### Lecture Notes of the Seventh International School on “Mathematical Theory in Fluid Mechanics”. Foreword

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

Using new results based on a convenient entropy condition, two types of algorithms for computing transonic flows are constructed. A sequence of solutions of the linearised problem with a posteriori control is constructed and its convergence to the physical solution of transonic flow in some special situations is proved. This paper contains also numerical results and their analysis for the case of flow past NACA 230012 airfoil. Some numerical improvements of the general algorithms, based on our...

We consider the two-dimesional spatially periodic problem for an evolutionary system describing unsteady motions of the fluid with shear-dependent viscosity under general assumptions on the form of nonlinear stress tensors that includes those with $p$-structure. The global-in-time existence of a weak solution is established. Some models where the nonlinear operator corresponds to the case $p=1$ are covered by this analysis.

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ${\rho}_{i}$ of the fluids and their velocity fields ${u}^{\left(i\right)}$ are prescribed at infinity: ${\rho}_{i}{|}_{\infty}={\rho}_{i\infty}>0$, ${u}^{\left(i\right)}{|}_{\infty}=0$. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ${\rho}_{i}\equiv {\rho}_{i\infty}$, ${u}^{\left(i\right)}\equiv 0$, $i=1,2$.

In this paper, we establish the large-data and long-time existence of a suitable weak solution to an initial and boundary value problem driven by a system of partial differential equations consisting of the Navier-Stokes equations with the viscosity $\nu $ polynomially increasing with a scalar quantity $k$ that evolves according to an evolutionary convection diffusion equation with the right hand side $\nu \left(k\right)|\U0001d5a3\left(\overrightarrow{v}\right){|}^{2}$ that is merely ${L}^{1}$-integrable over space and time. We also formulate a conjecture concerning regularity...

We prove the existence of regular solution to a system of nonlinear equations describing the steady motions of a certain class of non-Newtonian fluids in two dimensions. The equations are completed by requirement that all functions are periodic.

We study properties of Lipschitz truncations of Sobolev functions with constant and variable exponent. As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in [Frehse (2003) 1064–1083]. We also establish new existence results to a class of incompressible electro-rheological fluids.

Over a large range of the pressure, one cannot ignore the fact that the viscosity grows significantly (even exponentially) with increasing pressure. This paper concerns long-time and large-data existence results for a generalization of the Navier-Stokes fluid whose viscosity depends on the shear rate and the pressure. The novelty of this result stems from the fact that we allow the viscosity to be an unbounded function of pressure as it becomes infinite. In order to include a large class of viscosities...

We study the vibration of lumped parameter systems whose constituents are described through novel constitutive relations, namely implicit relations between the forces acting on the system and appropriate kinematical variables such as the displacement and velocity of the constituent. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables, which when substituted into the balance of linear momentum leads to a single governing ordinary...

**Page 1**