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In this paper we study the flow of incompressible Newtonian fluid through a helical pipe with prescribed pressures at its ends. Pipe's thickness and the helix step are considered as the small parameter ε. By rigorous asymptotic analysis, as ε → 0 , the effective behaviour of the flow is found. The error estimate for the approximation is proved. (2000). 35B40, 35Q30, 76D05. Mathematics Subject Classification

In this paper we prove the global existence of weak solutions to degenerate parabolic systems coupled with an integral condition arising from the fully coupled moisture movement, transport of dissolved chemical species and heat transfer through porous materials. The problem under consideration covers a large range of problems including hygro‐thermo‐chemical modelling of concrete at early ages taking into account hydration (memory) phenomena. In the model, all changes of material properties are expressed as functions of state variables and the memory function (the so called hydration degree). Physically relevant mixed Dirichlet‐Neumann boundary conditions and initial conditions are considered. Existence of global weak solutions of the problem is proved by means of semidiscretization in time, proving necessary uniform estimates and by passing to the limit from discrete approximations. Degeneration occurs in the nonlinear transport coefficients which are not assumed to be bounded below and above by positive constants. Degeneracies in transport coefficients are overcome by proving suitable a‐priori L∞‐estimates based on De Giorgi and Moser iteration technique.

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This paper is devoted to the mathematical justification of an asymptotic model of a viscous flow in a curved tube with moving walls by proving error estimates. To this aim, we first construct the space correctors near the pipe's inlet and outlet due to the boundary layer phenomenon. In order to guarantee the adequate properties for these correctors we study what we called modified Leray's problem defined in a semi-infinite strip. We ensure the existence and uniqueness of an exponential decaying solution when the axial variable tends to infinity. Then, by deriving a Poincaré's type inequality and other estimates for the boundary value problems taking into account the condition on the pipe's lateral boundary, we evaluate the difference between the asymptotic approximation and the exact solution of the problem. K E Y W O R D Sasymptotic analysis, curved pipe, error estimates, Navier-Stokes equations, time-dependent domain