In this paper, we first consider a nonlocal mixed diffusion equation ∂tu(t,x)=Lu(t,x) in multiparticle systems with different spatial distributions, where the combination of nonlocal diffusion operators L in a spatial variable is defined by the improper integrals with kernels Jj(x) and the low frequencies of Ĵj(ξ) have the same kind of asymptotic expansions. We use the energy method to establish the decay estimates of solutions to ∂tu(t,x)=Lu(t,x) with initial condition u(0, x) = u0(x). Finally, we consider an anisotropic single-particle equation with nonlocal operator L is defined by the improper integrals with kernels Jj(xj) and it is a generation of an anisotropic nonlocal operator −∑j=1N(−∂xjxj)αj.
In this paper, we consider the Cauchy problem to the tri-dimensional compressible Navier-Stokes-Korteweg system with a specific choice on the Korteweg tensor in the whole space and establish the global solutions to the tri-dimensional Navier-Stokes-Korteweg equations with a class of large initial data whose L 2 norm can be arbitrarily large.
In this paper, we will give the first result concerning the non-uniform dependence on initial data for the 2D magnetohydrodynamics (MHD)-Boussinesq equations as a hyperbolic-parabolic system. More precisely, we prove that the data-to-solution map of the Cauchy problem to the 2D MHD-Boussinesq equations is not uniformly continuous in Hs with s > 2.
In this paper, we consider the Cauchy problem of tri-dimensional compressible Navier–Stokes equations and construct global smooth solutions by choosing a class of new special initial velocity and density whose [Formula: see text]-norm can be arbitrarily large and improve the previous result in Li et al. [J. Math. Fluid Mech. 24, 22 (2022)]. Our main idea is splitting the linearized equations from the compressible Navier–Stokes equations and exploring the damping effect of the linearized system.
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