By Jan Prüss, Gieri Simonett
In this monograph, the authors strengthen a finished method for the mathematical research of a wide range of difficulties concerning relocating interfaces. It comprises an in-depth examine of summary quasilinear parabolic evolution equations, elliptic and parabolic boundary price difficulties, transmission difficulties, one- and two-phase Stokes difficulties, and the equations of incompressible viscous one- and two-phase fluid flows. the idea of maximal regularity, an important aspect, can be absolutely constructed. The authors current a latest process in keeping with strong instruments in classical research, sensible research, and vector-valued harmonic analysis.
The concept is utilized to difficulties in two-phase fluid dynamics and section transitions, one-phase generalized Newtonian fluids, nematic liquid crystal flows, Maxwell-Stefan diffusion, and various geometric evolution equations. The booklet additionally contains a dialogue of the underlying actual and thermodynamic rules governing the equations of fluid flows and part transitions, and an exposition of the geometry of relocating hypersurfaces.
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Additional resources for Moving Interfaces and Quasilinear Parabolic Evolution Equations
We derive all of the aforementioned geometric quantities for Γρ in terms of ρ and Σ. It is also important to study the mapping properties of these quantities in dependence of ρ, and to derive expressions for their variations. For instance, we show that κ (0) = tr L2Σ + ΔΣ , where κ = κ(ρ) denotes the mean curvature of Γρ , LΣ the Weingarten tensor of Σ, and ΔΣ the Laplace-Beltrami operator on Σ. This is done in Section 2. We also study the ﬁrst and second variations of the area and volume functional, respectively.
Here we assume 1 = 2 =: , σ nonconstant. Experience shows that σ is strictly decreasing and positive at melting temperature 32 Chapter 1. Problems and Strategies θm , and as σ is also concave, it has a unique zero θc > θm ; we call θc the critical temperature. As beyond the critical temperature there is no phase separation anymore, we restrict to the temperature range θ ∈ (0, θc ). Here the model equations read (∂t u + u · ∇u) − div S + ∇π = 0 div u = 0 [[u]] = 0, in Ω \ Γ(t), in Ω \ Γ(t), u=0 on ∂Ω, PΓ u Γ = P Γ u on Γ(t), −[[T νΓ ]] = σ(θΓ )HΓ νΓ + σ (θΓ )∇Γ θΓ u(0) = u0 on Γ(t), in Ω.
C) The critical points of the entropy functional for prescribed total mass and total energy are precisely the equilibria of the system. (d) The non-degenerate equilibria are zero velocities, constant pressures in the components of the phases, and the interface is a union of non-intersecting spheres which do not touch the outer boundary ∂Ω. If phase transition is present, then the spheres are of equal size. 32) holds. (f ) The set E of non-degenerate equilibria forms a real analytic manifold. This result shows that the models are thermodynamically consistent, hence are physically reasonable.