Related papers: Optimal time-decay estimates for an Oldroyd-B mode…
In this paper, we study the decay rate in time to solutions of the Cauchy problem for the one-dimensional viscous conservation law where the far field states are prescribed. Especially, we deal with the case that the flux function which is…
This work is concerned with the large time behavior of solutions to the barotropic compressible Navier-Stokes equations in $\mathbb{R}^{d}(d\geq2)$. Precisely, it is shown that if the initial density and velocity additionally belong to some…
In this paper, we are concerned with optimal decay rates for higher order spatial derivatives of classical solutions to the full compressible MHD equations in three dimensional whole space. If the initial perturbation are small in…
In this paper, we study the following Cauchy problem for linear visco-elastic damped wave models with a general time-dependent coefficient $g=g(t)$: \begin{equation} \label{EqAbstract} \tag{$\star$} \begin{cases} u_{tt}- \Delta u +…
In this paper, we present a refined framework for the global-in-time well-posedness theory for the pressureless Euler--Navier--Stokes system and the optimal temporal decay rates of certain norms of solutions. Here the coupling of two…
This paper is concerned with the Cauchy problem of Navier-Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in $\R^3$. For less regular data and weaker compatibility condition than those…
In this paper, we investigate the Cauchy problem for the compressible nematic liquid crystal flows in three-dimensional whole space. First of all, we establish the time decay rates for compressible nematic liquid crystal flows by the method…
This expository article is intended to give an overview about recently achieved results on asymptotic properties of solutions to the Cauchy problem $u_{tt}-\Delta u+b(t)u_t =0,\qquad u(0,\cdot)=u_1,\quad \mathrm{D}_tu(0,\cdot)=u_2$ for a…
In this work we study stochastic Oldroyd type models for viscoelastic fluids in $\mathbb{R}^d, d= 2, 3$. We show existence and uniqueness of strong local maximal solutions when the initial data are in $H^s$ for $s>d/2, d= 2, 3$.…
We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter epsilon in front of the second-order time-derivative. In a recent paper, under a suitable assumption on initial data, we proved decay-error estimates for…
This article deals with the behavior in time of the solution to the Cauchy problem for a fractional wave equation with a weighted $L^1$ initial data. Initially, we establish the global existence of the solution using Fourier methods and…
We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the…
In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising in the $2D$ Oldroyd model of viscoelastic fluids of order one with the forcing term…
In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in $\mathbb{R}$, where the viscosity depends on the density in a super-linear power law(i.e.,…
We consider the Cauchy problem for wave equations with localized damping in ${\bf R}^{2}$. The damping is effective only near spatial infinity. We obtain fast energy decay estimate such that $O(t^{-2}\log t)$ as $t \to \infty$. Unlike the…
In this paper, we prove the global existence of analytical solutions to the compressible Oldroyd-B model without retardation near a non-vacuum equilibrium in ${\mathbb R}^n$ $(n=2,3)$. Zero retardation results in zero dissipation in the…
We prove that there exists a~large-data and global-in-time weak solution to a~system of partial differential equations describing an unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up…
We consider damped wave equations with a potential and rotational inertia terms. We study the Cauchy problem for this model in the one dimensional Euclidean space and we obtain fast energy decay and L^2-decay of the solution itself as time…
We study the large-time behavior of solutions to the compressible Navier-Stokes equations for a viscous and heat-conductive gases in $\mathbb{R}^3$. More precisely, under a suitable additional condition involving only the low frequencies of…
Solutions of the Navier-Stokes and Euler equations with initial conditions for 2D and 3D cases were obtained in the form of converging series, by an analytical iterative method using Fourier and Laplace transforms \cite{TT10,TT11}. There…