Global solutions of a two-dimensional initial boundary-value problem for a system of semilinear magnetoelasticity equations

1996 ◽  
Vol 48 (2) ◽  
pp. 181-188 ◽  
Author(s):  
O. M. Botsenyuk
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Initial Boundary Value

Energy Decay of Global Solutions for a System of Petrovsky Equations

2013 ◽  
Vol 405-408 ◽  
pp. 3160-3164
Author(s):  
Yao Jun Ye
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Decay Estimate ◽  
Boundary Value ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Energy Decay ◽  
Difference Inequality ◽  
Initial Boundary Value

The initial-boundary value problem for a class of nonlinear Petrovsky systems in bounded domain is studied. We prove the energy decay estimate of global solutions through the use of a difference inequality.

scholarly journals Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term

2019 ◽  
Vol 39 (3) ◽  
pp. 395-414
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Global Solutions ◽  
Positive Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

We give an existence theorem of global solution to the initial-boundary value problem for \(u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)\) under some smallness conditions on the initial data, where \(\sigma (v^2)\) is a positive function of \(v^2\ne 0\) admitting the degeneracy property \(\sigma(0)=0\). We are interested in the case where \(\sigma(v^2)\) has no exponent \(m \geq 0\) such that \(\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0\). A typical example is \(\sigma(v^2)=\operatorname{log}(1+v^2)\). \(f(u)\) is a function like \(f=|u|^{\alpha} u\). A decay estimate for \(\|\nabla u(t)\|_{\infty}\) is also given.

scholarly journals On linear and nonlinear instability of the incompressible swept attachment-line boundary layer

1998 ◽  
Vol 355 ◽  
pp. 193-227 ◽  
Author(s):  
VASSILIOS THEOFILIS
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Small Amplitude ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Instability Waves ◽  
Linear And Nonlinear ◽  
Initial Boundary Value

The stability of an incompressible swept attachment-line boundary layer flow is studied numerically, within the Görtler–Hämmerlin framework, in both the linear and nonlinear two-dimensional regimes in a self-consistent manner. The initial-boundary-value problem resulting from substitution of small-amplitude excitation into the incompressible Navier–Stokes equations and linearization about the generalized Hiemenz profile is solved. A comprehensive comparison of all linear approaches utilized to date is presented and it is demonstrated that the linear initial-boundary-value problem formulation delivers results in excellent agreement with those obtained by solution of either the temporal or the spatial linear stability theory eigenvalue problem for both zero suction and a layer in which blowing is applied. In the latter boundary layer recent experiments have documented the growth of instability waves with frequencies in a range encompassed by that of the unstable Görtler–Hämmerlin linear modes found in our simulations. In order to enable further comparisons with experiment and, thus, assess the validity of the Görtler–Hämmerlin theoretical model, we make available the spatial structure of the eigenfunctions at maximum growth conditions.The condition on smallness of the imposed excitation is subsequently relaxed and the resulting nonlinear initial-boundary-value problem is solved. Extensive numerical experimentation has been performed which has verified theoretical predictions on the way in which the solution is expected to bifurcate from the linear neutral loop. However, it is demonstrated that the two-dimensional model equations considered do not deliver subcritical instability of this flow; this strengthens the conjecture that three-dimensionality is, at least partly, responsible for the observed discrepancy between the linear theory critical Reynolds number and the subcritical turbulence observed either experimentally or in three-dimensional numerical simulations. Further, the present nonlinear computations demonstrate that the unstable flow has its line of maximum amplification in the neighbourhood of the experimentally observed instability waves, in a manner analogous to the Blasius boundary layer. In line with previous eigenvalue problem and direct simulation work, suction is observed to be a powerful stabilization mechanism for naturally occurring instabilities of small amplitude.

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