On Analysis of some Nonlinear Systems of Partial Differential Equations of Continuum Mechanics
On Analysis of some Nonlinear Systems of Partial Differential Equations of Continuum Mechanics
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DissertationDate of Exam
11.09.2002Date of Publication
2003Advisor
Frehse, JensCo-Referee
Hildebrandt, StefanMetadata
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Abstract
In this thesis we consider systems of partial differential equations of continuum mechanics and analyze regularity properties of their weak solutions.
The first chapter contains a detailed introduction and reviews the contents of chapter two, three and four.
We start in chapter 2 with the local regularity problem related to the equations modelling the mechanical behaviour of elasto-perfect plastic materials respectively to an elasto-viscoplastic approximation of these materials, i. e. we consider the Norton-Hoff approximation to Hencky's law. These equations form a nonlinear systems of partial differential equations of second order and of elliptic type in the usual primal formulation, where one is interested in the displacement vector u=u(x) respectively the strain tensor ε(u) = ½ (∇u + (∇u)T).
We study these systems via a dual approach which was developed by A. Bensoussan and J. Frehse. In this approach we look for the stress tensor σ = σ(x) which solves the system of equations:
A σ + | σD |p-2 σD = ε(u)
div σ + f = 0
in the weak sense with mixed boundary conditions. We show local Hölder continuity of the stress tensor in two dimensions for the Norton-Hoff approximation of the Hencky law in plasticity theory and deduce also corresponding results for the strain tensor ε(u).
The main tool to achieve this result is a logarithmic Morrey estimate, which was developed by J. Frehse together with A. Bensoussan and G. Seregin in the here considered context of the dual theory of elliptic systems. These logarithmic Morrey estimates combined with a suitable adapted estimate on higher integrability a la Meyers-Necas-Gehring-Giaquinta-Modica give the final result.
We also deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for p > 2d / d+2. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.
Finally we survey and improve some results concering uniqueness and regularity of solutions to the instationary Navier-Stokes equations in three (and higher) dimensions. In particular we shall show that the class of weak solutions which additionally belong to the space L2 (0,T;BMO) guarantees uniqueness as well as regularity of the weak solution under consideration. We also discuss the related issue of controlling the blow-up phenomenon of smooth solutions to the Navier-Stokes equations. The method of proof which we present is elementary and depends deeply on the special structure of the nonlinear convective term u · ∇ u of the Navier-Stokes equations together with the property div u = 0; namely the convective term is a div-curl expression and according to Coifman, Lions, Meyer and Semmes it belongs to the Hardy space H1. This shows that this property respective method of proof is also applicable to other equations in hydrodynamics as for example the Boussinesq equations, the equations of Magneto-Hydrodynamics and the equations of higher grade type fluids.
The first chapter contains a detailed introduction and reviews the contents of chapter two, three and four.
We start in chapter 2 with the local regularity problem related to the equations modelling the mechanical behaviour of elasto-perfect plastic materials respectively to an elasto-viscoplastic approximation of these materials, i. e. we consider the Norton-Hoff approximation to Hencky's law. These equations form a nonlinear systems of partial differential equations of second order and of elliptic type in the usual primal formulation, where one is interested in the displacement vector u=u(x) respectively the strain tensor ε(u) = ½ (∇u + (∇u)T).
We study these systems via a dual approach which was developed by A. Bensoussan and J. Frehse. In this approach we look for the stress tensor σ = σ(x) which solves the system of equations:
A σ + | σD |p-2 σD = ε(u)
div σ + f = 0
in the weak sense with mixed boundary conditions. We show local Hölder continuity of the stress tensor in two dimensions for the Norton-Hoff approximation of the Hencky law in plasticity theory and deduce also corresponding results for the strain tensor ε(u).
The main tool to achieve this result is a logarithmic Morrey estimate, which was developed by J. Frehse together with A. Bensoussan and G. Seregin in the here considered context of the dual theory of elliptic systems. These logarithmic Morrey estimates combined with a suitable adapted estimate on higher integrability a la Meyers-Necas-Gehring-Giaquinta-Modica give the final result.
We also deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for p > 2d / d+2. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.
Finally we survey and improve some results concering uniqueness and regularity of solutions to the instationary Navier-Stokes equations in three (and higher) dimensions. In particular we shall show that the class of weak solutions which additionally belong to the space L2 (0,T;BMO) guarantees uniqueness as well as regularity of the weak solution under consideration. We also discuss the related issue of controlling the blow-up phenomenon of smooth solutions to the Navier-Stokes equations. The method of proof which we present is elementary and depends deeply on the special structure of the nonlinear convective term u · ∇ u of the Navier-Stokes equations together with the property div u = 0; namely the convective term is a div-curl expression and according to Coifman, Lions, Meyer and Semmes it belongs to the Hardy space H1. This shows that this property respective method of proof is also applicable to other equations in hydrodynamics as for example the Boussinesq equations, the equations of Magneto-Hydrodynamics and the equations of higher grade type fluids.
Subjects
Systems of partial differential equations, Regularity, Existence, Weak solution, Norton-Hoff Approximation, Incompressible Fluid, Shear-Dependent Viscosity, Navier-Stokes Equations
Classification (DDC)
510 MathematikSteinhauer, Mark: On Analysis of some Nonlinear Systems of Partial Differential Equations of Continuum Mechanics. - Bonn, 2003. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-02751
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-02751
@phdthesis{handle:20.500.11811/1941,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-02751,
author = {{Mark Steinhauer}},
title = {On Analysis of some Nonlinear Systems of Partial Differential Equations of Continuum Mechanics},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2003,
note = {In this thesis we consider systems of partial differential equations of continuum mechanics and analyze regularity properties of their weak solutions.
The first chapter contains a detailed introduction and reviews the contents of chapter two, three and four.
We start in chapter 2 with the local regularity problem related to the equations modelling the mechanical behaviour of elasto-perfect plastic materials respectively to an elasto-viscoplastic approximation of these materials, i. e. we consider the Norton-Hoff approximation to Hencky's law. These equations form a nonlinear systems of partial differential equations of second order and of elliptic type in the usual primal formulation, where one is interested in the displacement vector u=u(x) respectively the strain tensor ε(u) = ½ (∇u + (∇u)T).
We study these systems via a dual approach which was developed by A. Bensoussan and J. Frehse. In this approach we look for the stress tensor σ = σ(x) which solves the system of equations:
A σ + | σD |p-2 σD = ε(u)
div σ + f = 0
in the weak sense with mixed boundary conditions. We show local Hölder continuity of the stress tensor in two dimensions for the Norton-Hoff approximation of the Hencky law in plasticity theory and deduce also corresponding results for the strain tensor ε(u).
The main tool to achieve this result is a logarithmic Morrey estimate, which was developed by J. Frehse together with A. Bensoussan and G. Seregin in the here considered context of the dual theory of elliptic systems. These logarithmic Morrey estimates combined with a suitable adapted estimate on higher integrability a la Meyers-Necas-Gehring-Giaquinta-Modica give the final result.
We also deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for p > 2d / d+2. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.
Finally we survey and improve some results concering uniqueness and regularity of solutions to the instationary Navier-Stokes equations in three (and higher) dimensions. In particular we shall show that the class of weak solutions which additionally belong to the space L2 (0,T;BMO) guarantees uniqueness as well as regularity of the weak solution under consideration. We also discuss the related issue of controlling the blow-up phenomenon of smooth solutions to the Navier-Stokes equations. The method of proof which we present is elementary and depends deeply on the special structure of the nonlinear convective term u · ∇ u of the Navier-Stokes equations together with the property div u = 0; namely the convective term is a div-curl expression and according to Coifman, Lions, Meyer and Semmes it belongs to the Hardy space H1. This shows that this property respective method of proof is also applicable to other equations in hydrodynamics as for example the Boussinesq equations, the equations of Magneto-Hydrodynamics and the equations of higher grade type fluids.},
url = {https://hdl.handle.net/20.500.11811/1941}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-02751,
author = {{Mark Steinhauer}},
title = {On Analysis of some Nonlinear Systems of Partial Differential Equations of Continuum Mechanics},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2003,
note = {In this thesis we consider systems of partial differential equations of continuum mechanics and analyze regularity properties of their weak solutions.
The first chapter contains a detailed introduction and reviews the contents of chapter two, three and four.
We start in chapter 2 with the local regularity problem related to the equations modelling the mechanical behaviour of elasto-perfect plastic materials respectively to an elasto-viscoplastic approximation of these materials, i. e. we consider the Norton-Hoff approximation to Hencky's law. These equations form a nonlinear systems of partial differential equations of second order and of elliptic type in the usual primal formulation, where one is interested in the displacement vector u=u(x) respectively the strain tensor ε(u) = ½ (∇u + (∇u)T).
We study these systems via a dual approach which was developed by A. Bensoussan and J. Frehse. In this approach we look for the stress tensor σ = σ(x) which solves the system of equations:
A σ + | σD |p-2 σD = ε(u)
div σ + f = 0
in the weak sense with mixed boundary conditions. We show local Hölder continuity of the stress tensor in two dimensions for the Norton-Hoff approximation of the Hencky law in plasticity theory and deduce also corresponding results for the strain tensor ε(u).
The main tool to achieve this result is a logarithmic Morrey estimate, which was developed by J. Frehse together with A. Bensoussan and G. Seregin in the here considered context of the dual theory of elliptic systems. These logarithmic Morrey estimates combined with a suitable adapted estimate on higher integrability a la Meyers-Necas-Gehring-Giaquinta-Modica give the final result.
We also deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for p > 2d / d+2. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.
Finally we survey and improve some results concering uniqueness and regularity of solutions to the instationary Navier-Stokes equations in three (and higher) dimensions. In particular we shall show that the class of weak solutions which additionally belong to the space L2 (0,T;BMO) guarantees uniqueness as well as regularity of the weak solution under consideration. We also discuss the related issue of controlling the blow-up phenomenon of smooth solutions to the Navier-Stokes equations. The method of proof which we present is elementary and depends deeply on the special structure of the nonlinear convective term u · ∇ u of the Navier-Stokes equations together with the property div u = 0; namely the convective term is a div-curl expression and according to Coifman, Lions, Meyer and Semmes it belongs to the Hardy space H1. This shows that this property respective method of proof is also applicable to other equations in hydrodynamics as for example the Boussinesq equations, the equations of Magneto-Hydrodynamics and the equations of higher grade type fluids.},
url = {https://hdl.handle.net/20.500.11811/1941}
}
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