Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media
A modification of Maxwell’s equations is proposed to describe media with electric and magnetic properties changing essentially under electromagnetic field. It is shown that for such media the electromagnetic waves have finite speed of propagations property for a time depending on the initial energy...
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irk-123456789-692422014-10-10T03:01:39Z Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media Namlyeyeva, Yu.V. Taranets, R.M. Yurchenko, V.M. A modification of Maxwell’s equations is proposed to describe media with electric and magnetic properties changing essentially under electromagnetic field. It is shown that for such media the electromagnetic waves have finite speed of propagations property for a time depending on the initial energy of electromagnetic field and nonlinear parameters of the medium. Запропоновано модифікацію рівнянь Максвела, що описує середовища, в яких електричні та магнітні властивості суттєво змінюються під впливом зовнішнього електромагнітного поля. Для таких середовищ встановлено, що електромагнітні хвилі розповсюджуються зі скінченою швидкістю на протязі часу, який, в свою чергу, залежить від початкової енергії електромагнітного поля та нелінійних параметрів середовища. Предложена модификация уравнений Максвелла, описывающая среды, в которых электрические и магнитные свойства существенно изменяются под воздействием внешнего электромагнитного поля. Для таких сред установлено, что электромагнитные волны распространяются с конечной скоростью в течение времени, зависящего от начальной энергии электромагнитного поля и нелинейных параметров среды. 2009 Article Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media / Yu.V. Namlyeyeva, R.M. Taranets, V.M. Yurchenko // Физика и техника высоких давлений. — 2009. — Т. 19, № 4. — С. 44-56. — Бібліогр.: 26 назв. — англ. 0868-5924 PACS: 02.30.Jr, 02.90.+p, 41.20.Jb, 13.40.Hq http://dspace.nbuv.gov.ua/handle/123456789/69242 en Физика и техника высоких давлений Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України |
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A modification of Maxwell’s equations is proposed to describe media with electric and magnetic properties changing essentially under electromagnetic field. It is shown that for such media the electromagnetic waves have finite speed of propagations property for a time depending on the initial energy of electromagnetic field and nonlinear parameters of the medium. |
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Namlyeyeva, Yu.V. Taranets, R.M. Yurchenko, V.M. |
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Namlyeyeva, Yu.V. Taranets, R.M. Yurchenko, V.M. Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media Физика и техника высоких давлений |
| author_facet |
Namlyeyeva, Yu.V. Taranets, R.M. Yurchenko, V.M. |
| author_sort |
Namlyeyeva, Yu.V. |
| title |
Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media |
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Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media |
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Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media |
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Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media |
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Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media |
| title_sort |
finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media |
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Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/69242 |
| citation_txt |
Finite speed of the electromagnetic-field propagations in nonlinear isotropic dispersive media / Yu.V. Namlyeyeva, R.M. Taranets, V.M. Yurchenko // Физика и техника высоких давлений. — 2009. — Т. 19, № 4. — С. 44-56. — Бібліогр.: 26 назв. — англ. |
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Физика и техника высоких давлений |
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2025-07-05T18:53:07Z |
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| fulltext |
Физика и техника высоких давлений 2009, том 19, № 4
© Yu.V. Namlyeyeva, R.M. Taranets, V.M. Yurchenko, 2009
PACS: 02.30.Jr, 02.90.+p, 41.20.Jb, 13.40.Hq
Yu.V. Namlyeyeva1, R.M. Taranets1, V.M. Yurchenko2
FINITE SPEED OF THE ELECTROMAGNETIC-FIELD PROPAGATIONS
IN NONLINEAR ISOTROPIC DISPERSIVE MEDIA
1Institute of Applied Mathematics and Mechanics, NAS of Ukraine
74, R. Luxemburg Str., Donetsk, 83114, Ukraine
2Donetsk Institute for Physics and Engineering named after O.O. Galkin NAS of Ukraine
72, R. Luxemburg Str., Donetsk, 83114, Ukraine
Received June 11, 2009
A modification of Maxwell’s equations is proposed to describe media with electric and magnetic
properties changing essentially under electromagnetic field. It is shown that for such media the
electromagnetic waves have finite speed of propagations property for a time depending on the
initial energy of electromagnetic field and nonlinear parameters of the medium.
Keywords: Maxwell’s equations, nonlinear dispersive medium, finite speed of propaga-
tions, asymptotic behavior
1. Introduction
We consider a classical Maxwell system in the SI (see [12]):
0
curl (1.1)
( ) curl 0 (1.2)
div ρ div 0 (1.3)
t
tM
+ = ,⎧
⎪ + = ,⎨
⎪ = , = ,⎩
D J H
B E
D B
where E and H are electric and magnetic fields; D and B are electric and magnetic
inductions; ρ is the charge density. The current density J satisfies the Ohm’s law:
= σ ,J E (1.4)
where σ is the electric conductivity.
We consider isotropic media in which permittivity ( )x tε = ε , and magnetic
( )x tμ = μ , conductivity are functions of space and time. In this situation, equa-
tions of state have the following simple form (see [12]):
= ε , = μD E B H . (1.5)
Substituting (1.5) into equations (1.1) and (1.2) we obtain equations for 0E=E E/%
and 0H=H H/% for an isotropic medium in the following dimensionless form:
Физика и техника высоких давлений 2009, том 19, № 4
45
1 1
0
2 2
curl 0 (1.6)
( )
curl 0 (1.7)
t
t
a bM
a b
⎧ + − = ,⎪
⎨
+ + = ,⎪⎩
E E H
H EH
%
%
% % %%%%
% %%% %
where
0 0 0 0 0
i
i
xtt xt x
ε μ σ
= , = , ε = , μ = , σ =
ε μ σ
% % % %% , and
0 0 0 0 0 0
1 2 1 2
0 0 0 0 0 0 0
1 d 1 d 1 1
d d
t t H t Ea a b b
t t x E x H
⎛ ⎞σε μ
= + σ , = , = , =⎜ ⎟ε ε μ ε ε μ μ⎝ ⎠
% %
% %
% %% %
. (1.8)
Here the subscript 0 denotes the corresponding values for vacuum. Further, we
suppose that
0 0 0 0 0 0
0 0 0 0 0 0 0
1 1 1t t H t E
x E x H
σ
= , = , =
ε ε μ
,
whence we find that
0 0 0
0 0 0
0 0 0
H cc t x
E
ε ε
= ε , = , =
σ σ
, (1.9)
where
0 0
1c =
μ ε
is the velocity of light. In view of (1.9), omitting ’ ’, we ob-
tain the following system:
1 1
1
2 2
curl 0 (1.10)
( )
curl 0 (1.11)
t
t
a b
M
a b
+ − = ,⎧
⎨ + + = ,⎩
E E H
H H E
where
1 2 1 2
1 d 1 d 1 1,
d d
a a b b
t t
ε μ⎛ ⎞= +σ = , = , =⎜ ⎟ε μ ε μ⎝ ⎠
. (1.12)
Hyperbolic systems as (M1) are well investigated (see, e.g., [21]). For example, if
σ = 0, and
( ) ( )= ε , , = μ ,D E H E B E H H , (1.13)
then the system (M0), (1.13) is that considered in [6]. In particular, the authors
have shown that this system is Poincaré-invariant if and only if 2( ) ( ) cε , μ , =E H E H ,
and they found its various invariants.
Note, the system ( 1M ) can be reduced to the following equation relative toE :
3 4 1 2 2 2 2( div curl ) 0tt ta a b b b b a+ + + ∇ − Δ +∇ × +∇ × =E E E E E E H , (1.14)
where
( ) ( )( ) ( )3 4
d dd d d d d d1 1 1 1 1 1 1
d d d d d d d da at t t t t t t t
μ μ⎛ ⎞ε ε ε ε ε= + σ + + , = + σ + +σ +⎜ ⎟ε ε μ ε ε ε μ⎝ ⎠
.
Физика и техника высоких давлений 2009, том 19, № 4
46
Now, we consider the special case when ( ) ( )t tε = ε , μ = μ such that 2( ) ( ) vt tε μ =
is a constant. In this case, due to (1.13), the equation (1.14) has the form:
2
5( ) v 0tt ta −+ − Δ =E E E , (1.15)
where 5
1 d
d
a
t
ε⎛ ⎞= + σ⎜ ⎟ε ⎝ ⎠
. For example, if 5a is a constant, i.е. ( )tε =
5 5
0
(0) ( ) d
t
a t ae e− τ⎛ ⎞
= ⎜ε − σ τ τ⎟
⎜ ⎟
⎝ ⎠
∫ , then (1.15) has the following solution w =E
{ } { },
1
j j
Ni k x t
w j
j
E e +ω
=
= = , where N
jk R∈ , and
22
5 5 4v
2
j
j
a a k−− ± −
ω = . As a
consequence, this solution has spatiotemporal oscillations if 2 2
5 4v 0ja k−− | | < ,
and spatial oscillations only if 2 2
5 4v 0ja k−− | | ≥ .
In a more general situation, electric and magnetic inductions depend on electric
and magnetic fields (see [12]), i.e.
( ) ( )= , , = ,D D E H B B E H . (1.16)
Below, we consider the simplest case of the equations of state (1.16) when per-
mittivity and magnetic conductivity are some functions of space and time de-
pending on electric E and magnetic H fields and its gradients, precisely, of the
energy density w and w∇ . In the case, we arrive at the system ( 1M ), i.e. equa-
tions for E and H in an isotropic nonlinear medium, where ( )i ia a x t w w= , , ,∇ and
( )i ib b x t w w= , , ,∇ ( 1 2i = , ) satisfy relations (1.12). We will study media in
which nonlinear functions ia and ib satisfy the following conditions:
1
1 1( ) 0 1 2m p
ia x t w w d w w d i−, , ,∇ ≥ |∇ | , < < ∞, = , , (1.17)
1 2( ) ( )b x t w w b x t w w, , ,∇ = , , ,∇ , (1.18)
1
2 2( ) 0 1 2n
ib x t w w d w d i−| , , ,∇ |≤ , < < ∞, = , , (1.19)
2
3 3( ) 0 1 2n
ib x t w w d w w d i−| ∇ , , ,∇ |≤ | ∇ |, < < ∞, = , , (1.20)
where 2 2( )w w x t E H= , = + is the dimensionless energy density corresponding to
isotropic media with constant permittivity and magnetic conductivity;
1 0 and 0m R p n∈ , > > (1.21)
are parameters of medium. The special choice of structure conditions on ia and ib
allows us to apply methods from the theory of parabolic equations to the hyper-
bolic system ( 1M ). Conditions (1.17)–(1.20) result in the following restrictions on
ε and μ:
Физика и техника высоких давлений 2009, том 19, № 4
47
( )1 1 1 1
2 1 1
dd1 1
d d
n m m ppd w d w w d w wt t
− − − −μεε = μ ≥ , + σ ≥ |∇ | , ≥ | ∇ |
ε μ
,
whence we deduce that
1 1 1
2 10
0
max exp d
t
n m p
td w d w w− − −
=
⎧ ⎫⎛ ⎞⎪ ⎪ε = μ ≥ , ε ⎜ | ∇ | τ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
∫ .
The equations like 1( )M describe media in which permittivity and magnetic con-
ductivity are some nonlinear functions. The media have same structure to have to
appear in the simulation of various processes in laser optics and weakly ionized
plasma theory, where properties of medium are strongly dependent on energy
density of electromagnetic field, for example, ferroelectric, piezoelectric, multifer-
roic, etc.
In this paper, we study the propagation properties of solutions to Cauchy
problem for Maxwell’s equations in the following dimensionless form
1 1
2 2
0 0
curl 0 in (1.22)
( ) curl 0 in (1.23)
(0 ) ( ) (0 ) ( ) (1.24)
t T
t T
a b Q
M a b Q
x x x x
+ − = ,⎧
⎪ + + = ,⎨
⎪ , = , , = ,⎩
E E H
H H E
E E H H
where Q (0 ) N
T T R= , × , 2 3N = , , 0 T< < ∞ , and the functions ia =
( )ia x t E H E H= , , , ,∇ ,∇ , ( )i ib b x t E H E H= , , , ,∇ ,∇ ( 1 2i = , ) satisfy conditions
(1.17)–(1.20). The unknown functions are electric E and magneticH fields, which
depend on the time t and the space-variable x . Moreover, we suppose that the initial
electromagnetic field is located in half-space { }( ) 0N N
N NR x x x R x− ′:= = , ∈ : < , i.e.
supp ( 0) Nw R−., ⊂ , (1.25)
where 2 2( , )w x t E H= + .
Remark 1.1. If 1 2a a= , 1 2b b= , and div( ) 0× =E H then from ( )M we find
that 1
0
( ) ( 0)exp 2 d
t
w x t w x a
⎛ ⎞
, = , ⎜ − τ⎟
⎜ ⎟
⎝ ⎠
∫ , i.e. the energy density w decays in time.
Thus, the presented system (M) is obtained from the classical Maxwell’s sys-
tem (M0) taking into account the equations of state (1.16) for isotropic nonlinear
medium and the Ohm’s law for current density (1.4). Media are described to pos-
sess the finite speed propagations property. There are many papers in which en-
ergy decay was obtained for different problems concerning Maxwell’s equations.
Well-posedness and asymptotic stability results and decay of solutions are proved
making use of different techniques. Below, we mention some results concerning
energy decay and asymptotic of solutions.
Some linear evolution problems arise in the theory of hereditary electromag-
netism. Many authors studied the influence of dissipation due to the memory on
Физика и техника высоких давлений 2009, том 19, № 4
48
the asymptotic behavior of the solutions (see [2,4,5,7,13,14,20]). The polynomial
decay of the solutions when the memory kernel decays exponentially or polyno-
mially was shown in [15]. It is studied the asymptotic behavior of the solution of
the linear problem describing the evolution of the electromagnetic field inside a
rigid conducting material, whose constitutive equations contain memory terms
expressed by convolution integrals. These models were proposed in [19] where it
was shown that the exponential decay of the memory kernel is able to produce a
uniform rate decay of the energy in rigid conductors with electric memory.
The exact boundary controllability and stabilization of Maxwell’s equations
have been studied by many authors (see [17] and references therein). In [17] the
internal stabilization of Maxwell’s equations with the Ohm’s law for space vari-
able coefficients is studied. The authors give sufficient conditions on parameters
of the medium which guarantee the exponential decay of the energy of the system.
The result is based on observability estimate, obtained in some particular cases by
the multiplier method, a duality argument and a weakening of norm argument, and
argument used in internal stabilization of scalar wave equations.
The energy decay of solutions of the scalar wave equation with nonlinear
damping in bounded domains has been shown in [3,11,16,23–26]. In the case
when there is no damping term in the equation for the dielectric polarization, the
long-time asymptotic behavior of the solution of Maxwell’s equations involving
generally nonlinear polarization and conductivity is studied in [8].
The propagation of electromagnetic waves in gas of quantum mechanical sys-
tem with two energy levels is considered in [10]. The decay of the polarization
field in a Maxwell–Bloch system for t →∞ was shown.
The transient Landau–Lifschitz equations describing ferromagnetic media
without exchange interaction coupled with Maxwell’s equations is considered in
[9]. The asymptotic behavior of the solution of this mathematical model for mi-
cromagnetism is studied. It is shown the strong convergence of the electromag-
netic field with respect to the energy norm for t →∞ on bounded sets of nonvan-
ishing electrical conductivity.
Following the dominant trend in the literature, we can conclude that study of
the system (M) is not only of theoretical interest but is useful for applied re-
searches. Since these authors are not specialists in electromagnetism, we apolo-
gize in advance for the omissions and inaccuracies. We hope that there is an inter-
disciplinary audience which may find this useful, whether we do not know any
concrete media with the proposed properties.
The present paper is organized as follows. In Section 2 we formulate our main
result. In Sections 3 we prove the finite speed propagations property to some time
which depends on the parameters of the problem and the initial electromagnetic
field. The method of proof is connected with nonhomogeneous variants of Stam-
pacchia lemma, in fact, it is an adaptation of local energy or Saint–Venant princi-
ple like estimates method. Appendix A contains necessary interpolation inequali-
ties and important properties of nonhomogeneous functional inequalities.
Физика и техника высоких давлений 2009, том 19, № 4
49
2. Main results
We introduce the following concept of generalized solution of the system ( )M :
Definition 2.1. Let 1 1 ( 1)n p p m p n> , > , − < < − and 2 2w E H= + . A pair
( ( ) ( ))x t x t, , ,E H such that
1 1 1(0 ( )) (0 ( )) ( )
m p
pN p p N
t Tw C T L R w L T W R w L Q
+
,∈ , ; , ∈ , ; , ∈
is called a solution of problem ( )M if for a.e. 0t > the integral identities
2 2 2
1
1 1( ) ( )d ( ) ( )d d ( ) ( )d d
2 2N
T T
t
Q QR
E t x t x x E t x t x x t a E t x t x x t, η , − , η , + , η ,∫ ∫∫ ∫∫ –
2
1
1curl d d (0 ) (0 )d
2 N
TQ R
b x t E x x x− = , η ,∫∫ ∫E H , (2.1)
2 2 2
2
1 1( ) ( )d ( ) ( )d d ( ) ( )d d
2 2N
T T
t
Q QR
H t x t x x H t x t x x t a H t x t x x t, η , − , η , + , η , +∫ ∫∫ ∫∫
2
2
1curl d d (0 ) (0 )d
2 N
TQ R
b x t H x x x+ = , η ,∫∫ ∫H E (2.2)
are satisfied for every 1( )TC Qη∈ .
The main result is the following.
Theorem 1. Let the pair ( ( ) ( ))x t x t, , ,E H be a solution of the problem ( )M , in
the sense of Definition 2.1. Let 1 1p n> , > ( and ( 1)( )1 (2 )
p p Nn pN p
− +< +
−
if 2)p < ,
and
11 1max 1 1 ( 2) 11
n np p p m p nN p N p
⎧ ⎫⎛ ⎞ ⎛ ⎞−− , − + − , − + − < < − +⎨ ⎬⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠⎩ ⎭
.
Then there exists a time 0T ∗ > , depending on the known parameters only (in par-
ticular,
1( )( 0) NL Rw x, ), and a function ( ) [0 ] (0) 0t C TΓ ∈ , , Γ = such that
( )
( 1)
( )
( 1)
for 1
( ) max
for 1
p N m p n
p N m p
p N m p n
p N m p
t t
t K t t K
t t
κ+ + −
κ+ + −
+ + −
+ + −
⎧ ≤ ,⎧ ⎫⎪ ⎪ ⎪Γ = , =⎨ ⎬ ⎨
⎪ ⎪ ⎪⎩ ⎭ > ,⎩
(2.3)
where
( )[ ]
( )[ ]
1 ( ) ( 1)
( 1) ( ( 1) ) ( 1)( 1)
p p N m p n np N m p
p N m p p p n m N p m p
− + + − + + −
κ =
+ + − − − + − + −
,
and
{ }supp ( ) ( ) ( ) 0N
N Nw t x x x R x t t T ∗′,. ⊂ = , ∈ : < Γ ∀ < < , (2.4)
Физика и техника высоких давлений 2009, том 19, № 4
50
i.e. E(x,t) = H(x,t) = 0 for all { }( ) ( )N
N Nx x x x R x t′∈ = , ∈ : < Γ . Here K =
1( )( (0 ) )NL RK n m p N w x= , , , , , is a positive constant.
Remark 2.1. The statement of Theorem 1 stays true if we consider the problem
for system ( )M in some bounded domain. Then, instead of (1.25), we suppose
that a support of initial energy of electromagnetic field is contained in some ball
into the domain.
3. Proof of finite speed of propagations
Summing (2.1) and (2.2), in view of conditions (1.17) and (1.18), we find that
1
11 ( , ) ( , )d ( , ) ( , )d d ( , )d d2 2N
T T
m p
t
Q QR
w t x t x x w t x t x x t d w w t x x tη − η + |∇ | η∫ ∫∫ ∫∫ +
+ 1
1div ( ) ( , )d d (0, ) (0, )d2 N
TQ R
b x t x t w x x x× η ≤ η∫∫ ∫E H . (3.1)
Above we used the following relation:
div ( ) curl curl× = −E H H E E H . (3.2)
From (3.1), (1.19) and (1.20) we get
( ) ( )d ( ) ( )d d ( )d d
m p
p
N
T T
p
t
Q QR
w x T x T x w x t x t x t c w x t x t
+
, η , − , η , + | ∇ | η ,∫ ∫∫ ∫∫ ≤
≤ 1
2( 0) ( 0)d 2 ( ) d d
N
T
n
QR
w x x x d w x t x t−, η , + | × || ∇η , |∫ ∫∫ E H +
+ 2
32 ( )d d ( 0) ( 0)d
N
T
n
Q R
d w w x t x t w x x x− | ∇ || × | η , ≤ , η ,∫∫ ∫E H +
+
( 1)
1( )d d ( ) ( )d d
m p p n m
p p
T T
p
Q Q
w x t x t c w x t x t
+ − −
−ε | ∇ | η , + ε η ,∫∫ ∫∫ +
+ ( ) d d
T
n
Q
c w x t x t| ∇η , |∫∫ (3.3)
for every nonnegative function 1( ) ( )Tx t C Qη , ∈ , where 0 1 1p nε > , > , > ,
( 2) 1p m p n− < < − + (i.e. ( 1) 11
p n m
p
− − >
−
).
For an arbitrary 1s R∈ and 0δ > we consider the families of sets
{ }( ) ( ) ( ) (0 ) ( )
( ) ( ) \ ( ) ( ) (0 ) ( )
N
N N T
T
s x x x R x s Q s T s
K s s s K s T K s
′Ω = = , ∈ : ≥ , = , ×Ω ,
,δ = Ω Ω +δ , ,δ = , × ,δ .
Физика и техника высоких давлений 2009, том 19, № 4
51
Next we introduce our main cut-off functions 1( ) ( )N
s x C R,δη ∈ such that
0 ( ) 1 N
s x x R,δ≤ η ≤ ∀ ∈ and possess the following properties:
0 \ ( )( ) ( )
1 ( )
N
s s
cx R sx x K s
x s,δ ,δ
⎧ , ∈ Ω ,⎪η = |∇η |≤ ∀ ∈ ,δ .⎨ δ, ∈Ω + δ ,⎪⎩
Choosing 0ε > sufficiently small and
1( ) ( )exp 0sx t x tT T−⎛ ⎞
⎜ ⎟,δ ⎝ ⎠
η , = η − ∀ > (3.4)
in integral inequality (3.3), we find
(0 ) ( ) ( ) ( )
1sup ( )d ( )d d d d
m p
p
T T
p
t T s Q s Q s
w x t x w x t x t c w x tT
+
∈ , Ω +δ +δ +δ
, + , + | ∇ |∫ ∫∫ ∫∫ ≤
≤
( 1)
1
( ) ( ) ( )
( 0)d d d d d ( )
p n m
p
T T
n
T
s K s Q s
cw x x w x t c w x t R s
− −
−
Ω ,δ
, + + =: ,δ
δ∫ ∫∫ ∫∫ , (3.5)
where 1 0 0s R T∈ , δ > , > . Owing to (1.25), we have
( )
( 0)d 0 0
s
w x x s
Ω
, ≡ ∀ ≥∫ . (3.6)
We introduce the functions related to ( )w x t, :
( 1)
1
( ) ( )
( ) d d ( ) d d
p n m
p
T T
n
T T
Q s Q s
A s w x t B s w x t
− −
−:= , :=∫∫ ∫∫ .
Applying the interpolation inequality of Lemma A.2 in the domain ( )sΩ +δ to the
function
m p
pv w
+
= for n p pa d p bm p m p= , = , =
+ +
, 0 1i j= , = , and integrating
the result with respect to time from 0 to T , we obtain
11 11( ) ( )k
T TA s cT R s+β−+ δ ≤ ,δ , (3.7)
where 1 1
( 1) ( 1)1( 1) ( 1)
N n p nk p N m p p N m p
− −= < , β =
+ + − + + −
, ( )11m n p N> − + . Simi-
larly, applying the interpolation inequality of Lemma A.2 in the domain ( )sΩ +δ
to the function
m p
pv w
+
= for ( )( 1)
( 1)( )
p p n m pa d p bp m p m p
− −
= , = , =
− + +
, 0 1i j= , = ,
and integrating the result with respect to time, we find that
22 11( ) ( )k
T TB s cT R s+β−+ δ ≤ ,δ , (3.8)
Физика и техника высоких давлений 2009, том 19, № 4
52
where ( )
( )2
( 2) 1
1
( 1) ( 1)
N p n m
k
p p N m p
− − +
= <
− + + −
, ( )
( )2
( 2) 1
( 1) ( 1)
p p n m
p p N m p
− − +
β =
− + + −
, m >
( )( 1) 111
p n pp N
−> − +
−
. Next we define the function
2 11 1( ) ( ( )) ( ( ))T T TC s A s B s+β +β:= + .
Then
1 21 1( ) ( )[ ( ) ( )]T T TC s c F T C s C s+β +β−β+ δ ≤ δ + , (3.9)
where
{ }1 2 2 1(1 )(1 ) (1 )(1 )
1 2(1 )(1 ) ( ) max k kF T T T− +β − +ββ = +β +β , = , .
Below, we find some estimate 1L -norm of ( )w x t, by 1L -norm of ( 0)w x, , which
will be used in the next consideration.
Lemma 3.1. There exists some constant 0c > , depending on known parameters
of the problem, such that the following estimate
1( )d ( 0)d
N NR R
w x t x c w x x t T, ≤ , ∀ ≤∫ ∫ , (3.10)
is valid. Here 1T depends on m p n N, , , and 1( )( 0) NL Rw x, .
Proof. We set 2 0s s′= − δ, δ = > in (3.5) and pass to the limit as s′ → ∞
(0 )
1sup ( )d ( )d d d d
m p
p
N
T T
p
t T Q QR
w x t x w x t x t c w x tT
+
∈ ,
, + , + | ∇ |∫ ∫∫ ∫∫ ≤
≤
( 1)
1( 0)d d d
p n m
p
N
TQR
w x x w x t
− −
−, +∫ ∫∫ . (3.11)
Applying the interpolation inequality of Lemma A.2 in NR to the function
m p
pv w
+
= for ( )( 1)
( )( 1)
p p n m pa d p bm p p m p
− −
= , = , =
+ − +
, 0 1i j= , = , and Young’s
inequality, we find that
(1 )
( 1)
1 d d d
aa
p bp n m m p
p p
N N N
p
R R R
w x c w x w x
−θθ
− − +
−
⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟≤ | ∇ |
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∫ ∫ ∫ ≤
≤
(1 )
( )
d ( ) d 0
ap
b p am p
p
N N
p
R R
w x c w x
−θ
− θ+ ⎛ ⎞
⎜ ⎟ε | ∇ | + ε ∀ε >
⎜ ⎟
⎝ ⎠
∫ ∫ ,
Физика и техника высоких давлений 2009, том 19, № 4
53
where ( )
( )( )
( ) ( 2) 1
( 1) ( 1)
N m n p n m
N m p p p n m
+ − − +
θ =
+ − + − −
. Integrating this inequality with re-
spect to time from 0 to T , we obtain
(1 )
( )( 1)
1
0
d d d ( ) d d
ap
b p ap n m m p
p p
N
T T
T
p
Q Q R
w x t w x c w x t
−θ
− θ− − +
−
⎛ ⎞
⎜ ⎟≤ ε | ∇ | + ε
⎜ ⎟
⎝ ⎠
∫∫ ∫∫ ∫ ∫ . (3.12)
Choosing 0ε > sufficiently small, from (3.11), (3.12) we have
(0 )
1sup ( )d ( )d d d d
m p
p
N
T T
p
t T Q QR
w x t x w x t x t c w x tT
+
∈ ,
, + , + | ∇ |∫ ∫∫ ∫∫ ≤
≤
(1 )
( )
0
( 0)d d d
ap
b p a
N N
T
R R
w x x c w x t
−θ
− θ⎛ ⎞
⎜ ⎟, +
⎜ ⎟
⎝ ⎠
∫ ∫ ∫ . (3.13)
From the last inequality we deduce that for every 0t t T: < < the following ine-
quality is valid
0
( )d ( 0)d ( )d d
N N N
t
R R R
w x t x w x x c w x x
γ
⎛ ⎞
⎜ ⎟, ≤ , + , τ τ
⎜ ⎟
⎝ ⎠
∫ ∫ ∫ ∫ ,
where ( )
( )
( 1) ( 1) ( 1)( )
1 ( )
N p n m N p m p
p p N m p n
− − − + − +
γ =
− + + −
. Applying Lemma A.3 from
Appendix A we obtain (3.10) with
1
1 1
2 ( 0)d if 11
1 ( 0)d if 12( 1)
N
N
R
R
w x x
T
w x x
−γ
γ−
⎧ ⎛ ⎞⎪ ⎜ ⎟, γ < ,⎪ − γ ⎜ ⎟
⎪ ⎝ ⎠:= ⎨
⎪ ⎛ ⎞
⎪ ⎜ ⎟, γ > ,
γ − ⎜ ⎟⎪ ⎝ ⎠⎩
∫
∫
(3.14)
and 1 0T → as
1( )( 0) 0NL Rw x, → .
Further, using the definition of the functions ( )TC s and (3.10), we get
0 0 1( ) ( ) ,TC s K F T T T≤ ∀ ≤ (3.15)
where the positive constant 0K depends on n m p N, , , and 1( )( 0) NL Rw x, .
Now we choose the parameter 0δ > which was arbitrary up to now:
1
1
0
2( ) ( ) ( )
1 ( )T T
T
cs F T C s
H s
β
β⎡ ⎤
δ := ⎢ ⎥−⎣ ⎦
,
Физика и техника высоких давлений 2009, том 19, № 4
54
where the function 2( ) ( ) ( )T TH s cF T C sβ= is such that 0( ) 1TH s < at some point
0 0s ≥ , whence we obtain
2 22(1 )(1 ) (1 )(1 )(1 )1 2 2 1 2
2 0 0min k kT T c K K
β β
− +β − +β +β
− −⎧ ⎫⎪ ⎪≤ = , ,⎨ ⎬
⎪ ⎪⎩ ⎭
(3.16)
and 2T →∞ as
1( )( 0) 0NL Rw x, → .
We obtain the following main functional relation for the function ( )T sδ :
1
0
0
1 ( )( ( )) ( ) 0 0 12
T
T T T
H ss s s s s
β
β+⎛ ⎞δ + δ ≤ εδ ∀ ≥ ≥ , < ε = <⎜ ⎟
⎝ ⎠
(3.17)
{ }1 20 minT T T T∗∀ < < := , , with 1T of (3.14) and 2T of (3.16). Now we apply
Lemma A.1 to the function ( )T sδ of (3.17). As a result, we obtain
0 0
1( ) 0 ( )1T Ts s s sδ ≡ ∀ ≥ + δ
− ε
. (3.18)
Then, in view of (3.15), we find
( )
(1 )(1 )2 11 1 1
11 1 1 21 2
1 1
0 0( ) ( ) ( ) [ ( )] ( ) max
k
k
T Ts c C s F T c F T c F T c T T
− +β
+ββ β +ββ +β −⎧ ⎫⎪ ⎪⎡ ⎤δ ≤ ≤ ≤ = ,⎨ ⎬⎣ ⎦ ⎪ ⎪⎩ ⎭
0 T T ∗∀ < < . Choosing in (3.18) 0 0s = and
( )
( 1)
( )
( 1)
for 1
( ) max
for 1
p N m p n
p N m p
p N m p n
p N m p
T T
s T c T T c
T T
+ + −
+ + −
+ + −
+ + −
κ
κ
⎧ < ,⎧ ⎫⎪ ⎪ ⎪= Γ = , =⎨ ⎬ ⎨
⎪ ⎪⎩ ⎭ ⎪ >⎩
0 T T ∗∀ < < , ( )[ ]
( )[ ]
1 ( ) ( 1)
( 1) ( ( 1) ) ( 1)( 1)
p p N m p n np N m p
p N m p p p n m N p m p
− + + − + + −
κ =
+ + − − − + − + −
. Thus
( ) 0w T x, ≡ for all { ( ) ( )}N
N Nx x x x R x t′∈ = , ∈ : ≥ Γ . And Theorem 1 is proved
completely.
Roman Taranets acknowledges financial support from the INTAS under the
project Ref. No: 05-1000008-7921.
Appendix A
Lemma A.1. [ 22] Let the nonnegative continuous nonincreasing function f(s):
[s 1
0 ) R,∞ → satisfy the following functional relation:
( ) 0( ) ( ) 0 1f s f s f s s s+ ≤ ε ∀ ≥ , < ε < .
Физика и техника высоких давлений 2009, том 19, № 4
55
Then 1
0 0( ) 0 (1 ) ( )f s s s f s−≡ ∀ ≥ + − ε .
Lemma A.2. [18] If NRΩ⊂ is a bounded domain with piecewise-smooth
boundary, 1a > , (0 ) 1b a d∈ , , > , and 0 i j i j N≤ < , , ∈ , then there exist positive
constants 1d and 2d 2( 0d = if the domain Ω is unbounded ) that depend only on
d j bΩ, , , , and N and are such that, for any function ( ) ( ) ( )j b
dv x W L∈ Ω ∩ Ω , the
following inequality is true:
1
1 2( ) ( )( ) ( )
b ba d
i j
L LL L
D v d D v v d v
θ −θ
Ω ΩΩ Ω
≤ +
where
1 1
1
1 1
i
b N a i
j j
b N d
+ − ⎡ ⎞θ = ∈ , ⎟⎢⎣ ⎠+ −
.
Lemma A.3. [1] Suppose that v(t) is a nonnegative summable function on
[0,T] that, for almost all t [0 ]T∈ , , satisfies the integral inequality
( )
0
( ) ( ) ( ) d
t
v t k m h g v≤ + τ τ τ∫ ,
where 0 0k m≥ , ≥ , ( )h τ is summable on [0 ]T, , and ( )g τ is a positive function
for 0τ > . Then
1
0
( ) ( ) ( )d
t
v t G G k m h−
⎛ ⎞
≤ ⎜ + τ τ⎟
⎜ ⎟
⎝ ⎠
∫
for almost all [0 ]t T∈ , . Here
0
0
d( ) 0( )
v
v
G v v vg
τ= , > >
τ∫ .
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Ю.В. Намлєєва, Р.М. Таранець, В.М. Юрченко
СКІНЧЕНА ШВИДКІСТЬ ПОШИРЕННЯ ЗБУРЕНЬ
ЕЛЕКТРОМАГНІТНОГО ПОЛЯ У НЕЛІНІЙНИХ ІЗОТРОПНИХ
ДИСПЕРСНИХ СЕРЕДОВИЩАХ
Запропоновано модифікацію рівнянь Максвела, що описує середовища, в яких
електричні та магнітні властивості суттєво змінюються під впливом зовнішнього
електромагнітного поля. Для таких середовищ встановлено, що електромагнітні
хвилі розповсюджуються зі скінченою швидкістю на протязі часу, який, в свою
чергу, залежить від початкової енергії електромагнітного поля та нелінійних
параметрів середовища.
Ключові слова: рівняння Максвела, нелінійне дисперсне середовище, скінчена
швидкість поширення збурень, асимптотична поведінка
Ю.В. Намлеева, Р.М. Таранец, В.М. Юрченко
КОНЕЧНАЯ СКОРОСТЬ РАСПРОСТРАНЕНИЯ ВОЗМУЩЕНИЙ
ЭЛЕКТРОМАГНИТНОГО ПОЛЯ В НЕЛИНЕЙНЫХ ИЗОТРОПНЫХ
ДИСПЕРСИОННЫХ СРЕДАХ
Предложена модификация уравнений Максвелла, описывающая среды, в которых
электрические и магнитные свойства существенно изменяются под воздействием
внешнего электромагнитного поля. Для таких сред установлено, что электромаг-
нитные волны распространяются с конечной скоростью в течение времени, зависящего
от начальной энергии электромагнитного поля и нелинейных параметров среды.
Ключевые слова: уравнения Максвелла, нелинейная дисперсная среда, конечная
скорость распространения возмущений, асимптотическое поведение
|