This paper is concerned with curvature blow-up and peakons for the high-order Fokas-Qiao-Xia-Li equations, which includes the Camassa-Holm equation, the Novikov equation, the Fokas-Olver-Rosenau-Qiao equation, the Fokas-Qiao-Xia-Li equation as its special cases. Firstly, we show the local well-posedness for the Cauchy problem of the equation in the framework of Sobolev spaces and Besov spaces. Then, we establish the precise blow-up mechanism for the strong solutions by means of the transport equation theory, and study the formation of singularities and provide several types of sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solution. Finally, peakons are discussed. The results are helpful to understand how higher-order nonlinearities affect the dispersion dynamics and breakdown mechanism of solutions.
A fully discrete implicit-explicit scheme for stochastic Cahn-Hilliard equation driven by multiplicative noise in a two-dimensional setting is considered in this paper. The spatial discretization is a polynomial based spectral method and the temporal discretization is a tamed semi-implicit scheme which treats the nonlinear term explicitly. We show that the scheme is unconditionally stable under various norms, and establish optimal strong convergence rates. We also present numerical experiments to validate our theoretical results.
We study the large-time behavior of smooth solutions to the Cauchy problem for the multi-dimensional unipolar isentropic hydrodynamic model of semiconductors, represented by the Euler-Poisson equations with weak damping -1/(1+t)(lambda)nu for lambda is an element of (0, 1). When the doping profile is a positive constant, the system is proved to admit a unique smooth solution which converges to a constant steady-state in the sub-exponential form of O ((1 + t)(- & vartheta;+lambda/2) e(-eta(1+t)1-lambda)) for some number eta > 0. Here, the index & vartheta; is an element of [lambda, infinity) relies on the size of initial perturbation, and could be large enough once the initial perturbation is sufficiently close to zero, such that the convergence rate involving the part of algebraic decay can be arbitrarily fast. This is a new technical issue totally different from the previous studies for the N-D compressible Euler equations with weak time-dependent damping. The other new observation is that, the effect of the weak damping with lambda is an element of (0, 1) makes the exponential decay of the solution to the Euler-Poisson system with the regular damping for lambda = 0 slow down to the sub-exponential decay. This is also completely different from that on the compressible Euler system with time-dependent damping.
We study the large-time behavior of solutions to the outflow problem for onedimensional compressible barotropic Navier-Stokes equations with general pressure. The viscosity & micro; under consideration depends on the density rho according to & micro; = & micro;(-)rho(alpha) with 0 <= alpha <= 21, which is motivated by the kinetic theory. We prove that the stationary wave, the rarefaction wave, and the superposition of stationary and rarefaction waves are all time-asymptotically stable under large initial perturbation. The crucial step in the proof is to derive the uniform positive upper and lower bounds for the density by applying Kanel's argument. An intrinsic cancellation effect is explored to obtain certain nonlinear energy estimates. Depending on the spatial decay of the perturbation either being algebraic or exponential, we prove convergence with corresponding rate in time to the non-degenerate stationary wave by weighted energy method for large perturbation.
In this paper, we propose a non-isothermal kinetic model describing interactions between dust particles and gas molecules. We assume that gas-dust collisions follow a diffuse reflection mechanism at the surface of the dust particles. The surface temperature of the particles is treated as a function of time and space, satisfying a transport-like equation. The key novelty of this model is its ability to derive both the conservation of total energy in the system and an explicit expression for entropy. After adimensionalising the equations, we perform a formal diffusive asymptotic analysis leading to a coupled system of partial differential equations. In particular, we examine the limiting case in which the mass ratio between gas and dust, as well as the Mach and Knudsen numbers, tend to zero.
For the weak solutions to the continuity equation in two-dimensional whole spaces, the control of trajectories which blows up to infinity in the future or in the past is discussed. In other words, by utilizing the Poincare inequality, a new sufficient condition for the uniqueness is obtained which is quite different from the previous one due to Diperna-Lions [Invent. Math., 98(3):511-547, 1989].
It is well known that solutions of the incompressible Euler equations can grow rapidly in time and are thus unstable. However, when coupled with the magnetic field in the electron Hall-MHD system concerned here, we demonstrate that the Sobolev-norm of any perturbation near a background magnetic field actually decays algebraically in time. To the author's knowledge, this is the first stability finding of the inviscid electron inertia Hall-MHD equations even with small initial data. Our result especially confirms the stabilizing effects of the magnetic field on the electrically conducting fluids, a phenomenon that has been observed in physical experiments and numerical simulations.
In this paper, we propose a multiphysics finite element method for the thermoporoelasticity model with nonlinear convective transport term. To design some stable numerical methods and reveal the multi-physical processes of deformation, diffusion and heat, we introduce three new variables to reformulate the original model into a fluid coupled problem. Then, we propose a fully discrete multiphysics finite element method for P-2-P-1-P-1-P-1 element pair with the backward Euler method. In computation, we use the Newton method to solve the nonlinear system, which is equivalent to a stabilized method. Also, we prove that the fully discrete multiphysics finite element method has an optimal convergence order. Then, we give some numerical examples to show the above proposed method can effectively overcome not only the numerical oscillation caused by the nonlinear thermal convection term but also the pressure and temperature oscillations and the "locking" of the displacement u when lambda -> infinity. Finally, we draw conclusions to summarize the main results of this paper.
The asymptotic behavior of solutions has been addressed for a second-order abstract equation with time-varying delay here. The well-posedness of considered system has been established by using Kato's variable norm technique under appropriate assumptions on the memory kernel and the weight of the delay. The exponential stability of the visco elastic equation with time-varying delay has been presented by the flatness rate of the kernel, in the term of viscoelasticity with delay perturbations.
