> <\body> |<\doc-author-data|> \; > <\table-of-contents|toc> |.>>>>|> Introduction.> |.>>>>|> The Function Class >. Solutions of Class >.> |.>>>>|> The Boundary Condition of Vanishing. The Initial Value Problem.> |.>>>>|> Simplification of the Problem. The Approximation Procedure.> |.>>>>|> Proof of the Existence Theorem.> |.>>>>|> Proof of Lemma > |.>>>>|> \; \; Translation by Andreas Klöckner, . I would like to hear about any errors or other comments you may have. text is loosely translated. text marks spots where I was unsure. Let the points of -dimensional space be designated by , let ,x,\,x> be the coordiantes in a fixed cartesian coordinate system. Further, let x=\x\x\\x> be the volume element in -space. Let be a time-dependent vector field defined on an open subset > of --space with components > in the aforementioned coordinate system. We will not assume that > is connected, and only for brevity will be speaking of . Regions in -space will be denoted , in --space they will be denoted >. The fact that a vector field which is continuously -differentiable on an --region > is divergence-free is characterized by the differential equation <\equation> div u=u|\x>>=0, where we note that throughout this paper we will be using the common summation convention without the use of the sum symbol. There is also another well-known differential-less characterization of the fact that the divergence is zero. We say that a scalar or vector-valued function on > belongs to on >> iff 0> outside a suitable compact subset of this region. The functions of this class, which we will be referring to often, thus vanish on a boundary strip of >. This aforementioned characterization is then: A field which is continuously -differentiable on > is called on > iff <\equation> uh|\x>\x \t=0 for any function of class in > that is uniquely determined and continuously -differentiable on >. This fact is a consquence of Gauss' Theorem which is applicable because N> in > and because of the fundamental lemma of variational calculus. If we introduce the the scalar product of two vector fields and on > as <\equation*> vw*\x \t, we can say that ``a field which is continuously -differentiable in > is divergence-free in >'' means that is orthogonal in > to the gradient field of any function of class that is uniquely determined and continuously -differentiable in ><\footnote> The formulation of these terms in --space rather than just in -space is advantageous for our problem. Applications of Hilbert space theory can be found in the following works: Nikodym>, Sur un théorème de M.S. Zaremba concernant les fonctions harmoniques. J. Math. pur appl., Paris, Sér. IX, (1933), 95--109; Leray>, Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta math., Uppsala (1934), 193--248; Weyl>, The method of orthogonal projection in potential theory. Duke math J. (1940), 411--444. . The following counterpart of this fact is of interest here: It is necessary and sufficient for a field (x,t)> which is continuous in > (and which has components >) to be the gradient field =\h/\x> of an in > uniquely determined and continuously -differentiable function that it is orthogonal in > to any divergence-free field of class that is continuously -differentiable in >. Necessity is once more a consequence of the Integral Theorem. The following considerations show sufficiency. The consideration of fields of the form (t)\(x)> with scalar > first shows that we may constrain ourselves to the the corresponding claim for -regions . So, assume <\equation*> wh\x=0 for any smooth divergence-free field of class in . The claim follows if we can show that the circulation of the field > <\equation*> >h\x=>h\s vanishes along any closed path > in . It is easy to see that this needs to be shown only for continuously curved paths without self-intersections. We will obtain this vanishing through a suitable choice of fields . For any given small \0>, there is a vector field which is smooth and divergence-free in and which has the following properties: is non-zero only in a closed tube around > of thickness \>. On any plane tube section that cuts > orthogonally, the vector forms an angle \> with the normal direction (i.e. the direction of > in the section). The sectional flow of , which is independent of the exact shape of the section because is divergence-free, is equal to . This fact suffices to prove the vanishing of the circulation along >. We consider such a field that belongs to a given (but sufficiently small) >. If we let F> denote the hypersurface element on these tube sections and if we choose the arc length along > as the parameter transverse to the sections, we can write the volume element x> in the tube as (x)\F\s>, where we assume > to be continuous in a neighborhood of > and equal to on >. Then <\equation*> hw\x=h\|w\|\*\F\s. If we replace the component > by the component > taken at the intersection of > with the section, by the component (x)> taken in a direction normal to F> and > by 1, then the right-hand side integral becomes <\equation*> hw\F\s=h\s, i.e. the circulation. Based upon the aforementioned properties of the field , we can meanwhile easily prove that that the error introduced by these replacements goes to zero with >. Thereby the claim is proven. The basic equations of Navier-Stokes for the movement of a homogeneous, incompressible liquid are <\equation> u|\t>+u>u|\x>>=-p|\x>+\u|\x>\x>>, where > is a positive constant, namely the and <\equation*> div u=0. Let be a solution in an --region > which we assume to be continuous along with all the occurring derivatives >, >, >. We will now introdcue a new time-dependent vector field which is divergence-free in >. It is assumed to be of class in > and sufficiently smooth: and the derivatives ,a,a> should be continuous in >. Otherwise, there will be no no requirements on the field . Because N> in and because <\equation*> u>u|\x>>=uu>|\x>>, we have <\eqnarray*> au|\t>\x \t>||a|\t>u*\x \t,>>|au>u|\x>>\x \t>||a|\x>>u>u \x \t,>>|au|\x>\x>> \x \t>||a|\x>>u|\x>> \x \t=a|\x>\x>> \x \t>>>> and since and N>, we have <\equation*> ap|\x> \x \t=0. Thereby we find that the field satisfies the following condition <\equation> a|\t>u \x \t+a|\x>>u>u \x \t+\a|\x>\x>>u \x \t=0 for any sufficiently smooth field on > with the properties <\equation> div a=0 >>,a\N >>. In addition, we need to take into account that is divergence-free, i.e. <\equation> h|\x>u \x \t=0,h\N holds for any function of the mentioned class that is sufficiently smooth on >. We have thereby reduced the basic equations to the form of equations between linear functional operators of arbitrary fields and functions and . The essential part of this is that the unknown field on which these operators depend occurs without any derivatives. We still need to convince ourselves that we may revert from equations () and () to the differential form of the equations if we restrict ourselves to sufficiently smooth solution fields in >. We already know that under this assumption () goes back to in >. For a sufficiently smooth , we may undo all the integrations-by-parts. It follows that <\equation*> au|\t>+u>u|\x>>-\u|\x>\x>>\x \t must hold for any sufficiently smooth field of the form (). Using the theorem proved above, we may conclude that the curly braces must be the partial derivatives of a uniquely determined function , i.e. that the differential equations of motion must hold in >. We see that the above integral form of the equations exactly expresses the physical demand that the pressure be unique. It is quite natural to build the general mathematical theory on the integral form of the equations. But then it is appropriate to rid ourselves of the artificial restriction to smooth solution fields . The occurrence of the quadratic forms <\equation*> u*u \x,u|\x>>*u|\x>> \x in the energy equation leads us to base the problem on a Hilbert space of vector fields. It is a methodical advantage that in this broader framework the differentiability properties of the solutions become the subject of a problem that can be entirely separated from the problem of existence.<\footnote> Compare the treatment of quadratic variation and linear differential problems by methods of Hilbert spaces in Courant> and Hilbert>, Methoden der mathematischen Physik, Volume 2, Berlin 1937, Chapter VII. The common initial value problem of the basic equations of hydrodynamics is the following: We need to find the solution in a prescribed, moving region (0>) of -space, while in is prescribed (together with a suitably formulated condition of continuous continuation for 0>) and the boundary values at the boundary of , 0> are also given (with a suitably formulated sense of continuation). Leray> dedicated three sizable works to this problem in the early thirties<\footnote> Leray>, a) Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'Hydrodyamique. J.Math.pur.appl. Paris, Sér. IX (1933) 1--82; b) Essay sur les mouvements plans d'un liquide visqueux que limitent des parois. c) loc. cit. in footnote . . These inquiries had already forced Leray to use the methods of Hilbert space and the integral interpretation of the equations in three dimensions<\footnote> A long while before then, Oseen> had based his well-known hydrodynamic inquiries on a form of the basic equations that is free of second derivatives. However, he only succeeded in proving existence for sufficiently small times. Cf. his work Hydrodynamik (Leipzig 1927) . In his works, Leray solved the question of existence for all 0> in the following cases, a) > under the added condition of finite kinetic energy, b) is a fixed oval with zero boundary values, c) > under the added condition of finite kinetic energy. The remarkable analysis that Leray dedicates to the question of differentiability point to a strange difference between the dimensions and 2>. While, at least if in the first case is the entire plane, the proof of infinite differentiability is successful, the proof methods that one should view as natural fail for 3>. Even for arbitrary smoothness of all prescribed data, the proof of smoothness of the solution did not work out. The other strange thing is the failure of the uniqeness proff in three dimensions. These questions are still not answered satisfactorily. It is hard to believe that the initial value problem of viscous liquids for should have more than one solution, and more attention should be paid to the settling of the uniqueness question. However, newer research indicates that for nonlinear partial differential problems the number of independent variables has significant influence on the local properties of solutions. In the present work, which is also dedicated to the initial value problem and in which we assume the integral view of the equations as their primary form, we will leave aside the questions of differentiability and uniqueness. We hope to come back to these things as well as to the proof of the energy equation (which is easy in our context) in later memoranda. The main point of this work is that the construction of approximate solutions that takes such broad space in Leray's work is replaced here by simpler process, which may also be applied to a much broader classes of partial differential problems. We also hope to come back to this issue later. This method enables the solution of the initial value problem for all 0> in substantial generality, however in this first memorandum what matters to us is more the exposition of the basic idea of the method rather than the generality of the results. We will restrain ourselves to the case that the -region is fixed in time, but otherwise completely arbitrary, and where has vanishing boundary values. The boundary condition will be defined in terms of Hilbert space--broad enough to guarantee solvability, and narrow enough to guarantee the uniqueness of the solution, at least in two dimensions<\footnote> If is the entire -space, the boundary condition thus phrased becomes the condition of finite kinetic energy and finite dissipation integral. The phrasing of the boundary condition is suggested by the work of Courant> and Hilbert>, Methoden der mathematischen Physik, Vol. 2, Berlin 1937, Chap. VII, Ÿ1, 3rd section. . In pure existence theory, the number of space dimensions will not play any role. >. Solutions of Class >.> We will take the class with respect to an --region > to mean the class of all real, measurable functions defined on this region with finite norm <\equation*> f \x \t. is a real Hilbert space. Terms such as weak and strong convergence in > will be understood in the following with respect to the norm. We remind that a sequence of functions H> in > converges weakly if first, the norms of all remain below a fixed value and second, if <\equation*> f*g \x \t\f>g \x \t holds for any fixed function H> in >. While maintaining the first condition, the second one may be weakened to the effect that the sequence of numbers <\equation*> f*g \x \t converges for any fixed in a set that is strongly dense in . Then, there exists one, and essentially only one weak limit function >> in >. Besides these terms, for which we have assumed an --region, we will have to use the same terms for a purely spatial -region . In this case, we will base our considerations on the norm <\equation*> f\x. We remind the reader of the weak compactness of a sequence of functions with uniformly bounded norms (F.Riesz's Theorem). The following criterion for strong convergence, which was also used extensively by Leray, will also be necessary here. For a sequence of functions that converges weakly in > to a limit function >>, we have <\equation*> |\>f\x*\t\(f>)\x \t, where equality holds if and only if f>> in the strong sense. All these things transfer to vector fields , on > if we use the scalar product <\equation*> uv \x \t and the corresponding norm. <\lemma> If the vector fields converge weakly in > to a limit field >(x,t)>, then <\equation*> >uu \x \t\u>u> \x \t. Equality holds if and only if the convergence in > is strong. Like Leray, we need the term of a generalized (purely spatial) -derivative of functions and fields . <\definition> An defined on an --region > is defined to belong to the class > if and only if it has the following properties: belongs to in >. There exist functions i>> belonging to in > such that the relations <\equation> h*fi> \x \t=-h|\x>*f*\x \t(h\N ) are satisfied for any function which is continuous in > along with its derivatives and which belongs to class , and for any ,n>. The class > obviously contains any that is continuously -differentiable in > such that and all f/\x> belong to in >. For such an , we have f/\x=fi>>. This follows from the integral theorem and the demand that must belong to , i.e. that vanishes outside a certain compact subset of >. Obviously, generalized -derivatives i>> in are uniquely determined except for the values on an --zero set in the case of H> in >. <\lemma> If a sequence of functions of class > converge weakly to >> and for all the expressions <\equation*> f\x \t+fi>fi>\x \t are uniformly bounded, then >> also belongs to > in > and every -derivatives i>> converges weakly to the corresponding -derivative >i>>. <\proof> Every satisfies (), where is an arbitrary function that is admissible there. The right hand sides converge to <\equation*> -h|\x>*f*>\x \t. For a fixed and , the left hand sides converge along the sequence of the 's. The admissible functions in > lie strongly dense in the Hilbert space .Thus, for any fixed the sequence of the i>> is weakly convergent. If we let >> denote the limit function, then from (), we conclude that <\equation*> h*f> \x \t=-h|\x>*f*>\x \t holds for any admissible and . By Definition , >> belongs to > in >, and because of uniquness of the -derivative, we have >=f>i>>. A field is said to be of class > in > if this is the case for all components. In the above integral form of the basic equations of hydrodynamics, there are no derivatives on . It is however practical to make a weak differentiability assumption like membership in the class > on the solutions . We may then write for the friction term in () <\equation> \a|\x>\x>>u=-\a|\x>>u>. <\definition> A field is called a solution of class > of the basic equations of hydrodynamics in the --region > if it satisfies the following conditions: <\enumerate-alpha> H> in >. Vanishing divergence; any function which is of class in > and continuously -differentiable satisfies the relation (). Equations of motion; any field that is of class in >, divergence-free and continuous along with its derivatives >, >, > satisfies the relation (). Observe that under the condition a) the term in the basic equations () which is nonlinear in is a valid Lebesgue integral for any admissible field . That is already the case if H> in >. Because of a) the condition of incompressibility b) is equivalent with <\equation*> div u\u=0 for a.e. >. We will consider all integrands in the basic equations outside of > defined to zero. The integrals can then be extended over all --space. With this convention, the following theorem, which we would like to prove here even though it is not needed in this paper, holds: <\theorem> A solution of class > satisfies the equation <\equation> >au\x=\>a|\t>u \x \t+\>a|\x>>u>u \x \t-\\>a|\x>>u> for a.e. value of >. <\proof> Consider that along with , is also an admissible field if is an arbitrary continuously differentiable function for all . If we replace by in equation , which we write abbreviated as <\equation*> K[a,u]\x \t=>>>K[a,u]\\=0, it follows that the equation <\equation> >>h(\)>K \x\\+>>h(\)>au\x\\=0 is also satisfied. The terms in curly braces are Lebesgue-integrable functions of > on \\\\> that vanish for all large \|>. The validity of for abritray )> with continuous (\)> is equivalent with the fact that <\equation*> >au\x=>>K \x\t=\>K for a.e. >. In , the left hand side is defined for just a.e. >, while the right hand side is an (totalstetig) function of >. In fact, one can prove: A solution of lcass > in > can be changed on an --zero set such that the new satisfies without exception, i.e. for any admissible and any >. But we will not elaborate further on this here. The cross sections of the --region > are -region . By using just terms of Hilbert space, we need to get as close as possible to the boundary condition of vanishing of a function and a field for all on the boundary of . This can be achieved by obtaining the function from function of class in > by means of a limit process. In doing so, it is necessary to use sufficiently effective bounds on the spatial -derivatives (but not on the -derivatives) of the approximating functions, so that the ``vanishing'' remains intact along the boundaries of the -regions . We express the boundary condition by membership in the following function class (N)>. <\definition> A function is said to be of class (N)> in > if it is a weak limit function in > of a sequence of functions (x,t)>, which belong to in > and are continuous along with their -derivatives und for which the expressions <\equation> \+\i>\i> are uniformly bounded.<\footnote> Cf. , l.c. footnote , p. 218. The definition of the boundary condition of vanishing given there is only seemingly stronger than ours. By S. Saks' Theorem the sequence of arithmetic means of a weakly convergent sequence has a strongly convergent subsequence. It follows from this theorem and from Lemma that for any in (N)>, there exists a sequence of functions > of the above-mentioned kind such that <\equation*> \\g,\i>\gi> holds inthe strong sense. It follows from Lemma that for a given --region the class (N)> is contained in the class >. <\lemma> Let > by a cylinder set G>, t\T>. Let be the weak limit in > of a sequence of functions (x,t)> continuously -differentiable in > that are of the following kind: For each > there is a compact subset of the -region such that > vanishes for outside that set; \ also let the integrals be uniformly bounded. Then belongs to (N)> in >.<\footnote> If is all -space, the class *(N)> coincides with the class >. In this case the admissible > are strongly dense in the function space > in the sense of the norm . <\proof> Observe the difference between the class of the > admissible in this lemma and the narrower class of the > of Definition . Membership of > in in the --region > in the present case requires that > vanishes sufficiently close to and . But since only -derivatives occur in , this difference is inconsequential. If we replace the present > by functions (t)\(x,t)>, where > is continuous in ]> and <\equation*> \=| 0\t\\,T-\\t\T,>>|| 2\\t\T-2\,>>>>> and otherwise \\1> (\0>), then Definition applies to the new |~>=\\>. Thus belongs to (N)>. <\lemma> The relations <\equation*> gi>=-g*fi>(i=1,2,\,n) are satisfied by any f of class > in > and any of class (N)> in >. <\proof> By Definition , the relations hold for any specified and for any > that is continuously -differentiable and of class in >. By Definition , is a weak limit of a sequence of such > with uniforml bounded integrals By Lemma , besides \g>, we also have i>\gi>> weakly in >. The relations that hold for , > thus also hold for , . To facilitate a more convenient phrasing of the initial condition, we also introduc the class . In doing so, we restrict ourselves to -space and field that are defined in an -region . If we only consider functions that belong to both the classes and , then it is clear that the strong closure of these sets of functions is identical to . The same is true of vector fields in . However, a difference arises if we restrict ourselves to divergence-free fields in . <\definition> A divergence-free field in of class is said to be of class if it is a weak limit field of fields that belong to in , that are twice continuously differentiable and that are divergence-free.<\footnote> By Saks' Theorem, it is then also the strong limit field of just these fields. One easily proves the following: If the field is divergence-free and of class and if the function (x)> is of class >, then\ <\equation*> u\i> \x=0. Membership of a divergence-free field in obviously replaces the boundary condition of vanishing on the normal component. We may now state the existence theorem for the hydrodynamic initial value problem. <\theorem> Let be an arbitrary region of -space. Let the field be divergence-free in and of class , but otherwise arbitrary. Then there is a field defined for all 0> in with the following properties: <\enumerate-Alpha> In any --cylinder region G>, t\T>, is a solution of class > of the basic equations of hydrodynamics (cf. Definition ). ``Vanishing of the boundary values'' for 0>: In any of the above-mentioned cylinder regions, belongs to (N)>. Initial condition: For 0>, U(x)> converges strongly in . For the construction of the solution of the initial value problem for an -region constant in time, we start with the equation <\equation> au \x\|>-au\|>=>>a|\t>u+>>a|\x>>u>u+\>>a|\x>\x>>u. <\lemma> Let the field be defined in for all 0> and let it belong to class in any cylinder section G>, t\T> of --space. Let it satisfy Equation for all \\\0> and for any field such that: is twice continuously differentiable and <\equation> a=a(x),div a=0 G,a\N G, i.e. vansishes outside a suitable compact subset of . Then satisfies the basic equation for the half cylinder >: G>, 0> and for any field admissible there (cf. condition c) in the definition of a weak solution). <\proof> If we write in the abbreviated form <\equation*> f(\)-f(\)=>>g(t), we see that the equation <\equation*> >\(t)f(t) \t+>\(t)g(t) \t=0 must be satisfied for any > that is continuously differentiable in >) and which vanishes for all sufficiently small and large . If we once more write the equation out in full, we recognize that Equation is satisfied in said half cylinder by any filed (t)a(x)>, where is an arbitrary one of the fields permitted above and (t)> is an arbitrary one of the functions permitted above. But now any permitted by condition c) in the definition of a solution may be approximated in the half cylinder > by sums of fields of such special shape that in the basic equation integration and limit may be interchanged. E.g. one could always arrange that the convergnece of the fields and their derivatives up to a prescribed order in > is uniform and that the approximating fields all vanish outside a fixed compact subset of >. It is thereby clear that a field which satisfies to the extent specified in the lemma, and which is further divergence-free and which belongs to class > in any cylinder section satisfies the full scope of the definition of a solution on any cylinder section. The following fact yields an even better basic equation: <\lemma> There is a sequence of twice continuously differentiable and linearly independent fields in in the field space <\equation> a=a>(x),div a>=0 G,a>\N G with the following property: An arbitrary twice continuously differentiable field in of the form is the uniform limit field in of a sequence of finite linear combinations of the field >(x)>, with uniform convergence of even the derivatives up to second order in . For a given , only such linear combinations occur in this approximation that have the value zero outside a certain compact subset of which only depends on . Based upon this fact it is clear that a field which is of class in each cylinder section and which satisfies the basic equation for all \\\0> and for any field of the mentioned sequence automatically does the same for all fields admitted above. In summary, we can say that the basic equations can be replaced in their entirety by the equations with . In the function sapce of divergence-free vector fields , , is an affine coordinate representation of the basic equations of hydrodynamics. The affine system of coordinate vectors can, by means of a unique linear transformation of a simple kind, be transformed into a new one which is orthonormal in the sense of the bilinear form <\equation*> vw \x. We may additionally assume that the sequence satisfies this condition: <\equation> a>a> \x=\,\>. <\lemma> The orthonormal system of the fields >(x)> is complete in the field space of divergence-free fields of class in . The proof results from Definition and Lemma . . The th approximation step consists simply of only considering the first out of the infinitely many basic equations , , <\equation> a=a>(x)(\=1,2,\,k) and trying to solve those through the ansatz <\equation> u=u(x,t)==1>\>(t)a>(x) with as yet undetermined scalar factors >=\>>. This ansatz automatically satisfies the condition of freedom from divergence (because of ) and the boundary condition of vanishing: <\equation> div u=0G,u\NG. Since only differentiable (t)> need to be considered and since the admissible fields do not depend on , the first equations may be written in the form <\equation> au|\t> \x=a|\x>> u>u \x+\a|\x>\x>>u \x. Because of , the equations , together with represent a system of ordinary differential equations <\equation> \>|\t>=F>(\,\,\)(\=1,2,\,k) for the >, in which the right hand sides >=F>> are polynomials in > with constant coefficients. The equations , , or the equivalent equations share with the strict hydrodynamic equations the important property that for their solutions, the energy equation <\equation> |\t>*uu \x=-\u|\x>>*u|\x>> \x holds. Namely, since the equations hold for all fields , they also hold for their linear combinations >. The energy equation follows in the usual way (and without difficulties at the boundary) since because of <\equation*> u|\x>>u>u \x=K|\x>>*u> \x=0K=uu and <\equation*> u|\x>\x>>*u \x=-u|\x>>*u|\x>> \x(u=u). It follows from that <\equation*> uu \x=\+\+\(u=u) never increases. From this we conclude that any solution of the differential system begun at exists for all (). The approximation procedure may very easily be interpreted formally in the following manner. We think of both sides of the Navie-Stokes differential equations and the solution formally as if they were expanded in the orthonormal system of the fields >>: >a>>. We then obtain purely formally a system of infinitely many differential equations of first order for the infinitely many scalar Fourier coefficients >. Our th step then simply consists of only considering the first of these equations and setting all unknowns with indices \k> to zero. The way in which we subsequently prove our existence theorem simultaneously yields a statement regarding the convergence properties of this simplest and most natural approximation method. We choose the initial values of the >(t)> at to be the Fourier coefficients of the expansion of the given field in the >>. While the solutions (t)> in the th step generally depend on , these initial values are independent of them. By the assumption that H(N)> in and by the completeness lemma , we have <\equation> u(x,0)\U(x)>(k\\). We summarize the properties of the fields of the sequence which we will need in the following: <\enumerate-alpha> Each (x,t)> is twice continuously --differentiable and divergence-free for G>, 0>. (x,t)> vanishes if lies outside a compact subset of the -region that only depends on . (x,t)> satisfies the equation (0>) and the equation (\\\0>) in the cases (=1,2,\,k)>. The integrals <\equation*> uu \x,u|\x>>*u|\x>> \x \t(u=u(x,t)) remain beneath a bound which is independent of . The initial values (x,0)> satisfy the limit relationship . d) follows immediately from the temporally integrated energy equation in connection with . . Each field >(x)> is continuous in and different from zero only in a compact subset of . If we apply the first half of d) to the right hand side of (>>) by estimating the term linear in > by means of the Schwarz Inequality and the term quadratic in by means of an absolute bound for the derivatives of , we obtain the following: The right hand side of (>>, >, \>) is uniformly bounded for fixed > for all and . The same is true of the left hand side <\equation*> |\t>au \x. For fixed >, the time functions <\equation*> a>(x)u(x,t) \x satisfy a Lipschitz condition for all 0> that is independent of . Furthermore, they remain uniformly bounded for all and . So by a well-known (Auswahlsatz) there exists for an arbitary, fixed > a sequence of integers > such that <\equation> lim\\>a>(x)u(x,t) \x exists for any 0>, in fact uniformly so in any finite -interval. The sequence of > depends of the index >, but we may pick the sequence belonging to the index +1> as a subsequence of the previous one. By means of a diagonal argument we may thus form a fixed sequence of integers (which we will once again label as >) for which the limit statement above holds properly for any fixed =1,2,\>. In the sequel, we will operate on this sequence of >. We will now prove that the sequence of fields >(x,t)> converges weakly in the -region for each fixed 0>. For the purposes of our proof, we now fix an arbitrary, fixed value > of and observe that by the first half of 5d) the sequence of these fields (>) is weakly compact in . The claim will be proven when we show that that sequence may possess only a single weak limit field in . Let >(x,t)> be such a limit field and let > be a subsequence of the > (this subsequence will depend on >) such that <\equation*> lim\\>w(x)u>(x,t) \x=w(x)u>(x,t) \x for each field of class in . In the case >>, the value of the right hand side is already fixed by the limit . If >> and \>> are two weak limit fields and if is their difference field, then <\equation*> a>v \x=0 for each >. By Definition the fields >>, \>> and thus also belong to class in . However, by Lemma the fields >> span the same field space in . From this we conclude <\equation*> vv \x=0 and thus the claim. Consequently, there is a field >> which is well-defined in for all 0> such that <\equation> lim\\>w(x)u>(x,t) \x=w(x)u>(x,t) \x for each field (H> in G) and for each 0>. The field >> satisfies condition B) of the existence theorem at the end of Section . This follows from b) and the second half of 5d) by applying Lemma . One easily proves that >\u>> also holds weakly in and (t\T>). . The proof that the field >(x,t)> satisfies condition A) of the existence theorem. In each cylinder region >G, t\T>, >> belongs to class >, which is, as we remarked, a superclass of (N)> (and because of B) it also belongs to the latter class). By the arguments in the first half of Section we only need to show that >> satisfies the equations for every >> and for all \\\0>. By c), >> satifies these equations for the same ,\> and for the first > fields >>. We now fix >, > and the index > and pass to the limit \\>. It is clear that on the left hand side of may be replaced by >>. The same ist true of the third integral on the right hand side (the first one is zero). Consider that in <\equation*> >>w(x)u>(x,t) \x \t the inner integral is a uniformly bounded function with respect to \ > because of the first half of d) and that we may apply a well-known Lebesguian convergence theorem to the outer -integral. It requires some deeper thoughts that make use of the second half of d) to see that we may also interchange the limit \\> and the integration in the second integral on the right hand side of . For this, we need the following theorem which we will prove later. <\lemma> Let a sequence of functions (x,t)> which are continuously -differentiable for G>, t\T> have the following properties: For each fixed , > belongs to class . For each fixed , the (x,t)> converge weakly in to a function >(x,t)>. The integrals <\equation*> f(x,t) \x,fi>fi>(f=f) remain uniformly bounded with respect to and . Then the > converge strongly to >> on the --region Q G>, t\T>, where is an arbitrary finite cuboid in -space. In particular, the assertion holds for itself if is bounded. Because of a), b), because of the result of the second step and because of d), the assumptions of the lemma are satisfied for the components of the sequence of fields >(x,t)> for an arbitrary fixed . Thus, it follows that <\equation*> (u-u>)(u-u>)(u=u>) goes to zero for \\> if is an arbitry finite cuboid of -space. We can thus justify the passing to the limit in the second integral on the right hand side of (>>, > fixed). Recall that the factor of the integrand vanishes outside a fixed compact subset of . If we choose C> and \>, then for the integral <\equation*> >>(a>)(u>)(a=a>,u=u>) we have the following stuation. The first factor converges weakly in the area of integration to >u>>>, while the second one converges strongly to >>. As is well-known, this suffices to carry out the passing to the limit under the integral sign. We have thus shown that the field >> satisfies the equations for any field >(x)> and for all positive >, >. The condition A) of the existence theorem is thus verified except for the freedom from divergence. This latter property, however, is trivially true, even for any fixed 0>. To complete the proof of the existence theorem, we only need to show that the initial condition C) is also satisfied. From the energy equation follows <\equation> uu \x\|=uu \x\|+u|\x>>*u|\x>> for each field of our sequence. The left hand side tends to <\equation*> UU \x for \\> because of . For , the fields converge weakly to >> in . In an --cylinder section, we have <\equation*> u>>\u>> weakly because of Lemma and d). By applying Lemma , implies the inequality <\equation*> UU \x\u>u> \x\|+\u>>u>> for an arbitrary 0>. In particular, <\equation*> 0>|\>u>u> \x\UU \x. If we once again apply Lemma to this last inequality, we recognize that the initial condition C) is satisfied, which is what we wanted to show. We will not go into detail on the question of strong convergence for a fixed . > The lemma is closely related to the (Auswahlsatz) and is proven similarly as well<\footnote> Cf. , l.c. footnote , p. 218. In Rellich's Theorem, the boundedness of the -integrals of the squares of the derivatives is assumed. Our boundedness assumption merely concerns the --integral and is thus better adapted to the state of affairs in our problem. Leray proves and uses a lemma even closer to the (Auswahlsatz) l.c. Footnote , p. 214, Lemma 2, which, like this theorem, only works with the -integral. Our convergence proof is more direct. . Let us note up front that the lemma, just like Rellich's Theorem, need not hold for itself if is infinite. A counterexample is given by the case where is the entire -space and <\equation*> f(x,t)=f(x+k,x,\,x) with belonging to > and in . In this case, >=0>, but there is no strong covnergence to zero<\footnote> We may thus only conclude the strong convergence of the approximate fields to >(x,t)> in the cylinder sections if is bounded. However, strong convergence is clearly true for arbitrary . Leray deduced it for his aprpoximations in the case where is the entire -space using complicated estimates of the distribution of energy over . We hope to come back to the stronger convergence properties of our approximations at some later date. .\ The proof of Lemma arises from Friedrichs' Inequality: Let be a finite cuboid in -space. For any given \0>, there exists a finite number of fixed functions >(x)> which belong to in such that the inequality <\equation*> f \x\>f*\> \x+\fi>fi> \x is satisfied by any function belonging to > in <\footnote> The >> may be assumed to be orthogonal in . The inequality then represents an estimate of the difference in Bessel's inequality. You may find the proof of the inequality in , l.c. footnonte , p. 218, Chap. VII, Ÿ3, Section 1. We may easily convince ourselves that the proof that is given there in 2 dimensions also works in dimensions. Friedrichs' Inequality does not hold for arbitrary bounded regions. . For the proof of Lemma , we first note that for fixed the functions (x,t)> of the lemma are continuously differentiable in and of class . If we define the functions to be zero outside , then this statement remains valid if we relate it to the entire -space instead of to . In particular, any of the functions on any finite cuboid of -space belongs to class >. The extension of the functions and the last statement were made possible by the assumption of membership in lcass . This is however the only place where this assumption is used. We now fix a cuboid and a number \0> arbitrarily and pick the finitely many auxiliary functions >(x)> such that Friedrichs' Inequality holds in . We apply it to the functions <\equation> f(x,t)=f(x,t)-f(x,t), which surely belong to > in , for fixed . By integration in , we conclude that all the functions satisfy the inequality <\equation> f\>f*\> \x\t+\fi>fi>. By assumption (weak convergence for fixed ), we have <\equation*> lim\,l\\>f*\> \x=0 for each fixed . Because of the boundedness assumption (first half), furthermore the function of <\equation*> (f-f)*\> \x remains uniformly bounded w.r.t. , . Thus the first term on the right hand side in tends to zero for \>, \>. By assumption, the factor of > for the functions remains below a fixed bound. But <\equation*> >\,l\\>(f-f)\c*\ implies strong convergence of our our sequence in the --region Q>, t\T>, since > was arbitrary. We easily obtain that the limit function is the function >(x,t)> mentioned in the statement of the lemma. Thus, Lemma is proven. <\initial> <\collection> <\references> <\collection> > |1>> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||Table of contents> |.>>>>|> |math-font-series||1Introduction.> |.>>>>|> |math-font-series||2The Function Class |H>. Solutions of Class |H>.> |.>>>>|> |math-font-series||3The Boundary Condition of Vanishing. The Initial Value Problem.> |.>>>>|> |math-font-series||4Simplification of the Problem. The Approximation Procedure.> |.>>>>|> |math-font-series||5Proof of the Existence Theorem.> |.>>>>|> |math-font-series||6Proof of Lemma > |.>>>>|>