Nonlinear problems in mathematical physics and related topics : in honor of professor O. A. Ladyzhenskaya
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書誌事項
Nonlinear problems in mathematical physics and related topics : in honor of professor O. A. Ladyzhenskaya
（International mathematical series）
Kluwer Academic/Plenum, c2002
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- 2
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Includes bibliographical references and index
内容説明・目次
- 巻冊次
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1 ISBN 9780306473333
内容説明
The new series, International Mathematical Series founded by Kluwer / Plenum Publishers and the Russian publisher, Tamara Rozhkovskaya is published simultaneously in English and in Russian and starts with two volumes dedicated to the famous Russian mathematician Professor Olga Aleksandrovna Ladyzhenskaya, on the occasion of her 80th birthday.
O.A. Ladyzhenskaya graduated from the Moscow State University. But throughout her career she has been closely connected with St. Petersburg where she works at the V.A. Steklov Mathematical Institute of the Russian Academy of Sciences.
Many generations of mathematicians have become familiar with the nonlinear theory of partial differential equations reading the books on quasilinear elliptic and parabolic equations written by O.A. Ladyzhenskaya with V.A. Solonnikov and N.N. Uraltseva.
Her results and methods on the Navier-Stokes equations, and other mathematical problems in the theory of viscous fluids, nonlinear partial differential equations and systems, the regularity theory, some directions of computational analysis are well known. So it is no surprise that these two volumes attracted leading specialists in partial differential equations and mathematical physics from more than 15 countries, who present their new results in the various fields of mathematics in which the results, methods, and ideas of O.A. Ladyzhenskaya played a fundamental role.
Nonlinear Problems in Mathematical Physics and Related Topics I presents new results from distinguished specialists in the theory of partial differential equations and analysis. A large part of the material is devoted to the Navier-Stokes equations, which play an important role in the theory of viscous fluids. In particular, the existence of a local strong solution (in the sense of Ladyzhenskaya) to the problem describing some special motion in a Navier-Stokes fluid is established. Ladyzhenskaya's results on axially symmetric solutions to the Navier-Stokes fluid are generalized and solutions with fast decay of nonstationary Navier-Stokes equations in the half-space are stated. Application of the Fourier-analysis to the study of the Stokes wave problem and some interesting properties of the Stokes problem are presented. The nonstationary Stokes problem is also investigated in nonconvex domains and some Lp-estimates for the first-order derivatives of solutions are obtained. New results in the theory of fully nonlinear equations are presented. Some asymptotics are derived for elliptic operators with strongly degenerated symbols. New results are also presented for variational problems connected with phase transitions of means in controllable dynamical systems, nonlocal problems for quasilinear parabolic equations, elliptic variational problems with nonstandard growth, and some sufficient conditions for the regularity of lateral boundary.
Additionally, new results are presented on area formulas, estimates for eigenvalues in the case of the weighted Laplacian on Metric graph, application of the direct Lyapunov method in continuum mechanics, singular perturbation property of capillary surfaces, partially free boundary problem for parametric double integrals.
目次
- Area Formulas for sigma-Harmonic Mappings
- G. Alessandrini, V. Nesi. On a Variational Problem Connected with Phase Transitions of Means in Controllable Dynamical Systems
- V.I. Arnold. A Priori Estimates for Starshaped Compact Hypersurfaces with Prescribed mth Curvature Function in Space Forms
- J. L.M. Barbosa, J.H.S. Lira, V.I. Oliker. Elliptic Variational Problems with Nonstandard Growth
- M. Bildhauer, M. Fuchs. Existence and Regularity of Solutions of dw=f with Dirichlet Boundary Conditions
- B. Dacorogna. A Singular Perturbation Property of Capillary Surfaces
- R. Finn. On Solutions with Fast Decay of Nonstationary Navier-Stokes System in the Half-Space
- Y. Fujigaki, T. Miyakawa. Strong Solutions to the Problem of Motion of a Rigid Bodyin a Navier-Stokes Liquid under the Action of Prescribed Forces and Torques
- G.P. Galdi, A.L. Silvestre. The Partially Free Boundary Problem for Parametric DoubleIntegrals
- S. Hildebrandt, H. von der Mosel. On Evolution Laws Forcing Convex Surfaces to Shrink to a Point
- N.M. Ivochkina. Existence of a Generalized Green Function for Integro-Differential Operators of Fractional Order
- M. Kassmann, M. Steinhauer. Lq-Estimates of the First-Order Derivatives of Solutions to the Nonstationary Stokes Problem
- H. Koch, V.A. Solonnikov. Two Sufficient Conditions for the Regularity of Lateral Boundary for the Heat Equation
- N.V. Krylov. Bound State Asymptotics for Elliptic Operators with Strongly Degenerated Symbols
- A. Laptev, O. Safronov, T. Weidl. Nonlocal Problems for Quasilinear Parabolic Equations
- G.M. Lieberman. Boundary Feedback Stabilization of a Vibrating String with an Interior Point Mass
- W. Littman, S.W. Taylor. On Direct Lyapunov Method in Continuum Theories
- M. Padula.The Fourier Coefficients of Stokes' Waves
- P.I. Plotnikov, J.F. Toland. A Geometric Regularity Estimate via Fully Nonlinear Elliptic Equations
- R. Schatzle. On the Eigenvalue Estimates for the Weighted Laplacian on Metric Graphs
- M. Solomyak. Potential Theory for Nonstationary Stokes Problem in Nonconvex Domains
- V.A. Solonnikov. Stability of Axially Symmetric Solutions to the Navier-Stokes Equations in Cylindrical Domains
- W.M.Zaj czkowski.
- 巻冊次
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2 ISBN 9780306474224
内容説明
The main topics reflect the fields of mathematics in which Professor O.A. Ladyzhenskaya obtained her most influential results.
One of the main topics considered in the volume is the Navier-Stokes equations. This subject is investigated in many different directions. In particular, the existence and uniqueness results are obtained for the Navier-Stokes equations in spaces of low regularity. A sufficient condition for the regularity of solutions to the evolution Navier-Stokes equations in the three-dimensional case is derived and the stabilization of a solution to the Navier-Stokes equations to the steady-state solution and the realization of stabilization by a feedback boundary control are discussed in detail. Connections between the regularity problem for the Navier-Stokes equations and a backward uniqueness problem for the heat operator are also clarified.
Generalizations and modified Navier-Stokes equations modeling various physical phenomena such as the mixture of fluids and isotropic turbulence are also considered. Numerical results for the Navier-Stokes equations, as well as for the porous medium equation and the heat equation, obtained by the diffusion velocity method are illustrated by computer graphs.
Some other models describing various processes in continuum mechanics are studied from the mathematical point of view. In particular, a structure theorem for divergence-free vector fields in the plane for a problem arising in a micromagnetics model is proved. The absolute continuity of the spectrum of the elasticity operator appearing in a problem for an isotropic periodic elastic medium with constant shear modulus (the Hill body) is established. Time-discretization problems for generalized Newtonian fluids are discussed, the unique solvability of the initial-value problem for the inelastic homogeneous Boltzmann equation for hard spheres, with a diffusive term representing a random background acceleration is proved and some qualitative properties of the solution are studied. An approach to mathematical statements based on the Maxwell model and illustrated by the Lavrent'ev problem on the wave formation caused by explosion welding is presented. The global existence and uniqueness of a solution to the initial boundary-value problem for the equations arising in the modelling of the tension-driven Marangoni convection and the existence of a minimal global attractor are established. The existence results, regularity properties, and pointwise estimates for solutions to the Cauchy problem for linear and nonlinear Kolmogorov-type operators arising in diffusion theory, probability, and finance, are proved. The existence of minimizers for the energy functional in the Skyrme model for the low-energy interaction of pions which describes elementary particles as spatially localized solutions of nonlinear partial differential equations is also proved.
Several papers are devoted to the study of nonlinear elliptic and parabolic operators. Versions of the mean value theorems and Harnack inequalities are studied for the heat equation, and connections with the so-called growth theorems for more general second-order elliptic and parabolic equations in the divergence or nondivergence form are investigated. Additionally, qualitative properties of viscosity solutions of fully nonlinear partial differential inequalities of elliptic and degenerate elliptic type are clarified. Some uniqueness results for identification of quasilinear elliptic and parabolic equations are presented and the existence of smooth solutions of a class of Hessian equations on a compact Riemannian manifold without imposing any curvature restrictions on the manifold is established.
目次
- Nonhomogeneous Navier-Stokes Equations with Integrable Low Regularity Data
- H. Amann. On the Rectiability of Defect Measures Arising in a Micromagnetics Model
- L. Ambrosio, B. Kirchheim, M. Lecumberry,T. Riviere. Vorticity and Smoothness in Viscous Flows
- H. Beirao da Veiga. Absolute Continuity of the Spectrum of the Periodic Operator of ElasticityTheory for Constant Shear Modulus
- M.Sh. Birman,T.A. Suslina. Some Properties of Solutions of Fully Nonlinear Partial Differential Inequalities
- I. Capuzzo Dolcetta. On Time Discretizations for Generalized Newtonian Fluids
- L. Diening, A. Prohl, M. Ruzicka. A Stokes Like-System for Mixtures
- J. Frehse, S. Goj, J. Malek. Real Processes and Realizability of a Stabilization. Method for Navier Stokes Equations
- A.V. Fursikov. On the Inelastic Boltzmann Equation with Diffusive Forcing
- I.M. Gamba, V. Panferov, C. Villani. On Problems in Continuum Mechanics with the Maxwell Viscosity
- S.K. Godunov. Uniqueness of Recovery of SomeSystems of Quasilinear Elliptic and Parabolic Partial Differential Equations
- V. Isakov. Global Solution of Coupled Kuramoto Sivashinsky and Ginzburg-Landau Equations
- V.K. Kalantarov. On Skyrme's Model
- L. Kapitanski. Linear and Nonlinear Ultraparabolic Equations of Kolmogorov Type Arising in Diffusion Theory and in Finance
- E. Lanconelli, A. Pascucci, S. Polidoro. On the Dimension of theGlobal Attractor for the Modifed Navier-Stokes Equations
- J. Malek, D. Prazak. The Diffusion Velocity Method as a Tool for General Diffusion Equations
- S. Mas-Gallic. Harnack Inequalities on Scale Irregular Sierpinski Gaskets
- U. Mosco. Mean Value Theorems and Harnack Inequalities
- M. Safonov. The Navier-Stokes Equations and Backward Uniqueness
- G. Seregin, V.Sverak. Hessian Equations on Compact Riemannian Manifolds
- J. Urbas. Index.
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