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1. 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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2. Counterexamples to maximal regularity for operators in divergence form

   https://doi.org/10.1007/s00013-024-02014-9  

   Bechtel, Sebastian; Mooney, Connor; Veraar, Mark (August 2024, Archiv der Mathematik)

   Abstract In this paper, we present counterexamples to maximal$$L^p$$ $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal$$L^2$$ $L^2$-regularity on$$H^{-1}$$ $H^{-1}$under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal$$L^p$$ $L^p$-regularity on$$H^{-1}(\mathbb {R}^d)$$ $H^{-1}\left(R^d\right)$or$$L^2$$ $L^2$-regularity on$$L^2(\mathbb {R}^d)$$ $L^2\left(R^d\right)$. 

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3. 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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4. Sobolev extension in a simple case

   https://doi.org/10.1515/ans-2023-0132  

   Drake, Marjorie; Fefferman, Charles; Ren, Kevin; Skorobogatova, Anna (April 2025, Advanced Nonlinear Studies)

   Abstract In this paper, we establish the existence of a bounded, linear extension operator $T:L^{2,p}\left(E\right)\to L^{2,p}\left({\mathbb{R}}^2\right)$$$T :{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$$when 1 <p< 2 andEis a finite subset of ${\mathbb{R}}^2$$${\mathbb{R}}^{2}$$contained in a line. 

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5. Stability and Large-Time Behavior on 3D Incompressible MHD Equations with Partial Dissipation Near a Background Magnetic Field

   https://doi.org/10.1007/s00205-025-02100-4  

   Lin, Hongxia; Wu, Jiahong; Zhu, Yi (April 2025, Archive for Rational Mechanics and Analysis)

   Abstract Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in$$\mathbb R^3$$ $R^3$. The velocity equation in this system is the 3D Navier–Stokes equation with dissipation only in the$$x_1$$ $x_1$-direction, while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting$$H^3({\mathbb {R}}^3)$$ $H^3\left(R^3\right)$. In addition, explicit decay rates in$$H^2({\mathbb {R}}^3)$$ $H^2\left(R^3\right)$are also obtained. For when there is no presence of a magnetic field, the 3D anisotropic Navier–Stokes equation is not well understood and the small data global well-posedness in$$\mathbb R^3$$ $R^3$remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps to stabilize the fluid. 

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