Remarks on the smoothness of the C1,αC^{1,\alpha} asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations

We show that the constructions of $C^{1,\alpha}$ asymptotically self-similar singularities for the 3D Euler equations by Elgindi, and for the 3D Euler equations with large swirl and 2D Boussinesq equations with boundary by Chen-Hou can be extended to construct singularity with velocity $\mathbf{u} \in C^{1,\alpha}$ that is not smooth at only one point. The proof is based on a carefully designed small initial perturbation to the blowup profile, and a BKM-type continuation criterion for the one-point nonsmoothness. We establish the criterion using weighted H\"older estimates with weights vanishing near the singular point. Our results are inspired by the recent work of Cordoba, Martinez-Zoroa and Zheng that it is possible to construct a $C^{1,\alpha}$ singularity for the 3D axisymmetric Euler equations without swirl and with velocity $\mathbf{u} \in C^{\infty}(\mathbb{R}^3 \backslash \{0\})$.Comment: In the previous version, the initial data is not in the weighted Holder space. We modify the space and show that the initial data is in the new weighted Holder space. 20 page

Nearly self-similar blowup of the slightly perturbed homogeneous Landau equation with very soft potentials

We study the slightly perturbed homogeneous Landau equation $$\partial_t f = a_{ij}(f) \cdot \partial_{ij} f + \alpha c(f) f, \quad c(f) = - \partial_{ij} a_{ij}(f),$$ with very soft potentials, where we increase the nonlinearity from $c(f) f$ in the Landau equation to $\alpha c(f) f$ with $\alpha>1$. For $\alpha > 1$ and close to $1$, we establish finite time nearly self-similar blowup from some smooth initial data $f_0 \geq 0$, which can be both radially symmetric or non-radially symmetric. The blowup results are sharp as the homogeneous Landau equation $(\alpha=1)$ is globally well-posed, which was established recently by Guillen and Silvestre. To prove the blowup results, we build on our previous framework \cite{chen2020slightly,chen2021regularity} on sharp blowup results of the De Gregorio model with nearly self-similar singularity to overcome the diffusion. Our results shed light on potential singularity formation in the inhomogeneous setting.Comment: 35 page

Singularity formation and global Well-posedness for the generalized Constantin–Lax–Majda equation with dissipation

We study a generalization due to De Gregorio and Wunsch et al of the Constantin–Lax–Majda equation (gCLM) on the real line ω_t + auω_x = u_xω−νΛ^γω, u_x = Hω, where H is the Hilbert transform and Λ=(−∂_(xx))^(1/2) . We use the method in Chen J et al (2019 (arXiv:1905.06387)) to prove finite time self-similar blowup for a close to 1/2 and γ=2 by establishing nonlinear stability of an approximate self-similar profile. For a  >  −1, we discuss several classes of initial data and establish global well-posedness and an one-point blowup criterion for different initial data. For a ≤ -1, we prove global well-posedness for gCLM with critical and supercritical dissipation

An Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for solving convection and convection-diffusion equations

We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines approach. The scheme can be viewed as a generalization of the standard Runge-Kutta finite volume (RK-FV) scheme for which the space-time region is partitioned by approximate characteristics with zero velocity. The high-order spatial reconstruction is achieved using the recently developed weighted essentially non-oscillatory schemes with adaptive order (WENO-AO); and the high-order temporal accuracy is achieved by explicit RK methods for convection equations and implicit-explicit (IMEX) RK methods for convection-diffusion equations. Our algorithm extends to higher dimensions via dimensional splitting. Numerical experiments demonstrate our algorithm's robustness, high-order accuracy, and ability to handle extra large time steps.Comment: 35 pages, 21 figures, submitted to the Journal of Computational Physic