Related papers: Stable Singularity Formation for the Inviscid Prim…
The primitive equations (PEs) model planetary large-scale oceanic and atmospheric dynamics. While it has been shown that there are smooth solutions to the inviscid PEs (also called the hydrostatic Euler equations) with constant temperature…
Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial…
Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs…
We consider a special class of infinite energy solutions to the inviscid incompressible porous medium equations (IPM), introduced in Castro-C\'ordoba-Gancedo-Orive [9]. The (IPM) equations then reduce to a one-dimensional nonlocal nonlinear…
In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show…
We study the formation of singularity for the isothermal Euler-Poisson system arising from plasma physics. Contrast to the previous studies yielding only limited information on the blow-up solutions, for instance, sufficient conditions for…
We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at…
We study singularity formation in two one-dimensional nonlinear wave models with quadratic time-derivative nonlinearities. The non-null model violates the null condition and typically develops finite-time blow-up; the null-form model is…
The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent…
We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up…
The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth…
This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The…
Under the genuinely nonlinear assumption for 1-D $n\times n$ strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic…
In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping…
It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup…
We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a…
Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,\alpha}$ initial velocity develops a finite time…
We study the formation of singularity for the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic…
A class of semi-bounded solutions of the two-dimensional incompressible Euler equations satisfying either periodic or Dirichlet boundary conditions is examined. For smooth initial data, new blowup criteria in terms of the initial concavity…
We study the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite…