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1. On the regularity of solutions to the k-generalized Korteweg-de Vries equation

   Kenig, C.; Linares, F.; Ponce, G.; Vega, L. (April 2018, Proceedings of the American Mathematical Society)

   The authors establish an instantaneous gain of regularity to the left, for solutions og the k-generalized KdV equations with data in Sobolev spaces of fractional order 

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2. Global Regularity of Skew Mean Curvature Flow for Small Data in d ≥ 4 Dimensions

   https://doi.org/10.1093/imrn/rnad104  

   Huang, Jiaxi; Li, Ze; Tataru, Daniel (May 2023, International Mathematics Research Notices)

   Abstract The skew mean curvature flow is an evolution equation for a $$d$$ dimensional manifold immersed into $$\mathbb {R}^{d+2}$$, and which moves along the binormal direction with a speed proportional to its mean curvature. In this article, we prove small data global regularity in low-regularity Sobolev spaces for the skew mean curvature flow in dimensions $$d\geq 4$$. This extends the local well-posedness result in [7]. 

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3. Rough solutions of the relativistic Euler equations

   https://doi.org/10.1142/S0219891624500127  

   Yu, Sifan (June 2024, Journal of Hyperbolic Differential Equations)

   We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory. 

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4. Global Sobolev persistence for the fractional Boussinesq equation with zero diffusivity

   Kukavica, Igor; Wang, Weinan (January 2020, Pure and applied functional analysis)

   We prove the persistence of regularity for the 2D alpha-franctional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e. for (u_0,rho_0)\in W^s,q(R^2)\times W^s,q(R^2), where s>1 and 2 

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5. Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics

   https://doi.org/10.1007/s00205-021-01697-6  

   Deng, Yu; Zillinger, Christian (July 2021, Archive for Rational Mechanics and Analysis)

   Abstract In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity. 

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