Gamma $$\Gamma$is sufficiently smooth. We prove that, if$q\in \Gamma $$q\in \Gamma$is a point where the density of$T$$T$is strictly below$\frac{Q+1}{2}$$\frac{Q+1}{2}$, then the current is regular at$q$$q$. The regularity is understood in the following sense: there is a neighborhood of$q$$q$in which$T$$T$consists of a finite number of regular minimal submanifolds meeting transversally at$\Gamma $$\Gamma$(and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for$Q=1$$Q=1$. As a corollary, if$\Omega \subset \mathbf {R}^{2+n}$$\Omega \subset R^{2+n}$is a bounded uniformly convex set and$\Gamma \subset \partial \Omega $$\Gamma \subset \partial \Omega$a smooth 1-dimensional closed submanifold, then any area-minimizing current$T$$T$with$\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$$\partial T=Q〚\Gamma 〛$is regular in a neighborhood of $\Gamma $$\Gamma$.

    

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5. On the rigorous derivation of the incompressible Euler equation from Newton’s second law

   https://doi.org/10.1007/s11005-023-01630-w  

   Rosenzweig, Matthew ( January 2023 , Letters in Mathematical Physics)

   A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system ofNparticles interacting in$${\mathbb {T}}^d$$$T^d$,$$d\ge 2$$$d\ge 2$, via Newton’s second law through asupercritical mean-field limit. Namely, the coupling constant$$\lambda $$$\lambda$in front of the pair potential, which is Coulombic, scales like$$N^{-\theta }$$$N^{-\theta }$for some$$\theta \in (0,1)$$$\theta \in (0,1)$, in contrast to the usual mean-field scaling$$\lambda \sim N^{-1}$$$\lambda \sim N^{-1}$. Assuming$$\theta \in (1-\frac{2}{d(d+1)},1)$$$\theta \in (1-\frac{2}{d(d+1)},1)$, they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$N\rightarrow \infty $$$N\to \infty$. Han-Kwan and Iacobelli asked if their range for$$\theta $$$\theta$was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$N\rightarrow \infty $$$N\to \infty$for$$\theta \in (1-\frac{2}{d},1)$$$\theta \in (1-\frac{2}{d},1)$. Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta $$$\theta$. Additionally, we show that for$$\theta \le 1-\frac{2}{d}$$$\theta \le 1-\frac{2}{d}$, one cannot, in general, expect convergence in the modulated energy notion of distance.

    

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