638 research outputs found
Global existence and optimal decay estimates of strong solutions to the compressible viscoelastic flows
This paper is dedicated to the global existence and optimal decay estimates
of strong solutions to the compressible viscoelastic flows in the whole space
with any . We aim at extending those works by Qian \&
Zhang and Hu \& Wang to the critical Besov space, which is not related to
the usual energy space. With aid of intrinsic properties of viscoelastic fluids
as in \cite{QZ1}, we consider a more complicated hyperbolic-parabolic system
than usual Navier-Stokes equations. We define "\emph{two effective
velocities}", which allows us to cancel out the coupling among the density, the
velocity and the deformation tensor. Consequently, the global existence of
strong solutions is constructed by using elementary energy approaches only.
Besides, the optimal time-decay estimates of strong solutions will be shown in
the general critical framework, which improves those decay results due to
Hu \& Wu such that initial velocity could be \textit{large highly oscillating}.Comment: 44page
Global solutions to the Oldroyd-B model with a class of large initial data
Consider a global wellposed problem for the incompressible Oldroyd-B model.
It is shown that this set of equations admits a unique global solution provided
the initial horizontal velocity , the product \om u^d_0 of the
coupling parameter \om and initial the vertical velocity , and initial
symmetric tensor of constrains are sufficient small in the scaling
invariant Besov space
\dot{B}^{\fr{d}{2}-1}_{2,1}\times\dot{B}^{\fr{d}{2}}_{2,1}, d\ge2. In
particular, the result implies the global well-posedness of Oldroyd-B model
with large initial vertical velocity .Comment: 24pages. arXiv admin note: text overlap with arXiv:1410.748
Global well-posedness for the Phan-Thein-Tanner model in critical Besov spaces without damping
In this paper, we mainly investigate the Cauchy problem for the
Phan-Thein-Tanner (PTT) model. The PPT model can be viewed as a Navier-Stokes
equations couple with a nonlinear transport system. This model is derived from
network theory for the polymeric fluid. We study about the global well
posedness of the PTT model in critical Besov spaces. When the initial data is a
small perturbation over around the equilibrium, we prove that the strong
solution in critical Besov spaces exists globally.Comment: arXiv admin note: substantial text overlap with arXiv:1811.0746
Optimal Time Decay Rate for the Compressible Viscoelastic Equations in Critical Spaces
In this paper, we are concerned with the convergence rates of the global
strong solution to constant equilibrium state for the compressible viscoelastic
fluids in the whole space. We combine both analysis about Green's matrix method
and energy estimate method to get optimal time decay rate in critical Besov
space framework. Our result imply the optimal -time decay rate and only
need the initial data to be small in critical Besov space which have very low
regularity compared with traditional Sobolev space.Comment: 20 page
Global classical solutions of 3D compressible viscoelastic system near equilibrium
In this paper, we prove the global existence of general small solutions to
compressible viscoelastic system. We remove the "initial state" assumption
() and the "div-curl" structure assumption compared
with previous works. It then broadens the class of solutions to a great extent,
more precisely the initial density state would not be constant necessarily, and
no more structure is need for global well-posedness. It's quite different from
the elasticity system in which structure plays an important role. Since we can
not obtain any dissipation information for density and deformation tensor, we
introduce a new effective flux in the thought of regarding the wildest
"nonlinear term" as "linear term". Although the norms of solution may increase
now, we can still derive the global existence for it.Comment: 23 page
Global regularity for the 2D Oldroyd-B model in the corotational case
This paper is dedicated to the Oldroyd-B model with fractional dissipation
for any . We establish the global smooth
solutions to the Oldroyd-B model in the corotational case with arbitrarily
small fractional powers of the Laplacian in two spatial dimensions. The methods
described here are quite different from the tedious iterative approach used in
recent paper \cite{XY}. Moreover, in the Appendix we provide some a priori
estimates to the Oldroyd-B model in the critical case which may be useful and
of interest for future improvement. Finally, the global regularity to to the
Oldroyd-B model in the corotational case with replaced by
for are also collected in the Appendix.
Therefore our result is more closer to the resolution of the well-known global
regularity issue on the critical 2D Oldroyd-B model.Comment: 23 pages, Submitted August 201
Global existence in critical spaces for incompressible viscoelastic fluids
We investigate local and global strong solutions for the incompressible
viscoelastic system of Oldroyd--B type. We obtain the existence and uniqueness
of a solution in a functional setting invariant by the scaling of the
associated equations. More precisely, the initial velocity has the same
critical regularity index as for the incompressible Navier--Stokes equations,
and one more derivative is needed for the deformation tensor. We point out a
smoothing effect on the velocity and a decay on the difference between
the deformation tensor and the identity matrix. Our result implies that the
deformation tensor has the same regularity as the density of the
compressible Navier--Stokes equations.Comment: 17 page
Large time behavior in critical Besov spaces for compressible viscoelastic flows
We consider the large time behavior of global strong solutions to the
compressible viscoelastic flows on the whole space ,
where the system describes the elastic properties of the compressible fluid.
Adding a suitable initial condition involving only the low-frequency, we prove
optimal time decay estimates for the global solutions in the critical
regularity framework, which are similar to those of the compressible
Navier-Stokes equations. Our results rely on the pure energy argument, which
allows us to remove the usual smallness assumption of the data in the
low-frequency.Comment: 21 pages. arXiv admin note: substantial text overlap with
arXiv:1906.09119, arXiv:1907.1253
Global Well-posedness of Incompressible Elastodynamics in Two Dimensions
We prove that for sufficiently small initial displacements in some weighted
Sobolev space, the Cauchy problem of the systems of incompressible isotropic
elastodynamics in two space dimensions admits a uniqueness global classical
solution.Comment: To appear in CPAM 201
Global well-posedness for the incompressible viscoelastic fluids in the critical framework
We investigate global strong solutions for the incompressible viscoelastic
system of Oldroyd--B type with the initial data close to a stable equilibrium.
We obtain the existence and uniqueness of the global solution in a functional
setting invariant by the scaling of the associated equations, where the initial
velocity has the same critical regularity index as for the incompressible
Navier--Stokes equations, and one more derivative is needed for the deformation
tensor. Like the classical incompressible Navier-Stokes, one may construct the
unique global solution for a class of large highly oscillating initial
velocity. Our result also implies that the deformation tensor has the same
regularity as the density of the compressible Navier--Stokes equations.Comment: 20 page
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