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In Part II of this sequence to our previous paper for the 3-dimensional Euler equations \cite{zhang2022potential}, we investigate potential singularity of the $n$-diemnsional axisymmetric Euler equations with $C^\alpha$ initial vorticity…
We give sufficient conditions on the initial data so that a semilinear wave inequality blows-up in finite time. Our method is based on the study of an associated second order differential inequality. The same method is applied to some…
The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the…
We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb R^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb R$, with subconformal power nonlinearity. We…
In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery…
We study the relation between perverse stability conditions and geometric stability conditions under blow up. We confirm a conjecture of Toda in some special cases and show that geometric stability conditions can be induced from perverse…
Motivated by the need to control the exponential growth of constraint violations in numerical solutions of the Einstein evolution equations, two methods are studied here for controlling this growth in general hyperbolic evolution systems.…
This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a reaction diffusion equation with special diffusion processes. It is shown, under certain conditions on the initial data, that the…
We present new many-parameter families of strongly and symmetric hyperbolic formulations of Einstein's equations that include quite general algebraic and live gauge conditions for the lapse. The first system that we present has 30 variables…
We construct a new Godunov type relativistic hydrodynamics code in Milne coordinates, using a Riemann solver based on the two-shock approximation which is stable under the existence of large shock waves. We check the correctness of the…
A sufficient integral criterion for a blow-up solution of the Hopf equations (the Euler equations with zero pressure) is found. This criterion shows that a certain positive integral quantity blows up in a finite time under specific initial…
In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schr\"odinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for…
This paper concerns the study of the incompressible Euler equations with variable density, in the case of space dimension $d=2$. Contrarily to their homogeneous (constant density) counterpart, those equations are not known to be well-posed…
We derive and analyze shock-wave solutions of hydrodynamic equations describing repulsively interacting one dimensional Bose gas. We also use the number-conserving Bogolubov approach to verify accuracy of the Gross-Pitaevskii equation in…
In the present paper, we investigate blow-up and lifespan estimates for a class of semilinear hyperbolic coupled system in $\mathbb{R}^n$ with $n\geqslant 1$, which is part of the so-called Nakao's type problem weakly coupled a semilinear…
The nonlocal nonlinear aggregation equation in one space dimension is investigated. In the so-called attractive case smooth solutions blow up in finite time, so that weak measure solutions are introduced. The velocity involved in the…
We consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data. We prove that there exists an open set of initial data such that the corresponding solution exists in…
In hyperbolic reductions of the Einstein equations the evolution of gauge conditions or constraint quantities is controlled by subsidiary systems. We point out a class of non-linearities in these systems which may have the potential of…
This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u \nabla v) + \xi \nabla\cdot (u \nabla w) + \lambda u - \mu u^k,…
This paper is concerned with the wave breaking phenomena for a coupled periodic Camassa-Holm system. We establish a new blowup criterion for strong solutions by the method of characteristic and convolution estimates, and also give the…