Related papers: On a Generalized System with Applications to Ideal…
In this paper we deal with the existence of local strong solution for a perfect compressible viscous fluid, heat conductive and self gravitating, coupled with a first order kinetics used in astrophysical hydrodynamical models. In our…
In the framework of a mixed finite element method, a structure-preserving formulation for incompressible magnetohydrodynamic (MHD) equations with general boundary conditions is proposed. A leapfrog-type temporal scheme fully decouples the…
We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{\mu t}u^p$ ($\mu\in\mathbb{R}, p>1$), posed on cones of the hyperbolic space. Under a…
We present an extension to the special relativistic, ideal magnetohydrodynamics (MHD) equations, designed to capture effects due to resistivity. The extension takes the simple form of an additional source term which, when implemented…
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…
This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We…
We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical…
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist…
We propose a one-dimensional (1D) model for the three-dimensional(3D) incompressible ideal magnetohydrodynamics. We establish a regularity criterion of the Beale-Kato-Majda type for this 1D model. Without the stretching effect, the model…
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…
This paper estimates the blow-up time for the heat equation $u_t=\Delta u$ with a local nonlinear Neumann boundary condition: The normal derivative $\partial u/\partial n=u^{q}$ on $\Gamma_{1}$, one piece of the boundary, while on the rest…
In this paper, we consider the ideal magnetic B\'{e}nard problem in both two and three dimensions and prove local-in-time existence and uniqueness of strong solutions in $H^s$ for $s > \frac{n}{2}+1, n = 2,3$. In addition, a necessary…
We study an anisotropic system arising in magnetohydrodynamics (MHD) in the whole space R^3 , in the case where there are no diffusivity in the vertical direction and only a small diffusivity in the horizontal direction (of size…
In this paper, we deal with the $2\frac{1}{2}$ dimensional Hall MHD by taking the velocity field $u$ and the magnetic field $B$ of the form $u(t,x,y)=\left(\nabla^{\perp}\phi(t,x,y), W(t,x,y)\right)$ and…
We consider solutions $u$ to the 3d nonlinear Schr\"odinger equation $i\partial_t u + \Delta u + |u|^2u=0$. In particular, we are interested in finding criteria on the initial data $u_0$ that predict the asymptotic behavior of $u(t)$, e.g.,…
In this paper, we investigate the incompressible viscous and resistive Hall magnetohydrodynamic equations (Hall MHD in short). We first study the regularity of the magneto-vorticity field $B+\omega$. In three dimensions, we derive some…
We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the \textit{scale-invariant regime}, time-derivative nonlinearities, mass terms and Tricomi terms.…
This paper deals with the finite-time blow-up phenomena of classical solutions for Vlasov/Navier-Stokes equations under suitable assumptions on the initial configurations. We show that a solution to the coupled kinetic-fluid system may be…
In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of…
We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the…