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Global existence of solutions to Navier-Stokes equations in cylindrical domains

Bernard Nowakowski, Wojciech M. Zajączkowski (2009)

Applicationes Mathematicae

We prove the existence of global and regular solutions to the Navier-Stokes equations in cylindrical type domains under boundary slip conditions, where coordinates are chosen so that the x₃-axis is parallel to the axis of the cylinder. Regular solutions have already been obtained on the interval [0,T], where T > 0 is large, on the assumption that the L₂-norms of the third component of the force field, of derivatives of the force field, and of the velocity field with respect to the direction of...

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

Elisabetta Rocca, Riccarda Rossi (2008)

Applications of Mathematics

This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature ϑ , an evolution equation for the phase change parameter χ , including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable 𝐮 . The main novelty of the model...

Global in Time Stability of Steady Shocks in Nozzles

Jeffrey Rauch, Chunjing Xie, Zhouping Xin (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem.

Global regular nonstationary flow for the Navier-Stokes equations in a cylindrical pipe

Piotr Kacprzyk (2007)

Applicationes Mathematicae

Global existence of regular solutions to the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe with large inflow and outflow is shown. Global existence is proved in two steps. First, by the Leray-Schauder fixed point theorem we prove local existence with large existence time. Next, the local solution is prolonged step by step. The existence is proved without any restrictions on the magnitudes of the inflow, outflow, external force and initial...

Global regular solutions to the Navier-Stokes equations in a cylinder

Wojciech M. Zajączkowski (2006)

Banach Center Publications

The existence and uniqueness of solutions to the Navier-Stokes equations in a cylinder Ω and with boundary slip conditions is proved. Assuming that the azimuthal derivative of cylindrical coordinates and azimuthal coordinate of the initial velocity and the external force are sufficiently small we prove long time existence of regular solutions such that the velocity belongs to W 5 / 2 2 , 1 ( Ω × ( 0 , T ) ) and the gradient of the pressure to L 5 / 2 ( Ω × ( 0 , T ) ) . We prove the existence of solutions without any restrictions on the lengths of the...

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

We study the Cauchy problem for the 3D MHD system with damping terms ε | u | α - 1 u and δ | b | β - 1 b (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

Global solutions of quasilinear systems of Klein–Gordon equations in 3D

Alexandru D. Ionescu, Benoît Pausader (2014)

Journal of the European Mathematical Society

We prove small data global existence and scattering for quasilinear systems of Klein-Gordon equations with different speeds, in dimension three. As an application, we obtain a robust global stability result for the Euler-Maxwell equations for electrons.

Global solutions, structure of initial data and the Navier-Stokes equations

Piotr Bogusław Mucha (2008)

Banach Center Publications

In this note we present a proof of existence of global in time regular (unique) solutions to the Navier-Stokes equations in an arbitrary three dimensional domain with a general boundary condition. The only restriction is that the L₂-norm of the initial datum is required to be sufficiently small. The magnitude of the rest of the norm is not restricted. Our considerations show the essential role played by the energy bound in proving global in time results for the Navier-Stokes equations.

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