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1. 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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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)

   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. The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

   https://doi.org/10.1093/imrn/rnz174  

   Kenig, Carlos ; Mendelson, Dana ( November 2019 , International Mathematics Research Notices)

   We consider the focusing energy-critical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$, for any $s> 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$ for $s> 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.

    

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4. Well-posedness for the surface quasi-geostrophic front equation

   https://doi.org/10.1088/1361-6544/ad36a4  

   Ai, Albert ; Avadanei, Ovidiu-Neculai ( April 2024 , Nonlinearity)

   We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021Pure Appl. Anal.3403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation’s nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.

    

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5. Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity

   https://doi.org/10.1088/1361-6544/ac93e1  

   Friedman, Isaac ; Riaño, Oscar ; Roudenko, Svetlana ; Son, Diana ; Yang, Kai ( December 2022 , Nonlinearity)

   Abstract We consider two types of the generalized Korteweg–de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of H 1 that includes functions with polynomial decay, extending the result of Linares et al (2019 Commun. Contemp. Math. 21 1850056) to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. We include a comparison of solutions to both types of equations, in particular, we investigate soliton resolution for the positive and negative data with different decay rates. Finally, we study the interaction of various solitary waves in both models, showing the formation of solitons, dispersive radiation and even breathers, all of which are easier to track in nonlinearities with lower power. 

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