Parabolic Problems

Name: Parabolic Problems | The Herbert Amann Festschrift
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The Herbert Amann Festschrift

Joachim Escher Patrick Guidotti Matthias Hieber Piotr Mucha Jan W. Prüss Yoshihiro Shibata Gieri Simonett Christoph Walker Wojciech Zajaczkowski(Editor)

Birkhäuser (Publisher)

1st Edition

Published on 20. July 2011

XII, 717 pages

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978-3-0348-0075-4 (ISBN)

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Content

Intro
Parabolic Problems
Contents
Preface
References
Double Obstacle Limit for a Navier-Stokes/Cahn-Hilliard System
1. Introduction and main result
2. Notation and preliminaries
3. Double obstacle limit for the Cahn-Hilliard equation
3.1. Limit of the energy
3.2. Convective Cahn-Hilliard equation
3.3. Limit for the convective Cahn-Hilliard equation
4. Main result: Double obstacle limit for the model H
References
Flows of Generalized Oldroyd-B Fluids in Curved Pipes
1. Introduction
2. Governing equations
3. Equivalent formulation in polar toroidal coordinates
4. Numerical approximation and results
4.1. Setting of the approximated problem
4.2. Algorithm
4.3. Numerical results
4.3.1. Creeping generalized Oldroyd-B flows
4.3.2. Inertial generalized Oldroyd-B flows
5. Conclusion
References
Remarks on Maximal Regularity
1. Weighted estimates for the maximal regularity operator
2. Applications to the abstract Cauchy problem
3. A proof of maximal regularity via Kato's inequality for fractional powers
References
On the Classical Solvability of Boundary Value Problems for Parabolic Equations with Incompatible Initial and Boundary Data
1. Introduction. Statement of the problems. Main definitions
2. Main results
3. Auxiliary problems
4. Problem 1
5. Problem 2
Appendix
References
On the Maxwell-Stefan Approach to Multicomponent Diffusion
1. Introduction
2. Continuum mechanical modeling of multicomponent fluids
3. The Maxwell-Stefan equations
4. Inversion of the flux-force relations
5. Wellposedness of the Maxwell-Stefan equations
6. Final remarks
References
Global Existence vs. Blowup in a One-dimensional Smoluchowski-Poisson System
1. Introduction
2. Preliminaries
3. Finite time blowup
4. Global existence
References
Perturbation Results for Multivalued Linear Operators
1. Introduction
2. Resolvent approach
3. Modified resolvent approach
4. Application to PDEs
References
Semilinear Stochastic Integral Equations in Lp
1. Introduction
2. Fractional powers and fractional derivatives
3. The main result
4. Stochastic lemmas
5. Linear theory
6. The semilinear equation
7. Maximal regularity considerations
8. Krylov's approach versus B-space valued stochastic integration
References
On the Motion of Several Rigid Bodies in an Incompressible Viscous Fluid under the Influence of Selfgravitating Forces
1. Introduction
2. Formulation of the problem
2.1. Bodies and motions
2.2. The fluid motion
3. Weak formulation
4. Main result I
5. Approximate problem
6. Artificial viscosity limit
6.1. Identifying the position of the rigid bodies
6.2. Convergence of the selfgravitating force
6.3. Pointwise convergence of the velocities
6.4. Compactness of the velocity gradients
6.5. Conclusion
7. The passage to the limit for d . 0 and d . 0
8. An alternative proof
8.1. Definition of weak solution II
8.2.1. Estimates for transport equations and momentum equations
8.2.2. Passing to the limit in the rigid velocity
8.2.3. Strong convergence of ue.
Acknowledgment
References
Geometric Aspects of the Periodic Degasperis-Procesi Equation
1. Introduction
2. Geometric reformulation of the µDP equation
3. Short time existence of geodesics
4. The exponential map
References
Global Leray-Hopf Weak Solutions of the Navier-Stokes Equations with Nonzero Time-dependent Boundary Values
1. Introduction and main results
2. Preliminaries
3. The approximate system
4. Proof of Theorem 1.3
5. Construction of the vector field E
References
Time and Norm Optimality of Weakly Singular Controls
1. Introduction
2. The right translation semigroup
3. Weakly singular controls, I
4. Weakly singular controls, II
5. Weakly singular controls, III
6. Adjoints
References
Asymptotic Behavior of a Leray Solution around a Rotating Obstacle
1. Introduction
2. Notation
3. Preliminaries
4. Main theorem
References
A Remark on Maximal Regularity of the Stokes Equations
1. Introduction and main result
2. Proof of the main result
References
On Linear Elliptic and Parabolic Problems in Nikol'skij Spaces
0. Introduction and notations
1. Nikol'skij spaces
2. Elliptic problems depending on a parameter with nonhomogeneous boundary conditions
3. Parabolic problems
References
Parabolic Equations in Anisotropic Orlicz Spaces with General N-functions
1. Introduction
2. Useful facts about Orlicz spaces
3. Closures of compactly supported smooth functions
4. Existence result
References
Maximal Parabolic Regularity for Divergence Operators on Distribution Spaces
1. Introduction
2. Notation and general assumptions
3. Preliminaries
4. Mapping properties for (-. · µ. + 1)1/2
5. Maximal parabolic regularity for A
5.1. Auxiliaries
5.2. Core of the proof of Theorem 5.4
5.3. The operator A
6. Nonlinear parabolic equations
7. Examples
7.1. Geometric constellations
7.2. Nonlinearities and reaction terms
8. Concluding remarks
References
On the Relation Between the Large Time Behavior of the Stokes Semigroup and the Decay of Steady Stokes Flow at Infinity
1. Introduction
2. Results
3. Proof of Theorem 2.3
4. Proof of Corollary 2.4
5. Concluding remarks
References
Well-posedness and Exponential Decay for the Westervelt Equation with Inhomogeneous Dirichlet Boundary Data
1. Introduction
1.1. Main results
2. Strongly damped abstract wave equation
2.1. Regularity properties of damped wave equation
2.2. Decay rates for the homogeneous equation
2.2.1. Energy estimates for the variable coefficient model
2.3. Extension of nonhomogeneous boundary data to the interior
2.3.1. Weak solutions with nonhomogeneous boundary data
2.3.2. "Parabolic extension" of Cauchy data.
3. Back to the nonlinear problem
3.1. The Westervelt equation with source term
3.1.1. Local well-posedness
3.1.2. Global well-posedness
3.1.3. Decay rates
3.2. The Westervelt equation with nonhomogeneous Dirichlet boundary data
3.2.1. Local well-posedness
3.2.2. Global well-posedness
3.2.3. Decay rates
References
On Divergence Form Second-order PDEs with Growing Coefficients in W1 Spaces without Weights
1. Introduction
2. Setting of the problem
3. Main results
4. Differentiating compositions of generalized functions with differentiable functions
5. Proof of Theorem 3.1
6. Proof of Theorems 3.3 and 3.4
References
Global Properties of Transition Kernels Associated with Second-order Elliptic Operators
1. Introduction
2. Local regularity and integrability of transition densities
3. Uniform and pointwise bounds on transition densities
4. Regularity properties
References
Metric-induced Morphogenesis and Non-Euclidean Elasticity: Scaling Laws and Thin Film Models
1. Elastic energy of a growing tissue and non-Euclidean elasticity
2. The residual stress and a result on its scaling
3. The prestrained Kirchhoff model
4. A rigidity estimate
5. A hierarchy of scalings
6. The prestrained von K´arm´an model
7. The prestrained von K´arm´an equations
References
Compactness and Asymptotic Behavior in Nonautonomous Linear Parabolic Equations with Unbounded Coefficients in Rd
1. Introduction
2. Preliminaries: the evolution operator G(t, s)
3. Compactness in Cb(Rd)
4. Compactness and asymptotic behavior
References
Gradient Estimates and Domain Identification for Analytic Ornstein-Uhlenbeck Operators
1. Introduction
2. Gradient estimates: the H-invariant case
3. Gradient estimates: the analytic case
4. An example
5. Application to the stochastic heat equation
References
sectoriality of Cylindrical Boundary Value Problems
1. Introduction
2. Main result
3. R-sectoriality and parameter-ellipticity
4. Proof of the main result
4.1. Constant coefficients a1a
4.2. Slightly varying coefficients a1a
4.3. Variable coefficients a1
5. Mixed orders
References
Analytic Solutions for the Two-phase Navier-Stokes Equations with Surface Tension and Gravity
1. Introduction and main results
2. The transformed problem
3. The linearized two-phase Stokes problem with free boundary
4. The nonlinear problem
5. Appendix
References
On Conserved Penrose-Fife Type Models
1. Introduction and the model
2. The linear problem
3. Local well-posedness
4. Global well-posedness
5. Asymptotic behavior
6. Appendix
References
The Asymptotic Profile of Solutions of a Class of Singular Parabolic Equations
1. Introduction
2. Notation and preliminary results
3. Estimate from above and below
3.1. L2-estimate from below
3.2. L2-estimates from above
3.3. L8-estimates from above
3.4. L8 interior estimates from below
4. Estimates at the boundary from above and from below
5. Asymptotic behaviour
References
Linearized Stability for Nonlinear Partial Differential Delay Equations
1. Introduction
2. Linearized stability for (PFDE)
3. Proof of Theorem 2.4
References
Stochastic Equations with Boundary Noise
1. Introduction
2. Preliminaries
3. The abstract stochastic evolution equation
4. Boundary noise
References
A Note on Necessary Conditions for Blow-up of Energy Solutions to the Navier-Stokes Equations
1. Motivation
2. Some auxiliary things
3. Ancient solution
4. Spatial decay for ancient solutions
References
Local Solvability of Free Boundary Problems for the Two-phase Navier-Stokes Equations with Surface Tension in the Whole Space
1. Introduction and results
2. Analysis in a bent space for the Neumann problem
3. Analysis in a bent space for a problem with surface tension and gravity
4. Reduction of the boundary condition to linearized problems
5. Initial flow
6. The nonlinear problem
References
Inversion of the Lagrange Theorem in the Problem of Stability of Rotating Viscous Incompressible Liquid
1. Formulation of main result
2. Auxiliary propositions
3. On the problem (1.13)-(1.15)
4. The case of non-symmetric F
References
Questions of Stability for a Parabolic-hyperbolic System
1. Introduction
2. Notation
3. Local existence
4. Long term existence and stability
5. Excluding eigenvalues
References

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