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1. Uniqueness of the 2D Euler equation on rough domains

   https://doi.org/10.4171/jems/1676  

   Agrawal, Siddhant; Nahmod, Andrea R (July 2025, Journal of the European Mathematical Society)

   We consider the 2D incompressible Euler equation on a bounded simply connected domain\Omega. We give sufficient conditions on the domain\Omegaso that for any initial vorticity\omega_{0} \in L^{\infty}(\Omega), the weak solutions are unique. Our sufficient condition is slightly more general than the condition that\Omegais aC^{1,\alpha}domain for some\alpha>0, with its boundary belonging toH^{3/2}(\mathbb{S}^{1}). As a corollary, we prove uniqueness forC^{1,\alpha}domains for\alpha >1/2and for convex domains which are alsoC^{1,\alpha}domains for some\alpha >0. Previously, uniqueness for general initial vorticity inL^{\infty}(\Omega)was only known forC^{1,1}domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below theC^{1,1}regularity is the fact that for less regular domains, the velocity near the boundary is no longer log-Lipschitz. We overcome this barrier by defining a new change of variable which we then use to define a novel energy functional. 

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2. 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)

   Abstract 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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3. 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)

   Abstract 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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4. 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)

   Abstract 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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5. Continuum limit of 2D fractional nonlinear Schrödinger equation

   https://doi.org/10.1007/s00028-023-00881-3  

   Choi, Brian; Aceves, Alejandro (March 2023, Journal of Evolution Equations)

   Abstract We prove that the solutions to the discrete nonlinear Schrödinger equation with non-local algebraically decaying coupling converge strongly in$$L^2({\mathbb {R}}^2)$$ $L^2\left(R^2\right)$to those of the continuum fractional nonlinear Schrödinger equation, as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter$$\alpha \in (1,2)$$ $\alpha \in (1,2)$approaches the boundaries. 

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