A functional analytic method for the analysis of general partial differential equations

In this paper we give a short overview of operator semigroups. These objects are widely used for proving well-posedness of partial differential equations and for investigating qualitative properties of the solutions.

Saved in:

Bibliographic Details
Date:	2006
Main Author:	Sikolya, E.
Format:	Article
Language:	English
Published:	Інститут програмних систем НАН України 2006
Subjects:	Прикладне програмне забезпечення
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/1579
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	A functional analytic method for the analysis of general partial differential equations / E. Sikolya // Проблеми програмування. — 2006. — N 2-3. — С. 669-673. — Бібліогр.: 7 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-1579
record_format	dspace
spelling	Sikolya, E. 2008-08-26T13:22:15Z 2008-08-26T13:22:15Z 2006 A functional analytic method for the analysis of general partial differential equations / E. Sikolya // Проблеми програмування. — 2006. — N 2-3. — С. 669-673. — Бібліогр.: 7 назв. — англ. 1727-4907 https://nasplib.isofts.kiev.ua/handle/123456789/1579 004.75 In this paper we give a short overview of operator semigroups. These objects are widely used for proving well-posedness of partial differential equations and for investigating qualitative properties of the solutions. en Інститут програмних систем НАН України Прикладне програмне забезпечення A functional analytic method for the analysis of general partial differential equations Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	A functional analytic method for the analysis of general partial differential equations
spellingShingle	A functional analytic method for the analysis of general partial differential equations Sikolya, E. Прикладне програмне забезпечення
title_short	A functional analytic method for the analysis of general partial differential equations
title_full	A functional analytic method for the analysis of general partial differential equations
title_fullStr	A functional analytic method for the analysis of general partial differential equations
title_full_unstemmed	A functional analytic method for the analysis of general partial differential equations
title_sort	functional analytic method for the analysis of general partial differential equations
author	Sikolya, E.
author_facet	Sikolya, E.
topic	Прикладне програмне забезпечення
topic_facet	Прикладне програмне забезпечення
publishDate	2006
language	English
publisher	Інститут програмних систем НАН України
format	Article
description	In this paper we give a short overview of operator semigroups. These objects are widely used for proving well-posedness of partial differential equations and for investigating qualitative properties of the solutions.
issn	1727-4907
url	https://nasplib.isofts.kiev.ua/handle/123456789/1579
citation_txt	A functional analytic method for the analysis of general partial differential equations / E. Sikolya // Проблеми програмування. — 2006. — N 2-3. — С. 669-673. — Бібліогр.: 7 назв. — англ.
work_keys_str_mv	AT sikolyae afunctionalanalyticmethodfortheanalysisofgeneralpartialdifferentialequations AT sikolyae functionalanalyticmethodfortheanalysisofgeneralpartialdifferentialequations
first_indexed	2025-11-24T02:44:11Z
last_indexed	2025-11-24T02:44:11Z
_version_	1850840219485470720
fulltext	Прикладне програмне забезпечення © Eszter Sikolya, 2006 ISSN 1727-4907. Проблеми програмування. 2006. №2-3. Спеціальний випуск 669 UDC 004.75 A FUNCTIONAL ANALYTIC METHOD FOR THE ANALYSIS OF GENERAL PARTIAL DIFFERENTIAL EQUATIONS Eszter Sikolya Department of Applied Analysis, Eötvös Loránd University P.O. Box 120, H-1518 Budapest, Hungary E-Mail: seszter@cs.elte.hu In this paper we give a short overview of operator semigroups. These objects are widely used for proving well-posedness of partial differential equations and for investigating qualitative properties of the solutions. Key-words: partial differential equation, abstract Cauchy problem, operator semigroup, air pollution transport model 1. Operator (semi)groups Many physical phenomena can be described by so-called dynamical systems. Here we investigate the following model. The elements of the state space describe completely the temporal change of the system and they include all factors important for the observant. They also determine unambiguously the further motion of the system. The time is parameterized by R or +R (it depends whether we want to handle the past or not). We assume that to each time )( +∈ RRt belongs a state of the system Ztz ∈)( from the state space Z . We also assume that the motion is deterministic, that is, for every time instant 0t and initial state 0z there exists a unique motion Zz zt →R: 00 , such that .00, )( 00 ztz zt = We further assume that the system is autonomous that means )()( 1,0, 0100 htzhtz ztzt +=+ holds for any R∈htt ,, 10 and Zzz ∈10 , . This implies that the orbits of the motion do not intersect each other. Using this model, we can define the operators ZZtT →:)( for )( +∈ RRt acting as ),(:)( 0,0 ttzztT zt += where 0t can be chosen arbitrary since the system is autonomous. Then clearly zzT =)0( holds because .)( 0,0 ztz zt = In this way we have defined a one-parameter (semi)group of operators satisfying ( ) ZIdT stsTtTstT = ∈=+ + )0( ,),()()( RR since the system is deterministic. Looking for the solutions of the Cauchy functional equation in C    = ≥=+ 1)0( 0,),()()( )( T stsTtTstT CFE we find that tatT e)( = is a solution for any C∈a . It is easy to see that tatT e)( = satisfies the following differential equation, too.     = ≥= .1)0( 0),()( )( T ttaTtT dt d CDE Прикладне програмне забезпечення 670 If we suppose the solution )(tT of )(CFE to be continuous, we obtain that it is unique (see Engel-Nagel, 2000). Theorem—Assume that CR →⋅ +:)(T is a continuous solution of )(CFE . Then there exists a unique C∈a such that tatT e)( = . We now generalize the above result in an arbitrary Banach (complete normed) space X , e.g., in ,nX C= ],,[ baCX = or )(1 RLX = . By )(XL we denote the space of bounded linear operators on X . Let us look for solutions )(:)( XLT →⋅ +R of the following problem    = ≥=+ XIdT stsTtTstT FE )0( 0,),()()( )( Definition—Let )(:)( XLT →⋅ +R be a solution of )(FE satisfying .)(lim 0 XxxxtT t ∈∀= +→ Then 0))(( ≥ttT is called a strongly continuous (one-parameter) semigroup (or C0-semigroup). If these properties hold for R instead of +R , we call 0))(( ≥ttT a strongly continuous (one-parameter) group (or C0-group) . For details see Engel-Nagel (2000) and Pazy (1983). 2. Generator If )(XLA∈ – e.g. n n XMA CC =∈ ),( – then using the exponential series we can define )(e XLtA ∈ . It is easy to see that the operator family 0,e:)( ≥= ttT tA forms a C0-semigroup satisfying )(FE . Furthermore, )(tT is a solution of the following differential equation:     = ≥= XIdT ttATtT dt d DE )0( 0),()( )( In this case 0 )( == t tT dt d A and A is called the generator of the semigroup. In general, we can define the generator of a strongly continuous semigroup as follows (see Engel-Nagel, 2000 and Pazy, 1983). Definition—Let 0))(( ≥ttT be a strongly continuous semigroup. The linear (but not necessarly bounded) operator 00 0 ))(( )( lim: )( lim::)( =+→ +→ =−=       ∈−∃∈= tt t xtTt dt d t xxtT Ax X t xxtT XxAD a is called the generator of 0))(( ≥ttT . Since ))(,( ADA is defined as the derivative of the orbits of the semigroup in 0 , )(tT is in some ways the generalization of the exponential function of A . Of course, in this case tAe can not be defined by the exponential series because ))(,( ADA is not bounded and the series not necessarily converges in norm. But one can prove that )(AD is always dense in X and ))(,( ADA is closed. 3. Abstract Cauchy problems Up to now it is not clear how operator semigroups can be used for solving problems in the applications. The clue is the abstract Cauchy problem. It is well-known that many physical phenomena can be formulated mathematically as a system of partial differential equations, see e.g. the air pollution transport model in the next section. These systems can often be rewritten as an abstract Cauchy problem, that is    = ≥= 0)0( 0),()( )( uu ttAutu ACP & Прикладне програмне забезпечення 671 The operator A on the right-hand side is usually an (unbounded) differential operator on a function (Banach) space X , 0,)( ≥∈ tXtx . One can prove the following (see e.g. in Engel-Nagel, 2000). Theorem—Let ))(,( ADA be a closed, densely defined linear operator on X and let )(ACP be the associated abstract Cauchy problem defined as above. Then the following assertions are equivalent. a) For every )(0 ADx ∈ there exists a unique solution of )(ACP depending continuously on the initial data 0x . b) ))(,( ADA is the generator of a strongly continuous semigroup 0))(( ≥ttT on X . In this case the solution is .0,)()( 0 ≥= txtTtx Hence, to prove well-posedness of a problem written in the form of an abstract Cauchy problem one has to verify that the operator on the right-hand side is the generator of a C0-semigroup. In general it is not easy, but in many important cases it is possible. 4. Examples The next examples can be found in Engel-Nagel (2000). Diffusion semigroup Let us take a look at the one-dimensional heat conduction equation with Neumann boundary conditions: .0,0)1,()0,( ]1,0[),(),0( )1,0(,0),,(),( 2 2 ≥= ∂ ∂= ∂ ∂ ∈= ∈≥ ∂ ∂= ∂ ∂ ttu s tu s ssfsu ststu s stu t We can rewrite it as    = ≥= 0)0( 0),()( )( xx ttAxtx ACP & with }.0)1()0(:]1,0[{:)( : 2 =′=′∈= ′′= ffCfAD fAf Here the Banach space is ]1,0[CX = and ),()( ⋅= tutx . Observe that the boundary conditions appear in the domain of A hence the operator becomes unbounded – but still it is closed and densely defined in X . Using the eigenvalues 22nπ− and eigenfunctions 2,cos2,1 ≥nnsπ of A and the theory of linear ordinary differential equations, one can prove the following. Theorem—The operator ))(,( ADA defined above generates a strongly continuous semigroup 0))(( ≥ttT on ]1,0[CX = with .coscose21:),( ]1,0[],1,0[,)(),())()(( 0 1 0 22 nrnsrsk sCfdrrfrsksftT n tn t t πππ ⋅∑+= ∫ ∈∈= ∞+ = − This semigroup is called the one-dimensional diffusion semigroup. In nR one can prove the following. Theorem—Consider the closure of the Laplace operator ,),,,(),,,(∆ 1 212 2 21 ∑ ∂ ∂= = n j n j n sssf s sssf KK defined for every f from the Schwartz space of rapidly decreasing, infinitely many times differentiable functions on nR . It generates a strongly continuous semigroup 0))(( ≥ttT on )(1 nLX R= with .)0( ,0,)(e 4 1 ))()(( 4 \|\| 2 IdT tdf t ftT n t n = ∈>∫= − − Rsrrs rs Rπ This semigroup is called the n-dimensional diffusion semigroup. Прикладне програмне забезпечення 672 Translation semigroup Let us investigate the closure of the following first order differential operator ).(:)( : 1 n cCAD fAf R= ∇= Here )(1 n cC R denotes the space of continuously differentiable functions having compact support in nR . One can easily prove that ))(,( ADA generates a strongly continuous semigroup 0))(( ≥ttT on )(0 nCX R= (the space of continuous functions vanishing at infinity on nR ) with ,),())()(( ntfftT Rss1s ∈+⋅= called the translation semigroup on nR . Multiplication semigroup Let CR →nq : be a continuous function. We can define the following closed, densely defined linear operator on )(0 nCX R= . )}.(:)({:)(M :M 00 nn q q CqfCfD qff RR ∈∈= = If ∞< ∈ )(Resup sq s nR then )(,0,e:)( 0 ntq q CftfftT R∈≥= defines the strongly continuous multiplication semigroup, generated by ))(M,(M qq D . Air pollution transport model We now turn to a concrete problem that is treated in details in Csomós, Faragó (2005), Dimov, Faragó, Havasi, Zlatev (2001, 2006). Air pollution transport can be modeled by the following partial differential equation.      ∈= ∈+−++−∇= ∂ ∂ .),()0,( ],0(),(∆)( )( 0 ncc TtcRcEcc t c APM Rxxx u σ Here ),( tcc x= denotes the concentration of the air pollutant, ),( txuu = describes the wind velocity, ),( tEE x= is the emission function, ),( txσσ = the deposition and )(cR the chemistry operator. For the sake of simplicity we assumed the diffusion coefficient to be 1 . If we look at the right-hand side of )(APM we find that all the operators acting on c are of type discussed above, hence generate strongly continuous semigroups on appropriate spaces. Using the perturbation theory of semigroups (see Engel-Nagel, 2000) we obtain well-posedness for )(APM . 5. Qualitative behaviour The importance of the operator semigroup theory is revealed especially in proving qualitative properties of solutions of partial differential equations (abstract Cauchy problems, resp.). A rich theory for qualitative properties of C0 –semigroups has been developed in the last 50 years that can be useful also in the applications. Here we mention only one example. Let us recall the famous Liapunov Stability Theorem for matrices (1892). Theorem—Let )(CnMA∈ an be an nn × matrix. Then the following assertions are equivalent. a) 0\|\|e\|\|lim = ∞→ tA t b) All eigenvalues of A have negative real part, i.e., 0Re <λ for all )(Aσλ ∈ . This result can be generalized for the asymptotic of semigroups having bounded generator (see Engel-Nagel, 2000 and Pazy, 1983). Theorem—Let )(XLA∈ on some Banach space X and 0,e:)( ≥= ttT tA the strongly continuous semigroup generated by A . Then the following assertions are equivalent. Прикладне програмне забезпечення 673 a) 0\|\|)(\|\|lim = ∞→ tT t b) 0Re <λ for all )(Aσλ ∈ . If the semigroup is regular enough, we also can characterize stability with the spectrum of the unbounded generator A . Theorem—Let A be the generator of an eventually norm-continuous semigroup 0))(( ≥ttT on X , that is, there exists 00 ≥t such that the function )(tTt a is norm continuous from ),( 0 ∞t into )(XL . Then the following assertions are equivalent. a) 0\|\|)(\|\|lim = ∞→ tT t b) 0)}(:sup{Re <∈ Aσλλ . The same holds if the semigroup is positive on a function space, that is, it maps positive (i.e. greater or equal to zero) functions into positive functions. This is the case in many important applications such as heat diffusion etc. Hence, to prove that the solutions of an abstract Cauchy problem converge to 0 if ∞→t it is enough to investigate the spectrum of the operator on the right-hand side. 6. A few words about applications in numerical analysis In the numerical solution of (complicated) partial differential equations the operator splitting method is often used. Here we divide the spatial differential operator of the system into simpler operators and solve the corresponding problems one after the other, by connecting them through their initial conditions (see e.g. Faragó, 2005). To use this method one has to assume that the sub-problems are well-posed which in practice is often hard to prove. We also have to know the error caused by the operator splitting and by the use of numerical methods. Applying operator semigroup techniques helps a lot to answer these questions. 1. Csomós, P. and Faragó I., 2005: Error analysis of the numerical solution obtained by applying operator splitting. Preprint. 2. Csomós, P., Faragó, I. and Havasi, Á., 2005: Weighted sequential splittings and their analysis. Comput. Math. Appl. 50, 1017–1031. 3. Dimov, I., Faragó, I., Havasi, Á. and Zlatev, Z., 2001: L–commutativity of the operators in splitting methods for air pollution models. Annales Univ. Sci. Sec. Math. 44, 127–148. 4. Dimov, I., Faragó, I., Havasi, Á. and Zlatev, Z., 2006: Different splitting techniques with application to air pollution models, Int. J. Environmental Pollution (to appear). 5. Engel, K.-J. and Nagel, R., 2000: One-parameter Semigroups for Linear Evolution Equations, Springer, New York. 6. Faragó, I., 2005: Splitting methods for abstract Cauchy problems, Lect. Notes Comp. Sci. 3401, Springer Verlag, Berlin, 35–45. 7. Pazy, A., 1983: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York. Прикладне програмне забезпечення © K. Georgiev, E. Donev, 2006 ISSN 1727-4907. Проблеми програмування. 2006. №2-3. Спеціальний випуск 674 1.

A functional analytic method for the analysis of general partial differential equations

Institution

Similar Items