Related papers: On the Lagrangian Trajectories for the One-Dimensi…
We present a numerical method of analyzing possibly singular incompressible 3D Euler flows using massively parallel high-resolution adaptively refined numerical simulations up to 8192^3 mesh points. Geometrical properties of Lagrangian…
This letter studies the Euler-alignment system with weakly singular influence functions by introducing a novel technique to bound the density. Instead of resorting to a nonlinear maximum principle used in [C. Tan, Nonlinearity, 33:…
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…
We are concerned with the critical threshold phenomena in the Restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for 3D and the 4D restricted Euler equations. It is well known that the…
We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness…
We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and…
Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $C^{1,\alpha}$ initial velocity,…
The Standard Model of elementary particle physics is one of the most successful models of contemporary theoretical physics being in full agreement with experiments. However, its mathematical structure deserves further investigations both…
We consider a well known model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field with finite curvature elasticity. We prove the existence of a plethora of equilibria, corresponding to…
Point vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the streamfunction. Special focus is given to the case of the surface…
Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite…
We investigate global solutions to the Euler-alignment system in $d$ dimensions with unidirectional flows and strongly singular communication protocols $\phi(x) = |x|^{-(d+\alpha)}$ for $\alpha \in (0,2)$. Our paper establishes global…
We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary.…
We study a one-dimensional thin film equation combining competitive effects of aggregation and repulsion, where repulsion is modeled by fourth-order diffusion and aggregation by backward second-order degenerate diffusion with exponent…
Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in…
For the compressible Euler equations, even when the initial data are uniformly away from vacuum, solution can approach vacuum in infinite time. Achieving sharp lower bounds of density is crucial in the study of Euler equations. In this…
Euler alignment systems appear as hydrodynamic limits of interacting self-propelled particle systems such as the (generalized) Cucker-Smale model. In this work, we study weak solutions to an Euler alignment system on smooth, bounded,…
We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere $C^1$…
In this paper, we consider global existence of classical solutions to the following kinetic model of pattern formation \begin{equation} \begin{cases} u_t=\Delta (\gamma (v)u)+\mu u(1-u) -\Delta v+v=u \end{cases} \qquad (0.1)…
In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that…