52 research outputs found
A note on the Prodi-Serrin conditions for the regularity of a weak solution to the Navier-Stokes equations
The paper is concerned with the regularity of weak solutions to the
Navier-Stokes equations. The aim is to investigate on a relaxed Prodi-Serrin
condition in order to obtain regularity for t > 0. The most interesting aspect
of the result is that no compatibility condition is required to the initial
data v_0\in J^2(\OO) J2({\Omega})
Higher regularity of solutions to the singular p-Laplacean parabolic system
We study existence and regularity properties of solutions to the singular
-Laplacean parabolic system in a bounded domain . The main purpose
is to prove global , ,
integrability properties of the second spatial derivatives and of the time
derivative of the solutions. Hence, for suitable and exponents , by
Sobolev embedding theorems, we deduce global regularity of and
in H\"older spaces. Finally we prove a global pointwise bound for the solution
under the assumption
On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem
We prove space-time decay estimates of suitable weak solutions to the
Navier-Stokes Cauchy problem, corresponding to a given asymptotic behavior of
the initial data of the same order of decay. We use two main tools. The first
is a result obtained by the authors in the paper "A remark on the partial
regularity of a suitable weak solution to the Navier-Stokes Cauchy problem"
(submitted), on the behavior of the solution in a neighborhood of in the
-norm, which enables us to furnish a representation formula for
a suitable weak solution. The second is the asymptotic behavior of the
norm of for . Following a Leray's
point of view, roughly speaking our result proves that a possible space-time
turbulence does not perturb the asymptotic spatial behavior of the initial data
of a suitable weak solution
Existence of regular time-periodic solution to shear-thinning fluids
In this note we investigate the existence of time-periodic solutions to the
-Navier-Stokes system in the singular case of , that describes
the flows of an incompressible shear-thinning fluid. In the space-periodic
setting and for we prove the existence of a regular
time-periodic solution corresponding to a time periodic force data which is
assumed small in a suitable sense. As a particular case we obtain `regular'
steady solutions
Navier-Stokes flow past a rigid body: attainability of steady solutions as limits of unsteady weak solutions, starting and landing cases
Consider the Navier-Stokes flow in 3-dimensional exterior domains, where a
rigid body is translating with prescribed translational velocity
with constant vector .
Finn raised the question whether his steady slutions are attainable as limits
for of unsteady solutions starting from motionless state when
after some finite time and (starting problem). This was
affirmatively solved by Galdi, Heywood and Shibata for small . We
study some generalized situation in which unsteady solutions start from large
motions being in . We then conclude that the steady solutions for small
are still attainable as limits of evolution of those fluid motions
which are found as a sort of weak solutions. The opposite situation, in which
after some finite time and (landing problem), is also
discussed. In this latter case, the rest state is attainable no matter how
large is
On the - estimates of the gradient of solutions to the Stokes problem
The paper is concerned with estimates of the gradient of the solutions to the
Stokes IBVP both in a bounded and in an exterior domain. More precisely, we
look for estimates of the kind , for all , where function is independent of .Comment: There is an improvement of Proposition
Global L^r-estimates and regularizing effect for solutions to the p(t, x) -Laplacian systems
We consider the initial boundary value problem for the p(t, x)-Laplacian
system in a bounded domain \Omega. If the initial data belongs to L^{r_0}, r_0
\geq 2, we give a global L^{r_0}({\Omega})-regularity result uniformly in t>0
that, in the particular case r_0 =\infty, implies a maximum modulus theorem.
Under the assumption p- = \inf p(t, x) > 2n/(n+r_0), we also state L^{r_0}- L^r
estimates for the solution, for r \geq r_0. Complete proofs of the results
presented here are given in the paper [F. Crispo, P. Maremonti, M. Ruzicka,
Global L^r-estimates and regularizing effect for solutions to the p(t, x)
-Laplacian systems, accepted for publication on Advances in Differential
Equations, 2017]
Singular p-Laplacian parabolic system in exterior domains: higher regularity of solutions and related properties of extinction and asymptotic behavior in time
We consider the IBVP in exterior domains for the p-Laplacian parabolic
system. We prove regularity up to the boundary, extinction properties for p \in
( 2n/(n+2) , 2n/(n+1) ) and exponential decay for p= 2n/(n+1)
On the high regularity of solutions to the p-Laplacian boundary value problem in exterior domains
In this note, we consider the boundary value problem in exterior domains for the
p-Laplacian system, p ∈ (1, 2). For suitable p and Lr -spaces, r > n, we furnish existence of
a high-regular solution that is a solution whose second derivatives belong to L r (Ω ). Hence,
in particular we get λ-Hölder continuity of the gradient of the solution, with λ = 1 − n/r
Further, we improve previous results on W2,2-regularity in a bounded domain
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