Abstract
In this paper, we consider the initial value problem for the compressible viscoelastic flows with self-gravitating in $\mathbb{R}^n(n≥ 3)$. Global existence and decay rates of classical solutions are established. The corresponding linear equations becomes two similar equations by using Hodge decomposition and then the solutions operator is derived. The proof is mainly based on the decay properties of the solutions operator and energy method. The decay properties of the solutions operator may be derived from the pointwise estimate of the solution operator to two linear wave equations.
Highlights
The compressible viscoelastic flows with self-gravitating in multidimensional space is governed by∂tρ + ∇ · = 0,∂t(ρu) + ∇ · + ∇P (ρ) = μ1∆u + μ2∇(∇ · u) + ∇ · + ρ∇Φ,∂tF + u · ∇F = ∇uF,∆Φ = ρ − ρ, lim|x|→∞ Φ = 0 (1)The variables are the density ρ, the velocity u, the deformation tensor F and the electrostatic potential Φ
The compressible viscoelastic flows with self-gravitating have strong physical background, we may refer to [20]
The three dimension Hookean elastodynamics has been studied and global classical solutions have been established by Hu [2]
Summary
The variables are the density ρ, the velocity u, the deformation tensor F and the electrostatic potential Φ. We investigate global existence and optimal decay of classical solutions to (2) with the following initial value t = 0 : ρ = ∂x1 ρ0(x), m = m0(x), F = F0(x), x ∈ Rn (3). There are two-folds in our present paper: firstly, it is very difficult to obtain the solutions operator to (2) since (2) has n2 + n + 1 equations and the unknown functions are coupled To overcome this difficulty, we introduce the Hodge decomposition and use some special relations (8), (9), (2) may be written as the system (13) whose linear system are decoupled; Secondly, we clarify the decay property of the solutions operator by investigating the solutions operator to two linear wave equation in (15) and (20).
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