scholarly journals A well-posedness theory for the Prandtl equations in three space variables

2017 ◽  
Vol 308 ◽  
pp. 1074-1126 ◽  
Author(s):  
Cheng-Jie Liu ◽  
Ya-Guang Wang ◽  
Tong Yang
Keyword(s):  
Well Posedness ◽  
Prandtl Equations ◽  
Three Space

Stability of Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Well Posedness ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Three Space

In this chapter we intend to investigate the stability of the Leray solutions constructed in the previous chapter. It is useful to start by analyzing the linearized version of the Navier–Stokes equations, so the first section of the chapter is devoted to the proof of the well-posedness of the time-dependent Stokes system. The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes equations, and the more delicate case of three space dimensions will be dealt with in Sections 3.3–3.5.

Well-posedness of Prandtl equations with non-compatible data

Nonlinearity ◽  
2013 ◽  
Vol 26 (12) ◽  
pp. 3077-3100 ◽  
Author(s):  
M Cannone ◽  
M C Lombardo ◽  
M Sammartino
Keyword(s):  
Well Posedness ◽  
Prandtl Equations

scholarly journals ON THE WAVE EQUATION WITH QUADRATIC NONLINEARITIES IN THREE SPACE DIMENSIONS

2011 ◽  
Vol 08 (01) ◽  
pp. 1-8 ◽  
Author(s):  
AXEL GRÜNROCK
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Parameter Range ◽  
Well Posedness ◽  
Quadratic Nonlinearities ◽  
The Cauchy Problem ◽  
Three Space

The Cauchy problem for the nonlinear wave equation [Formula: see text] in three space dimensions is considered. The data (u0, u1) are assumed to belong to [Formula: see text], where [Formula: see text] is defined by the norm [Formula: see text] Local well-posedness is shown in the parameter range 2 ≥ r > 1, [Formula: see text]. For r = 2 this coincides with the result of Ponce and Sideris, which is optimal on the Hs-scale by Lindblad's counterexamples, but nonetheless leaves a gap of ½ derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for □u = ∂u2 are obtained, too.

scholarly journals The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

2019 ◽  
Vol 29 (10) ◽  
pp. 1819-1851 ◽  
Author(s):  
K. Disser ◽  
J. Rehberg
Keyword(s):  
Charge Carrier ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Divergence Form ◽  
Well Posedness ◽  
Elliptic Regularity ◽  
Surfaces And Interfaces ◽  
Interfacial Recombination ◽  
Semiconductor Equations ◽  
Three Space

We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.

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