Time Existence Of Strong Solutions Research Articles

A regularity criterion for 3D shear thinning fluids in terms of the direction of vorticity

We prove that a weak solution for unsteady 3D shear thinning flows is strong under the certain assumption about the direction of the vorticity for 53<p≤2. As an auxiliary result, we show the short time existence of strong solution for whole space.

Local Existence of Strong Solutions of a Fluid\u2013Structure Interaction Model

We are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.

Analysis of the adiabatic piston problem via methods of continuum mechanics

We consider a system modelling the motion of a piston in a cylinder filled by a viscous heat conducting gas. The piston is moving longitudinally without friction under the influence of the forces exerted by the gas. In addition, the piston is supposed to be thermally insulating (adiabatic piston). This fact raises several challenges which received a considerable attention, essentially in the statistical physics literature. We study the problem via the methods of continuum mechanics, specifically, the motion of the gas is described by means of the Navier–Stokes–Fourier system in one space dimension, coupled with Newton's second law governing the motion of the piston. We establish global in time existence of strong solutions and show that the system stabilizes to an equilibrium state for t→∞.

Regularity for Ostwald-de Waele type shear thickening fluids

We obtain local in time existence of strong solution for non-Newtonian fluid with shear thickening viscosity. We also obtain the Hausdorff dimension of time singular set, and a Serrin type regularity criterion.

Asymptotic Behavior of Solutions to the Compressible Navier--Stokes Equation Around a Time-Periodic Parallel Flow

The global in time existence of strong solutions to the compressible Navier--Stokes equation around time-periodic parallel flows in $\mathbb{R}^n$, $n\geq 2$, is established under smallness conditions on Reynolds number, Mach number, and initial perturbations. Furthermore, it is proved for $n=2$ that the asymptotic leading part of solutions is given by a solution of the one-dimensional viscous Burgers equation multiplied by the time-periodic function. In the case $n\geq 3$ the asymptotic leading part of solutions is given by a solution of the $n-1$-dimensional heat equation with the convective term multiplied by the time-periodic function.

Large time existence of strong solutions to micropolar equations in cylindrical domains

We examine the so-called micropolar equations in three dimensional cylindrical domains under Navier boundary conditions. These equations form a generalization of the ordinary incompressible Navier–Stokes model, taking the structure of the fluid into account. We prove that under certain smallness assumption on the rate of change of the initial data and the external data there exists a unique and strong solution for any finite time T.

Existence of Strong Solutions for Incompressible Fluids with Shear Dependent Viscosities

Certain rheological behavior of non-Newtonian fluids in engineering sciences is often modeled by a power law ansatz with p ∈ (1, 2]. In the present paper the local in time existence of strong solutions is studied. The main result includes also the degenerate case (δ = 0) of the extra stress tensor and thus improves previous results of [L. Diening and M. Růžicka, J. Math. Fluid Mech., 7 (2005), pp. 413–450].

Uniform and optimal estimates for solutions to singularly perturbed parabolic equations

The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L2-estimates as well as certain decay properties of bounded classical solutions to it are established. These do not depend on the value of the small perturbation parameter, and allow to prove global in time existence of strong solutions to certain boundary-value problems for ultraparabolic equations with unbounded coefficients. Optimal decay estimates are proved for such solutions. All results concerning ultraparabolic equations apply, in particular, to the Kolmogorov equation for diffusion with inertia, to the (linear) Fokker-Planck equation, to the linearized Boltzmann equation, and to some nonlinear integro-differential ultraparabolic equations of the Fokker-Planck type, arising from biophysics. Optimal decay estimates are derived for global in time strong solutions to such equations.

Modeling, Mathematical and Numerical Analysis of Electrorheological Fluids

Many electrorheological fluids are suspensions consisting of solid particles and a carrier oil. If such a suspension is exposed to a strong electric field the effective viscosity increases dramatically. In this paper we first derive a model which captures this behaviour. For the resulting system of equations we then prove local in time existence of strong solutions for large data. For these solutions we finally derive error estimates for a fully implicit time-discretization.