Well-posedness of the microwave heating problem Текст научной статьи по специальности «Математика»

Discrete & Continuous Models

& Applied Computational Science__2024' 32 (2) 222-233

ISSN 2658-7149 (Online), 2658-4670 (Print) httW/purnak.rnd^u^h

Research article

UDC 519.872, 519.217

PACS 07.05.Tp, 02.60.Pn, 02.70.Bf

DOI: 10.22363/2658-4670-2024-32-2-222-233 EDN: CDJVIL

Well-posedness of the microwave heating problem

Baljinnyam Tsangia

Mongolian University of Science and Technology, Ulaanbaatar, Mongolia (received: May 1, 2024;revised: May 10, 2024;accepted: May 15, 2024)

Abstract. A number of initial boundary-value problems of classical mathematical physics is generally represented in the linear operator equation and its well-posedness and causality in a Hilbert space setting was established. If a problem has a unique solution and the solution continuously depends on given data, then the problem is called well-posed. The independence of the future behavior of a solution until a certain time indicates the causality of the solution. In this article, we established the well-posedness and causality of the solution of the evolutionary problems with a perturbation, which is defined by a quadratic form. As an example, we considered the coupled system of the heat and Maxwell's equations (the microwave heating problem).

Key words and phrases: evolutionary problems, nonlinear perturbation, Lipschitz continuous, quadratic form, coupled problems

Citation: TsangiaB., Well-posedness of the microwave heating problem. Discrete and Continuous Models and Applied Computational Science 32 (2), 222-233. doi: 10.22363/2658-4670-2024-32-2-222-233. edn: CDJVIL (2024).

1. Introduction

Here we consider a non-linear, coupled system in thermoelectricity. Thermoelectric effects are viewed as the result of the mutual interference of heat flow and electric flow in a system. The interaction of thermal and electric processes is modeled by the heat equation

pCpd09 + divq = Q

and Maxwell's equations

-curlH + J + d0D = J1 curlE + d0B = 0.

Here q is the thermal current flux, p is the volumetric mass density, Cp is the specific heat density, 9 is the absolute temperature, J is the electric current flux, E, H are the electric and magnetic fields, respectively, D is the displacement current, B is the magnetic induction and Jx is the given electric source. Q describes the production of internal energy by various mechanisms, such as the Joule heating, radioactive decay, etc. In our system the Joule heating Q = (E \J) produces the internal energy. This term governs the non-linearity in the system and, moreover, it couples the heat and Maxwell's equations. The system of these equations has to be supplemented by so-called constitutive equations, which describe the material's properties and effects. As constitutive equations, we deal with the following thermoelectric material relations

J = oE

©TsangiaB., 2024

j/v§> I This work is licensed under a Creative Commons "Attribution-NonCommercial 4.0 International" license.

q = -Agrad# D = eE B = llH.

Here o is the electric conductivity, X is the thermal conductivity, e is the electric permittivity, ¡j. is the magnetic permeability. The coupled system of the heat and Maxwell's equations with these constitutive equations becomes the microwave heating problem. The microwave heating problem has wide industrial applications and it has been studied theoretically and numerically in various situations (see e.g. [1-3] and the references therein). We study the coupled systems in the three-dimensional case. Moreover, we consider this system with the physical coefficients defined as 3-by-3-matrix-valued functions depending on the spatial variables only. We assume (homogeneous) Dirichlet boundary conditions for 9, (homogeneous) electric boundary conditions for E and non-vanishing initial values. We say that a problem is well-posed if the problem has a unique solution and the solution continuously depends on the given data. The independence of the future behavior of a solution until a certain time indicates the causality of the solution. In our solution theory the well-posedness and causality of a given problem are discussed.

The idea of tackling well-posedness and causality of the problem just discussed is to frame the above system in the theory of evolutionary equations: In [4, 5] it has been found that a number of initial boundary-value problems of classical mathematical physics is represented by the following general form

(doM(d-1)+A)u = F. (3)

Here d0 is the (continuously invertible) derivative with respect to time in a suitable weighted Hilbert space, A is a skew-selfadjoint operator in a suitable Hilbert space; the mapping (z ^ M(z)) is bounded operator valued and holomorphic in an open ball Bc (r, r) with some positive radius r centered at r. The operator Mid-1) is interpreted in the sense of a function calculus by establishing d0 as a normal operator in a suitable Hilbert space. The solution theory associated to (3) was established in [4, 5] and many diverse problems were studied there. For applications, we focus on a particular case of M(d-1), namely,

M(d-1) = M0 + d-1M1.

Here M0 is a selfadjoint, bounded, linear operator with M0lN(Mo) ^ c0 >0, M1 is a bounded, linear operator satisfying %M1lR(Mo) ^ c1 > 0. In the next section we establish the solution theory of the following problem

(d0M0 +M1 +A)u+F(u) = F, (4)

which covers the aforementioned non-linear coupled system. Here Fis a quadratic form. The nonlinear problem (1) yields a fixed point problem. In our approach the well-posedness of (4) is based on the strict positive definiteness of the operators % (d0M0 +M1 +A) and % (d0M0 +M1 + A)* and a Lipschitz continuous approximation of F. Due to the strict positive definiteness result, the inverse operator (d0M0 +M1 +A)_1 becomes Lipschitz continuous in a suitable Hilbert space. Thus, (4) amounts to be an evolutionary problem in the sense of (3) with a Lipschitz continuous perturbation, which is eventually solved by the contraction mapping principle. As an application we shall consider the microwave heating problem in the third section.

2. Solution theory

We start by establishing time differentiation d0 as a normal operator. It is initially considered on

o

CTO(IR), which is the set of infinitely often differentiable, complex-valued functions defined on the real line I having compact support. Hence d0 is a densely defined, closed linear operator on I? (I), moreover, it is an essentially skew-selfadjoint operator on L2 (I). We define the following weighted L2-space

Hvfi (I) : = L2 (I, exp (-2vx) dx) ■.= {<p G L\oc (I) | exp (-vm0) <p G L2 (I)} equipped with the norm

:=\ I l<P(x)l2 exp (-2vx)dx, <pGHvfi (R). ' V ■'R

Here m0 is the closure of the following operator

cL (R)CL2 (R) ^ L2 (R)

<p ^ (x^- x<p (x)).

The operator m0 is called the multiplication by argument operator and it is densely defined, Hermitian and moreover, it is self-adjoint. Let v g R. We also define the operator exp (-vm0) such that exp (-vm0) <p := (x^ exp (-vx) <p (x)) for <pGL)0C (R). Note that exp (-vm0) [cL (R)] = CL (R). Due

o

to the density of CL (R) in both the spaces L2 (R) and Hvfi (R), exp (-vm0) can be extended to a unitary operator from L2 (R) onto Hvfi (R) and the unitary extension is denoted again by exp (-vm0). The inverse of exp (-vm0) is

exp (vm0) : L2 (R) ^ Hvfi (R).

Note that we may utilize the notation H0fi (R) for the space L2 (R) with the inner product {■ \ -)0 0 and the norm H00. The following operator

3V := exp (vm0) d0 exp (-vm0)

is unitarily equivalent to d0 on Hv, 0 (R). The operator dv + v is the time derivative on Hv, 0 (R) and we denote it again by d0. Moreover, for all v g R>0, d0 : D (d0) C Hv 0 (R) ^ Hv 0 (R) is continuously invertible on Hvfi (R), that is

Ml <1

11 nL(Hvfi(R)) V

and a normal operator for all v G R\ {0} on Hvfi (R). Furthermore, d_ 1 : (R) ^ Hv,o (R) is a normal operator (see e.g. [6, Theorem 5.42]) and there is the Sobolev chain

HvM1 (do)^HvJc (do), kGN

with respect to d0, where Hv k (d0) := D (d*) is the Hilbert space with the norm

Hv,k = mv,o

for each kGN. Furthermore, we have

Hv,-k (do)^Hv-k_! (do), kGN,

where Hv,_k (d0) are completions of Hvfi (R) for all k e N with the norms l~lv_k ■ = ld_k-we can unitarily extend the following operator

Note that

Hv,0 (R)QHv,_i (do) ^ Hv,0 (R) cp » d_lcp.

We denote its extension again by 3-1. This motivates the unitary extension of d0 from Hvfi (R) onto Hv—1 (d0) for each veR\ {0} and we denote the extension again by d0. In the same manner we obtain unitary operators

HvM1 (do)

Hv,k (do) docp

for k e Z, as appropriate unitary extension/restriction of the originally discussed operator d0 defined on Hvo (R).

2.1. On skew-selfadjoint operator

Let H1 and H2 be Hilbert spaces. For a densely defined, closed linear operator C ■ D(C) CH1 ^ H2 and a block operator matrix B defined as follows

B

_( 0 -C* )

( C 0 J

(5)

is skew-selfadjoint and so is the following diagonal operator matrix

0

A _

B1 0 0 ••.

••. 0

0 Bn )

where each Bu i = 1.n is defined as in (5).

In a coupled system of the heat and Maxwell's equations with Dirichlet boundary condition and electric boundary condition A has the following form

0 div 0 0

grad 0 0 0

0 0 0 curl

0 0 curl 0

where div, grad, curl and curl are defined as follows. Let O c R3 be an open set. Consider the following vector analytical differential operators

n

gradc ■ cm (Q)cL2 (a) ^ ®l2 (a)

k=1

Ф » (дкФ)кф,...,п]

and

n n

t2.

divc ■ Qc^ (o)c^L2 (a) ^ l2 (a) k=1 k=l

n

^k^-M » Z k=\

The operators grad„ and -divc are formally adjoint to each other and closable. Denoting

grad := gradc, div := div,

and

grad := (-div) , div := (-gradc) we can construct the following skew-selfadjoint operator

Л :=( 0 diV ).

\ grad 0 )

where grad, div and grad, div are all together densely defined, closed linear operators. The operator A1? is not only skew-selfadjoint but also encode Dirichlet boundary condition, that is, <p being in

D (grad) means that cp satisfies a generalized homogeneous Dirichlet boundary condition.

Due to the skew-selfadjointness of A, we have a long Sobolev chain with respect to A + 1. Since ±1 e p (A), the domains of the operators A and A + 1 coincide. There is the Sobolev chain

Hk+1 (A + 1) ^ Hk (A + 1) for keZ.

Here H = ■ H0 (A + 1), Hk (A + 1) is the domain of (A + 1)k and it is a Hilbert space with the norm l-lk ■=l(A+1)k ] for each ke N and H_k (A + 1) is the completion of H for each k eN under the

norm |-|_fc ■ = |(A + 1)_k . For the sake of brevity, we also denote HkA ■ = Hk (A+ 1). Now we are in the position to construct the Sobolev lattices

(Hv,k ®HnA)kneZ

for the chains (Hv k (d0))keZ and (Hn (A + 1))neZ with respect to the operators d0 ® IH and IHv 0 ® A. Here IH ■ H ^ H and IHv 0 ■ Hvfi (R) ^ Hvfi (R) are the identity operators. Note that Hv^k ® H can be interpreted as the completion of the linear space generated by H-valued functions of the special form

t ^ $(t)w =■ (ip®w) (t)

o

for each k eN, where $ eC^ (R), w e H. In fact, Hvk ®H is unitarily equivalent to Hvk (R, H) for each k e N.

The operators d0 ® IH and IHv 0 ®A are well-defined and have essentially the same properties as the operators d0 and A, respectively. Therefore, we also write A and d0 for their canonical extensions

A0IH and IHv 0 do in Hv,o

2.2. The material law operator

The Fourier-Laplace transform

Cv :=Texp (-vm0) : Hvfi (R,H)^L2 (R,H)

given as a composition of the (temporal) Fourier transform T and the unitary weight operator exp (-vm0), is a spectral representation associated with d0. It is

do = £% (imo +v)Cv.

This observation allows us to consistently define an operator function calculus associated with d0 in a standard way and we can even extend this calculus to operator-valued functions by letting

M(d__1) : = £IM' 1

irn0 + v

Here the linear operator M^—— ) : L2 (R,H) ^ L2 (R,H) is determined uniquely via

(mI^-1—)<pW) ^mI^-1—)cp(A.) \ \imo + vr) \imo +W

.... in H

'i0 + v / / \imo + V /

o

for every AgR, <p g Cœ(R, H) by an operator-valued function M. For a material law the operator-valued function M needs to be bounded and an analytic function z ^ M(z) in an open ball Bc(r, r) with some positive radius r centered at r. Here we will concentrate on the following particular form of the material law

M(d-1)=M0 +d-1Mu

where M0 is selfadjoint, bounded linear and M0 ^ c0 >0 on the range M0 [H], the null space [{0}] M0 is non-trivial and M1 G L (H) with '¡RM1 ^ c1 > 0 on [{0}] M0.

This is not an artificial assumption, rather a necessary constraint enforced by the requirement of causality and strictly positive definite condition

X(u 1 (d0M(d-1))u)HvAKH) >c(u\ u)HvAKH)

for c g I>0 and all sufficiently large v g I>0 and all ugD (d0). The strict positive definite condition implies

%(u \ (Ô0M0 +Ml +A)u)Hv^H) >c(u\ u)HvAKH) for c G R>0 and all sufficiently large v g R>0 and all ugD (d0). Moreover,

d0M0 +M1 +A

has dense range in Hvfi (I,H). For all sufficiently large v g I>0, we have

\w-^11 1

\\(d0M0 +M1 +A) \\ <—, 0<cv <c1

\\ yi(Hv>0(R,H)) Cv

and this also implies the solution theory of the following evolutionary problems

(d0M0 +M1 +A)u = S(u) + f, where S is a suitable Lipschitz mapping.

2.3. Well-posedness of evolutionary problems with a non-linear perturbation term

After making some reformulations in the microwave heating problem, the problem gets the following shape

(d0M0 +M1 + A)u+F(u) = F,

where F is a quadratic form and it is not Lipschitz continuous. For a Lipschitz continuous approximation of the quadratic form we recall the following Lemma and Theorem in [7].

Lemma 1. Let f : R>0 ^ R>0 be differentiable, and such that (z ^ \^zf (z)\) is bounded. Let £ g Cnxn be selfadjoint with £ >0. Then there exists C >0 such that

\f((u\£u)cn ) — f((v\£v)

c»)L ^ C\u — v\cn

for all u, v g Cn.

Theorem 4. Let (Q, y.) be a a—finite measure space. Let £ G (lx (Q))nxn and

F : D(f) ^ Hvfl (R)0L2 (Ü)

u ^ (Rx Ü 3 (t,a¡) ^ (u(t,a¡) \ £(a>)u(t,a>))cn)

with maximal domain. Here D (F) c Hvfl (R) 0 (L2 (Q))". We assume that £ (a) g Cnxn is selfadjoint and positive for a.e. w G Q. Let f : R>0 ^ R>0 be differentiable with f (0) = 0 and such that

Z » \ Jzf (z)\

is bounded. Define

Ff : D (Ff) C Hvfl (R) 0 (L2 (Q))n ^ Hvfl (R) 0 L2 (a)

u » (Rx Q 3 (t,v) » f(F(u)(t,M)))

with maximal domain. Then D (Ff) = Hvfl (R) 0 (L2 (Q))" and Ff is Lipschitz continuous.

Hence, it suffices to find a specific function f which satisfies all the assumptions in Lemma 1 and approximates the quadratic form.

Example 1. We consider the following function

% : R>0 ^ R>0 2 Í

for | e R>0. The function is infinitely differentiable for all x > -1 and as % ^ 0+, it is approximated by the argument, that is,

ft (x) « x.

The mapping (z ^ (z)|) is uniformly bounded. Indeed,

^1 + lz

Let £ (co) G Cnxn be selfadjoint and positive for a.e. w G Q. By Lemma 1, the following holds

((u \ £u)cn) — f% ((v \ £v)cn)| ^C\u — v\cn for C G R>o. Since (0) = 0 for all £ G R>o, the mapping defined by

Ff( : Hv,o (R)^(L2 (Q))n ^ Hv,o (R)^L2 (a)

u » ((t,a>)»f% ((u(t,<u)\ £(a)u(t,a))cn))

is Lipschitz continuous for all £ G R>0 and (i,ffl)GRxfi (Theorem 4) and the Lipschitz constant of

the mapping Ff is . Furthermore, the following holds

4 11

pf( (u) = h «u \ £u)cn) ~ (u \ £u)cn

for sufficiently small % G R>0. We have obtained the Lipschitz continuous mapping Ff = (u ^ ft ((u \ £u)cn)) which approximates F = (u ^ ((u \ £u)c„)) as £ ^ 0+.

Hence, the solution theory of the perturbed problem

(d0M0 +M1 +A)u + Ff( (u) = F

provides an approximate solution of (4).

Theorem 5. Let (O, p) be a a—finite measure space. Let H = L2 (0)n. Let M0 G L (H) be selfadjoint, positive definite and M0Imq[h] ^ c0 > 0, and M1 g L(H) with №M1\[{o^]Mo ^ c1 > 0. Let A : HlyA c

H ^ H be a skew-selfadjoint operator. Assume that £ G (L» (0))nxn is selfadjoint and positive definite, and 0 < —||V£|| < 1 for some % G R>0. Let u0 g D(A) and F g jr>0 (m0)[Hv0 ®H] begivendata.

Then there exists a unique solution u g Hvfi ®H of

= Ff( (u)+F + S ®MoUo

for all v > v0 for some v0 g R>o The solution depends continuously and causally on the data. Moreover, the initial condition

(Mou) (0+) = MoUo

is attained in H-lyA.

3. The microwave heating problem

The microwave heating problem has wide industrial applications and it has been studied theoretically and numerically in various situations (see e.g. [1-3] and the references therein). In the study of the microwave heating problem the electric conductivity and/or thermal conductivity are considered as an operator, which may depend on the temperature (see e.g. [3]). But we will study the microwave heating problem with temperature independent thermal conductivity, electric conductivity, magnetic permeability and electric permittivity. These material coefficients are defined as 3x3 matrix-valued functions depending only on the spatial variables. This may describe material properties more substantially. The equations are introduced in the introduction. The set of originally given equations turns into the following equations

pCpdo& + divq = (E \ oE)t

c3

grad# +X-lq = 0 - curlH + ctE + 30eE = J1 curlE + дфН = 0.

A formal reformulation of these equations yields

30M0 + Mi +

0

grad 0 0

div

0 0 0

0 0 0 curl

0 0

-curl

0

E H

■■F +

(E \ оЕ)сз 0 0 0

(7)

where

Mo

and (E \ аЕ)г3 = (и \ £u)cw with

pcp 0 0 0

000 000 0 £ 0 0 0 ß J

M,

0 0 0 0

0 Ä-1 0 0

0 0 CT 0

V 0 0 0 0 J

£ : =

0000 0000 0 0 CT 0

V 0 0 0 0

Now we reformulate (7) to the proper evolutionary problem. Assume that S satisfies the (generalized) Dirichlet boundary condition and E satisfies the (generalized) homogeneous electric boundary condition. Then we have the following skew-selfadjoint operator

0 div 0 0

grad 0 0 0

0 0 0 -curl

0 0 curl 0

A :

in Н1Л : = H (grad, fl) ф H (div, fl) ф H (curl, fl) ®H(cur\,Q), where fl с R3 is an open set. As

in the preceding application, we assume that pCp : L2 (fl) ^ L2 (fl), s : (l2 (fl))3 ^ (L2 (fl))3 and у : (L2 (O))3 ^ (L2 (O))3 are selfadjoint, bounded linear and strictly positive definite operators. Hence, M0 is selfadjoint, bounded linear and strictly positive definite in M0 [(L2 (fl))10]. LetЛ-1 be in L ((L2 (fl))3) and ЖЛ-1 strictly positive definite. Furthermore, assume that a is selfadjoint, positive definite and it is in (L™ (Q))3x3. Then, M1 is in L ((L2 (fl))10) and ПМ1 is strictly positive definite on [|0}]M0 = {0} ф (L2 (fl))3 ф {0}6. Since a e (L™ (fl))3x3 is selfadjoint and positive definite, so is £ e (L™ (fl))10x10. Moreover, there exists a selfadjoint, positive definite operator V£. Hence, the quadratic

form (u \ £u)c10 is approximated by the Lipschitz continuous mapping Ff = (u^ ft ((u \ £u)c10)) as I ^ 0+, where ft is defined in Example 1. Let 90 e H(grad,o), E0 e H(curl,o), H0 eH(curl,Q)

and V0 : = (pCp90,0, eE0,¡iH0). Then u0 : = (&0,0,E0,H0) e D(A), V0 = M0u0 and V0 eM0 [D (A)]. The initial boundary-value problem with respect to the microwave heating problem is presented in our framework as follows

30M0 + Mi +

0

grad 0 0

div

0 0 0

0

0

0 o

curl

0 0

-curl

0

\\ / \ % (u \ £u)cw ' pCp$o '

q 0 + 5 ® 0

= F +

E 0 sE0

)) , H , , 0 \

(8)

This problem yields the following initial value evolutionary problem with the Lipschitz continuous perturbation

30M0 +Mi +

0 o

grad

0 0

div

0 0 0

0

0

0 o

curl

0 0

-curl

0

\\ ' Ff| (u) ' ' PCP&0 "

q = F + 0 + 5® 0

E 0 sE0

)) < H ) , 0 ) \ )

(9)

for sufficiently small £ e R>0. In the next theorem we sum up the solution theory of (9), which concerns the approximation solution of (8).

Theorem 6. Let Pr1 e L ((L2 (O)) ), and R2T1 be strictly positive definite. Let z, p e L ((L2 (O)) be selfadjoint and strictly positive definite operators. Let pCp e L (L2 (O)) be selfadjoint and strictly positive definite. Let a e (L™ (Q))3x3 be selfadjoint, positive definite. Let (90, 0,E0,H0) e D(A) and F e jR>0 (m0)[Hv0 (R)®(L2 (O))10] be given data. Let v0 e R>0. Furthermore, assume that

0 <

vjl

llVêll

< 1 for all v >v0 and for some parameter Ç e R>0 and some cv > 0. Then there exists

a unique solution u e HVoo (R) ® (L2 (Of) of (8) for all v > v0. The solution depends continuously and causally on the given data.

4. Conclusions

We have obtained a Lipschitz continuous function approximating the quadratic form

(u^(u\ £u)cn)

for a selfadjoint, positive definite operator £ in (L(ü))nxn, n e N. This gives us an opportunity to conclude the well-posedness and causality of the evolutionary problems with non-linear term consisting of the quadratic form with the help of the solution theory associated to the evolutionary problems with a Lipschitz continuous perturbation.

The quadratic form can be found in the heat equation coupled with Maxwell's equations. One of these coupled systems is the microwave heating problem. Here we assumed that the physical coefficients describing the properties of the underlying material, s, ^ e L ((L2 (ü))3), X,o,a e (L™ (Q))3x3 are 3x3 matrix-valued functions depending only on the spatial variables.

Author Contributions: The author has read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Data sharing is not applicable.

Acknowledgments: I would like to express my heartfelt gratitude to my supervisor, Prof. Dr. Rainer Picard for his continuous support, guidance and his established solution theory of evolutionary problems. My especial thanks to Prof. Dr. Marcus Waurick who made numerous comments and corrections on my research work. I would also like to thank academician O.Chuluunbaatar for advising and his suggestions correcting some errors in the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Hill, J. M. & Marchant, T. R. Modelling microwave heating. Appl. Math. Model. 20, 3-15 (1996).

2. Yin, H. M. Regularity of weak solution to Maxwell's equations and applications to microwave heating. J. Differ. Equ. 200,137-161 (2004).

3. Yin, H. M. Existence and regularity of a weak solution to Maxwell's equations with a thermal effect. Math. Methods Appl. Sci. 29,1199-1213 (2006).

4. Picard, R. A structural observation for linear material laws in classical mathematical physics. Math. Methods Appl. Sci. 32, 1768-1803 (2009).

5. Picard, R. & McGhee, D. Partial Differential Equations: A unified Hilbert Space Approach 469 pp. (Berlin/New-York, 2011).

6. Weidmann, J. Linear Operators in Hilbert Spaces 402 pp. (Springer-Verlag, New-York, 1980).

7. Tsangia, B. Evolutionary problems: Applications to Thermoelectricity PhD thesis (TU Dresden, 2014).

Information about the authors

Baljinnyam Tsangia—Dr.rer.nat, Lecturer of Department of Mathematics, School of Applied Sciences, Mongolian University of Science and Technology (e-mail: [email protected], ORCID: 0000-0002-3331-2516)

UDC 519.872, 519.217

PACS 07.05.Tp, 02.60.Pn, 02.70.Bf

DOI: 10.22363/2658-4670-2024-32-2-222-233

Корректность задачи о микроволновом нагреве

Балжинням Цангиа

Монгольский университет науки и технологий, Улан-Батор, Монголия

Аннотация. Ряд начально-краевых задач классической математической физики формулируется в виде линейного операторного уравнения, а его корректность и причинность в гильбертовом пространстве были установлены ранее. Если задача имеет единственное решение и решение постоянно зависит от заданных параметров, то задача называется корректной. Независимость дальнейшего поведения решения до определенного момента указывает на причинность решения. В данной работе установлены корректность и причинность решения эволюционных задач с возмущением, определяемым квадратичной формой. В качестве примера рассмотрена связанная система уравнений теплопроводности и Максвелла (задача микроволнового нагрева).

Ключевые слова: Эволюционные задачи, нелинейное возмущение, Липшицева непрерывность, квадратичная форма, связанные задачи