- Email: [email protected]
Linear and non-linear equations of parabolic type
Book reviews
intended and also by students and post-graduates methods. The book should be translated.
345
specializing
in numerical
N. P. Zhidkov 0. A. LADYZHENSKAYA, V. A. SOLONNIKOV and N. N. URAL’TSEVA. Lineinye i kvazilineinye wauneniya parabolickdwgo tipa. (Linear and non-linear equations of parabolic type). 376 p. “Nauka”, Moscow, 1967. THE monograph is mainly devoted to second-order equations of parabolic type and contains seven chapters. Linear equations are studied in Chapters III, IV, and quasilinear equations in Chapters V, VI. In Chapter VII some results for parabolic equations of arbitrary order and for systems, are presented. Chapter II is devoted to auxiliary propositions. Chapter I is of an introductory nature. First the classical formulations of the initial and boundary value problems considered in the book are stated, and proofs are given of the maximum principle and its generalizations and corollaroes (including the uniqueness theorem for classical solutions). Then (section 3) the questions of permissible extensions of the concept of the solution of a problem, and of the connection between the properties of generalized solutions and the properties of the functions occurring in the equation, which are important later, are discussed. A generalization is considered permissible if the uniqueness theorem holds for it. Examples are given showing the exact limits of such generalizations, and also the conditions necessary for preserving the fundamental properties inherent in the classical solutions of the simplest parabolic equations. For quasilinear equations the precision of those conditions which, as is described later, are sufficient for the solvability uas a wholen of the initial and boundary value problems, is demonstrated by examples. Therefore, the conditions for which the fundamental statements of the book are proved cannot be improved in terms of the scale of functional spaces considered. In Section 4 of Chapter I the authors explain the history of the origin and solution of the problems dealt with in the book, and give a short description of the fundamental results of the monograph. Chapters I-III, V-VII and Sections l-7 of Chapter VII mainly contain thr results of papers by 0. A. Ladyzhenskaya and N. N. Ural’tseva, and were written by them. Chapter IV and Sections S-10 of Chapter VII, devoted to linear equations with smooth coefficients, were written by V. A. Solonnikov. Methodologically the book is close to the monograph of 0. A. Ladyzhenskaya and N. N. Ural’tseva “Linear and quasilinear equations of elliptic type”.
346
Book reviews
In Chapter II the various analytic facts used later are collected. The and greater part of the chapter is concerned with the functional classes defined in the papers of 0. A. Ladyzhenskaya and N. N. Ural’tseva. It is proved that functions of the class are bounded, and functions of the classes (there are several of them) satisfy a Holder condition. The separation of these classes and their investigation is one of the important achievements of the author, enabling a number of important problems for quasilinear parabolic equations to be solved. These classes play the same part in the investigation introduced by 0. A. Ladyzhenskaya of parabolic equations as the classes and N. N. Ural’tseva in the study of equations of elliptic type, and are, in turn, far from being a generalization of the class of functions studied by De Georgi (see, “Matematika”, 4, 6, 19601. Chapter HI is devoted to linear equations, the principal parts of which have divergent form, the coefficients in the principal parts being only measurable and bounded, and the coefficients of the lowest order terms belong to some space L, $QT) (QT is the cylinder Q x [O, ~1). For such equations the unique soivabolity of the initial boundary value problems in a wide class of fun&ions (known as the V,(Q,) class) is first established. Then the utmost possible improvement of the smoothness of the solution in relation to an improvement of the smoothness and matching conditions in the given problem are traced in detail step by step. The existence of generalized solutions of V2(QT) is proved by Galerkin’s method. In Chq._ter III other functional-analytic and approximate methods of obtaining different fundamental solutions of initial boundary value problems, and methods of proving uniqueness theorems are also described. In particular, the general methods of solving the abstract Cauchy problem for equations with operator coefficients, are illustrated by the simplest parabolic problems. These methods were proposed by 0. A. Ladyzhenskaya and M. I. Bishik at the beginning of the fifties, and were later considerably extended. Chapter IV considers linear parabolic equations of the second order with smooth coefficients in the Holder spaces H **,'(Q,$, and equations the highest order coefficients of which are continuous, and the lowest order coefficients are summable in QT with definite powers - in the spaces of S. L. Sobolev W,‘*‘(Q,). By means of the method proposed by V. A. Solonnikov for investigating general boundary value problems for parabolic systems (Trudy MIAN, 83, 1965) the solvability of the fundamental initial boundary value problems is established for these equations, and exact bounds of the solutions in these spaces are deduced. The essence of this method consists of the construction and investigation of the socalled regularizer of the problem. Several paragraphs of Chapter IV are devoted
Book reviews
347
to the method of potentials. In Chapter V quasilinear principal part
uniformly parabolic equations with divergent
d Ut -~W7
1
6 4 u,)+ a(x, t, U, &)= 0.
are investigated.
As in other chapters the solvability as a whole of the initial boundary value problem is proved for them, and the various properties of the whole set of their solution are also established. The constraints with which this is done are caused (as shown in Cnapter I) by the essence of the matter. A somewhat less detailed analysis is carried out for the solution of the general quasilinear equations
(Chapter VI). For these equations the question of solvability classical sense is mainly considered.
as a whole in the
Among the numerous results of Chapters IV and V we here distinguish one theorem belonging to the whole class of quasilinear equations (*) and not requiring any assumptions, apart from the parabolicity of the equation on the solution U(X, t) considered, ml.1 smoothness of the coefficients aij(x, t, U, u,) of the derivatives U, ,z ,. Indeed, if the function U(X, t) is continuous, the derivatives
ut and xX2:, belong to L,(Q,),and the iJ
U,
1
are bounded and continuous
with respect to t as elements of L,(Q), then the function uX, is continuous with respect to (x, t) in Holder’s sense, where the Holder norm f& u, can be bounded by max 1u I, max 1u, 1 and known quantities. The crucial theorem reduces the problem of obtaining prior estimates of the derivatives of the solutions of specific quasilinear parabolic equations, to the derivation of estimates for only max j u 1, max j 24, I and to already known estimates of the solutions of linear equations with smooth coefficients. In Chapter VII a class of quasilinear systems of differential equations is considered for which some form of the maximum principle is valid. Fo: such systems the same questions as are considered in the previous chapters for a single equation, are considered with exhaustive completeness. The most difficult and important part, Chaptlsrs V-VII constructs prior estimates of the solutions of qu;i-,l!inear equations. Here the authors apply a variety of delicate methods to prove the integrodifferential inequalities
348
Book reviews
satisfied :iy l,he solutions themselves, their derivatives, or some combinations of fictions of the derivatives, thereby establishing that they belong to the classes and . When prior estimates of the necessary strength have been obtained for all possible solutions, the solvability of the initial boundary value problem is established fairly simply by invoking theorcrns on the fixed points of continuous mappings in Banach spaces. In Chapter VII results due to V. A. Solonnikov ou the solvability of general boundary value problems for linear parabolic systems in the classes If2 bz*l(Q,) and W2b”*‘(Qq,l are presented. In Sections S-9 general information is given on types :f parabolic systems and formulations of fundamental initial boundary valu13 problems for them, the proofs are indicated in Section 10. The book contains extensive and varied material. Numerous important results obtained up to the present time on general linear and quasilinear uniformly parabolic equations are presenlo 1 in it. The greater part of the results are d 13 to the authors. The corresponding material has not previously been publish~ in Russian or foreign literature.
M. Biman and V. Maz’ya R. A. NEWING and J. CU~INGH~. London Edinburgh, Interscience,
Q~r~n~~ naechanics. Oliver and Boyd, New York, ix + 6235p., l%7.
THIS book is a textbook on elementary quantum mechanics has;! 1 on lectures delivered at the University of Bangor (North Wales), and intended for students specializing in mathematical physics. The authors have apparently set themselves the task of giving the students a brief explana: ‘on of the formal structure and working apparatus of modern quantum mechanics, without going into fundamentals or a detailed physical interpretation. The authors also do no go into the instructive history of tha creation of quantum mechanics. A careful selection of the maierial and of the m&hods of deriving the formulas has enabled the authors to fulfil the? intentions in the extremely modest volurnz of the book. The book begins with an enumeration of the necessary information on the theory of linear spaces and operators. Then the phenomenological bases of quota mechanics are fo~ulated (states - vectors in Hilbert space, observables - selfconjugate opera:ors) and Bohr’s co~‘~:espondence principle in naive form (it is proposed for the h:~~::it,ion to quantum mechanics to replace functions by operators i:i ihe Hamiltonian equations, and Poisson bracks!.s !>y commutators).
intended and also by students and post-graduates methods. The book should be translated.
345
specializing
in numerical
N. P. Zhidkov 0. A. LADYZHENSKAYA, V. A. SOLONNIKOV and N. N. URAL’TSEVA. Lineinye i kvazilineinye wauneniya parabolickdwgo tipa. (Linear and non-linear equations of parabolic type). 376 p. “Nauka”, Moscow, 1967. THE monograph is mainly devoted to second-order equations of parabolic type and contains seven chapters. Linear equations are studied in Chapters III, IV, and quasilinear equations in Chapters V, VI. In Chapter VII some results for parabolic equations of arbitrary order and for systems, are presented. Chapter II is devoted to auxiliary propositions. Chapter I is of an introductory nature. First the classical formulations of the initial and boundary value problems considered in the book are stated, and proofs are given of the maximum principle and its generalizations and corollaroes (including the uniqueness theorem for classical solutions). Then (section 3) the questions of permissible extensions of the concept of the solution of a problem, and of the connection between the properties of generalized solutions and the properties of the functions occurring in the equation, which are important later, are discussed. A generalization is considered permissible if the uniqueness theorem holds for it. Examples are given showing the exact limits of such generalizations, and also the conditions necessary for preserving the fundamental properties inherent in the classical solutions of the simplest parabolic equations. For quasilinear equations the precision of those conditions which, as is described later, are sufficient for the solvability uas a wholen of the initial and boundary value problems, is demonstrated by examples. Therefore, the conditions for which the fundamental statements of the book are proved cannot be improved in terms of the scale of functional spaces considered. In Section 4 of Chapter I the authors explain the history of the origin and solution of the problems dealt with in the book, and give a short description of the fundamental results of the monograph. Chapters I-III, V-VII and Sections l-7 of Chapter VII mainly contain thr results of papers by 0. A. Ladyzhenskaya and N. N. Ural’tseva, and were written by them. Chapter IV and Sections S-10 of Chapter VII, devoted to linear equations with smooth coefficients, were written by V. A. Solonnikov. Methodologically the book is close to the monograph of 0. A. Ladyzhenskaya and N. N. Ural’tseva “Linear and quasilinear equations of elliptic type”.
346
Book reviews
In Chapter II the various analytic facts used later are collected. The and greater part of the chapter is concerned with the functional classes defined in the papers of 0. A. Ladyzhenskaya and N. N. Ural’tseva. It is proved that functions of the class are bounded, and functions of the classes (there are several of them) satisfy a Holder condition. The separation of these classes and their investigation is one of the important achievements of the author, enabling a number of important problems for quasilinear parabolic equations to be solved. These classes play the same part in the investigation introduced by 0. A. Ladyzhenskaya of parabolic equations as the classes and N. N. Ural’tseva in the study of equations of elliptic type, and are, in turn, far from being a generalization of the class of functions studied by De Georgi (see, “Matematika”, 4, 6, 19601. Chapter HI is devoted to linear equations, the principal parts of which have divergent form, the coefficients in the principal parts being only measurable and bounded, and the coefficients of the lowest order terms belong to some space L, $QT) (QT is the cylinder Q x [O, ~1). For such equations the unique soivabolity of the initial boundary value problems in a wide class of fun&ions (known as the V,(Q,) class) is first established. Then the utmost possible improvement of the smoothness of the solution in relation to an improvement of the smoothness and matching conditions in the given problem are traced in detail step by step. The existence of generalized solutions of V2(QT) is proved by Galerkin’s method. In Chq._ter III other functional-analytic and approximate methods of obtaining different fundamental solutions of initial boundary value problems, and methods of proving uniqueness theorems are also described. In particular, the general methods of solving the abstract Cauchy problem for equations with operator coefficients, are illustrated by the simplest parabolic problems. These methods were proposed by 0. A. Ladyzhenskaya and M. I. Bishik at the beginning of the fifties, and were later considerably extended. Chapter IV considers linear parabolic equations of the second order with smooth coefficients in the Holder spaces H **,'(Q,$, and equations the highest order coefficients of which are continuous, and the lowest order coefficients are summable in QT with definite powers - in the spaces of S. L. Sobolev W,‘*‘(Q,). By means of the method proposed by V. A. Solonnikov for investigating general boundary value problems for parabolic systems (Trudy MIAN, 83, 1965) the solvability of the fundamental initial boundary value problems is established for these equations, and exact bounds of the solutions in these spaces are deduced. The essence of this method consists of the construction and investigation of the socalled regularizer of the problem. Several paragraphs of Chapter IV are devoted
Book reviews
347
to the method of potentials. In Chapter V quasilinear principal part
uniformly parabolic equations with divergent
d Ut -~W7
1
6 4 u,)+ a(x, t, U, &)= 0.
are investigated.
As in other chapters the solvability as a whole of the initial boundary value problem is proved for them, and the various properties of the whole set of their solution are also established. The constraints with which this is done are caused (as shown in Cnapter I) by the essence of the matter. A somewhat less detailed analysis is carried out for the solution of the general quasilinear equations
(Chapter VI). For these equations the question of solvability classical sense is mainly considered.
as a whole in the
Among the numerous results of Chapters IV and V we here distinguish one theorem belonging to the whole class of quasilinear equations (*) and not requiring any assumptions, apart from the parabolicity of the equation on the solution U(X, t) considered, ml.1 smoothness of the coefficients aij(x, t, U, u,) of the derivatives U, ,z ,. Indeed, if the function U(X, t) is continuous, the derivatives
ut and xX2:, belong to L,(Q,),and the iJ
U,
1
are bounded and continuous
with respect to t as elements of L,(Q), then the function uX, is continuous with respect to (x, t) in Holder’s sense, where the Holder norm f& u, can be bounded by max 1u I, max 1u, 1 and known quantities. The crucial theorem reduces the problem of obtaining prior estimates of the derivatives of the solutions of specific quasilinear parabolic equations, to the derivation of estimates for only max j u 1, max j 24, I and to already known estimates of the solutions of linear equations with smooth coefficients. In Chapter VII a class of quasilinear systems of differential equations is considered for which some form of the maximum principle is valid. Fo: such systems the same questions as are considered in the previous chapters for a single equation, are considered with exhaustive completeness. The most difficult and important part, Chaptlsrs V-VII constructs prior estimates of the solutions of qu;i-,l!inear equations. Here the authors apply a variety of delicate methods to prove the integrodifferential inequalities
348
Book reviews
satisfied :iy l,he solutions themselves, their derivatives, or some combinations of fictions of the derivatives, thereby establishing that they belong to the classes and . When prior estimates of the necessary strength have been obtained for all possible solutions, the solvability of the initial boundary value problem is established fairly simply by invoking theorcrns on the fixed points of continuous mappings in Banach spaces. In Chapter VII results due to V. A. Solonnikov ou the solvability of general boundary value problems for linear parabolic systems in the classes If2 bz*l(Q,) and W2b”*‘(Qq,l are presented. In Sections S-9 general information is given on types :f parabolic systems and formulations of fundamental initial boundary valu13 problems for them, the proofs are indicated in Section 10. The book contains extensive and varied material. Numerous important results obtained up to the present time on general linear and quasilinear uniformly parabolic equations are presenlo 1 in it. The greater part of the results are d 13 to the authors. The corresponding material has not previously been publish~ in Russian or foreign literature.
M. Biman and V. Maz’ya R. A. NEWING and J. CU~INGH~. London Edinburgh, Interscience,
Q~r~n~~ naechanics. Oliver and Boyd, New York, ix + 6235p., l%7.
THIS book is a textbook on elementary quantum mechanics has;! 1 on lectures delivered at the University of Bangor (North Wales), and intended for students specializing in mathematical physics. The authors have apparently set themselves the task of giving the students a brief explana: ‘on of the formal structure and working apparatus of modern quantum mechanics, without going into fundamentals or a detailed physical interpretation. The authors also do no go into the instructive history of tha creation of quantum mechanics. A careful selection of the maierial and of the m&hods of deriving the formulas has enabled the authors to fulfil the? intentions in the extremely modest volurnz of the book. The book begins with an enumeration of the necessary information on the theory of linear spaces and operators. Then the phenomenological bases of quota mechanics are fo~ulated (states - vectors in Hilbert space, observables - selfconjugate opera:ors) and Bohr’s co~‘~:espondence principle in naive form (it is proposed for the h:~~::it,ion to quantum mechanics to replace functions by operators i:i ihe Hamiltonian equations, and Poisson bracks!.s !>y commutators).