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We show that the log-Hölder coefficients of a solution to the transport equation is preserved in time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial value problem for the Euler–Riesz system, where the interaction forcing is given by ∇ (− Δ) s ρ for some − 1 < s < 0 , with s = − 1 corresponding to the classical Euler–Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler–Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler–Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler–Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler–Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support. [ABSTRACT FROM AUTHOR]

Abstract We show existence of self-similar solutions satisfying Kolmogorov's scaling for generalized dyadic models of the Euler equations, extending a result of Barbato, Flandoli, and Morandin [1]. The proof is based on the analysis of certain dynamical systems on the plane. [ABSTRACT FROM AUTHOR]

Abstract We consider the stationary problem for the generalized Proudman–Johnson equation ψ ψ x x x − a ψ x ψ x x = ν ψ x x x x + f , where a ∈ R and ν > 0 are parameters and ψ and f satisfy the periodic boundary conditions on [ − π , π ]. We establish the existence of a time-dependent solution and the uniqueness of the stationary solution for some large ν. Then, we investigate the behavior of solution sequences in the limit ν → 0 + , fixing a and f , where bifurcations appear and new solutions emerge. In particular, we analyze if there is convergence towards stationary solutions for the corresponding inviscid equation. [ABSTRACT FROM AUTHOR]

We provide a simple proof that the Cauchy problem for the incompressible Euler equations in R d with any d ≥ 3 is ill-posed in critical Sobolev spaces, extending an earlier work of Bourgain and Li (2021) [5] in the case d = 3. The ill-posedness is shown for certain critical Lorentz spaces as well. [ABSTRACT FROM AUTHOR]

Abstract: The ubiquitin-like (Ubl) system has been shown to be ubiquitous in all three kingdoms of life following the very recent characterization of ubiquitin-like small archaeal modifier proteins (SAMP1 and 2) from Haloferax volcanii. The ubiquitin (Ub) and Ubl molecules in eukaryotes have been studied extensively and their cellular functions are well established. Biochemical and structural data pertaining to prokaryotic Ubl protein (Pup) continue to be reported. In contrast to eukaryotes and prokaryotes, no structural information on the archaeal Ubl molecule is available. Here we determined the crystal structure of SAMP1 at 1.55Å resolution and generated a model of SAMP2. These were then compared with other Ubl molecules from eukaryotes as well as prokaryotes. The structure of SAMP1 shows a β-grasp fold of Ub, suggesting that the archaeal Ubl molecule is more closely related to eukaryotic Ub and Ubls than to its prokaryotic counterpart. The current structure identifies the location of critical elements such a single lysine residue (Lys4), C-terminal di-glycine motif, hydrophobic patches near leucine 60, and uniquely inserted α-helical segments (α1 and α3) in SAMP1. Based on the structure of SAMP1, several Ub-like features of SAMPs such as poly-SAMPylation and non-covalent interactions have been proposed, which should provide the basis for further investigations concerning the molecular function of archaeal Ubls and the large super-family of β-grasp fold proteins in the archaeal kingdom. [Copyright &y& Elsevier]

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: { (x 1 , x 2 , x 3) : 0 < x 3 < x 2 < x 1 }. In this domain, we prove local well-posedness for C α vorticities not necessarily vanishing on the boundary with any 0 < α < 1 , and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of R 3 via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in R 3 with bounded and piecewise smooth vorticities. [ABSTRACT FROM AUTHOR]