Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations
We construct the global in time Mandelbrot transformation relating the solution of the transport equation at nonsingular points for the tunnel in the sense of Maslov LPDE to the global generalized deltashock wave type solution to the continuity equation in the discontinuous velocity field.
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| Cite this: | Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations / V.G. Danilov // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 46-58. — Бібліогр.: 10 назв. — англ. |
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Danilov, V.G. 2017-09-23T16:48:53Z 2017-09-23T16:48:53Z 2008 Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations / V.G. Danilov // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 46-58. — Бібліогр.: 10 назв. — англ. 1810-3200 2000 MSC. 35C20, 35D05, 35C10 https://nasplib.isofts.kiev.ua/handle/123456789/124296 We construct the global in time Mandelbrot transformation relating the solution of the transport equation at nonsingular points for the tunnel in the sense of Maslov LPDE to the global generalized deltashock wave type solution to the continuity equation in the discontinuous velocity field. This work was supported by the Russian Foundation for Basic Research under grant 05-01-00912 and DFG project 436 RUS 13/895/0-1. en Інститут прикладної математики і механіки НАН України Український математичний вісник Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations Article published earlier |
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Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations |
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Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations Danilov, V.G. |
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Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations |
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Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations |
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global mandelbrot transformation for maslov tunnel pseudodifferential equations |
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We construct the global in time Mandelbrot transformation relating the solution of the transport equation at nonsingular points for the tunnel in the sense of Maslov LPDE to the global generalized deltashock wave type solution to the continuity equation in the discontinuous velocity field.
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Global Mandelbrot transformation for Maslov tunnel pseudodifferential equations / V.G. Danilov // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 46-58. — Бібліогр.: 10 назв. — англ. |
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Український математичний вiсник
Том 5 (2008), № 1, 46 – 58
Global Mandelbrot transformation for Maslov
tunnel pseudodifferential equations
Vladimir G. Danilov
(Presented by E. Ya. Khruslov)
Abstract. We construct the global in time Mandelbrot transformation
relating the solution of the transport equation at nonsingular points for
the tunnel in the sense of Maslov LPDE to the global generalized delta-
shock wave type solution to the continuity equation in the discontinuous
velocity field.
2000 MSC. 35C20, 35D05, 35C10.
Key words and phrases. Kolmogorov–Feller equation, Maslov canon-
ical operator, global asymptotic solution, Mandelbrott transformation,
delta-shock.
1. Introduction
The goal of the present paper is to present a new approach to con-
structing singular (i.e., containing the Dirac δ-function as a summand)
solutions to the continuity equation and to show how these results can
be used to construct the global in time solution of the Cauchy problem
for Kolmogorov–Feller-type equations.
In general, the relation between the solutions of the continuity equa-
tion and the system consisting of the Hamilton–Jacobi equation plus the
transport equation is well known. The velocity field u is determined as
the velocities of points on the projections of the trajectories of the Hamil-
tonian system corresponding to the Hamilton–Jacobi equation. In this
velocity field, as it was mentioned by B. Mandelbrot, the squared solution
of the transport equation satisfies the continuity equation
ρt + 〈∇, uρ〉 + aρ = 0 (1.1)
Received 04.12.2007
This work was supported by the Russian Foundation for Basic Research under grant
05-01-00912 and DFG project 436 RUS 13/895/0-1.
ISSN 1810 – 3200. c© Iнститут математики НАН України
V. G. Danilov 47
with some additional term aρ, which is defined below.
But this relation between the transport and continuity equations is
known only in the domain where the action function is sufficiently smooth.
We generalize this relation to the case in which the singular support of
the velocity field is a stratified manifold transversal to the velocity field.
This holds, for example, in the one-dimensional case under the condition
that, for any t ∈ [0, T ], the singularity support is a discrete set without
limit points.
2. Generalized solutions of the continuity equation
Here we follow the approach developed in [1], where the solution is
understood in the sense of integral identity, which, in turn, follows from
the fact that relation (1.1) is understood in the sense of D(Rn+1
x,t ).
We specially note that the integral identities in [1] can be derived
without using the construction of nonconservative product [2, 4] (or the
measure solutions [5]), and the value of the velocity on the discontinuity
lines (surfaces) is not given a priori but is calculated. Of course, in the
case considered in [1], the integral identities exactly coincide in form with
the identities derived using the construction of nonconservative product
(measure solutions) under the above assumptions.
First, we consider an n−1-dimensional surface γt moving in R
n
x, which
is determined by the equation
γt = {x; t = ψ(x)},
where ψ(x) ∈ C1(Rn), and ∇ψ 6= 0 in the domain in R
n
x where we work.
This is equivalent to determining a surface by an equation of the form
S(x, t) = 0
(S(x, t) ∈ C1, S(x, t) = 0, ∇x,tS|S=0 6= 0) under the condition that
∂S
∂t
6= 0.
But if ∂S
∂t = 0, then we can make the change x′i = xi − cit, solve the
problem with moving surface (it will be considered with appropriately
chosen c1, c2, . . . ), and then return to the original variables. Possible
generalizations are considered later.
Next, we assume that ζ(x, t) ∈ C∞
0 (Rn × R
1
+). Then, by definition,
〈δ(t− ψ(x)), ζ(x, t)〉 =
∫
Rn
ζ(x, ψ(x)) dx,
48 Global Mandelbrot transformation...
where δ(z) is the Dirac delta function.
Let δ(t− ψ(x)) be applied to the test function η(x) ∈ C∞
0 (Rn), then
〈δ(t− ψ(x)), η(x)〉 =
∫
γt
η(x) dωψ,
where dω is the Leray form [6] on the surface {t = ψ(x)} such that
−dψdωψ = dx1 . . . dxn.
One can show that (see [1, 6])
〈δ(t− ψ(x)), η(x)〉 =
∫
γt
η(x)
|∇ψ| dσ.
First, we assume that the solution ρ to Eq. (1.1) has the form
ρ = R(x, t) + e(x)δ(t− ψ(x)), (2.1)
where R(x, t) is a piecewise smooth function with possible discontinuity
at {t = ψ(x)}:
R = R0(x, t) +H(t− ψ)R1(x, t),
e(x) ∈ C(Rn) and has a compact support, ψ(x) ∈ C2 and ∇ψ 6= 0 for
x ∈ supp e(x), and H(z) is the Heaviside function.
It is clear that the term
e(x)δ′(t− ψ(x))
appears in (1.1) if we differentiate the δ(t− ψ) with respect to t. Hence
it is necessary to have in (1.1)
〈∇, ρu〉 = −e(x)δ′(t− ψ) + more smooth summands,
since ∇δ(t− ψ) = −∇ψδ′(t− ψ). Then we must have
ρu =
e∇ψ
|∇ψ|2 δ(t− ψ) + more smooth summands.
Now we formulate an integral identity, defining a generalized solution.
We denote Γt = {(x, t); t = ψ(x)}; this is an n-dimensional surface in
R
n × R
1
+. Let
u(x, t) = u0(x, t) +H(t− ψ)u1(x, t),
where ψ is the same function as previously, and u0, u1 ∈ C(Rn × R
1
+).
V. G. Danilov 49
Let us consider Eq. (1.1) in the sense of distributions. For all ζ(x, t) ∈
C∞
0 (Rn × R
1
+), ζ(x, 0) = 0, we have
〈∂p
∂t
+ 〈∇, ρu〉, ζ
〉
= −〈ρ, ζt〉 − 〈ρu,∇ζ〉.
Substituting the singular terms for ρ and ρu calculated above, we come
to the following definition.
Definition 2.1. A function ρ(x, t) determined by relation (2.1) is called
a generalized δ-shock wave type solution to (1.1) on the surface {t =
ψ(x)} if the integral identity holds
∞
∫
0
∫
Rn
(Rζt + (uR,∇ζ) + aRζ) dx dt+
∫
Γt
e
|∇ψ|
d
dn⊥
ζ(x, t) dx = 0 (2.2)
for all test functions ζ(x, t) ∈ D(Rn×R
1
+), ζ(x, 0) = 0, d
dn⊥
=
( ∇ψ
|∇ψ| ,∇
)
+
|∇ψ| ∂∂t .
We have the relation
∫
Rn
e
|∇ψ|
d
dn
ζ(x, ψ) dx =
∫
Γt
e
|∇ψ|
d
dn⊥
ζ(x, t) dx,
where
d
dn⊥
=
( ∇ψ
|∇ψ| ,∇
)
+ |∇ψ| ∂
∂t
.
We note that the vector n⊥ is orthogonal to the vector (∇ψ,−1),
which is the normal on the surface Γt, i.e., d
dn⊥
lies in the plane tangent
to Γt.
We can give a geometric definition of the field d
dn⊥
. The trajectories
of this vector field are curves lying on the surface Γt, and they are or-
thogonal to all sections of this surface produced by the planes t = const.
Furthermore, it is clear that the expression 1
|∇ψ| is an absolute value of
the normal velocity of a point on γt, i.e., on the cross-section of Γt by
the plane t = const, and the expression 1
|∇ψ| ·
∇ψ
|∇ψ|
def
= ~Vn is the vector of
normal velocity of a point on γt. Thus, we have another representation:
∫
Γt
e
|∇ψ|
d
dn⊥
ζ(x, t) dx =
∫
Γt
e
(
(~Vn,∇) +
∂
∂t
)
ζ(x, t) dx,
where Vn = π∗(vn), vn is the normal velocity of a point on γt, and π∗ is
induced by the projection mapping π : Γt → Rnx .
50 Global Mandelbrot transformation...
It follows from the last definition that the following relations must
hold:
Rt + (∇, Ru) + aRζ = 0, (x, t) 6∈ Γt,
([R] − |∇ψ|[Run]) +
( d
dn
)∗ e
|∇ψ| = 0, (x, t) ∈ Γt,
The last relation can be rewritten in the form
KE +
d
dn
E = [Run]|∇ψ| − [R], (2.3)
where E = e/|∇ψ|, the factor K = (∇, ∇ψ
|∇ψ|) = ÷ν (ν is the normal on
the surface {t = ψ(x)}) and, as is known, is the mean curvature of the
cross-section of the surface Γt by the plane t = const, d
dn = ( ∇ψ
|∇ψ| ,∇).
Now we assume that there are two surfaces
Γ
(i)
t = {(x, t); t = ψi(x)}
in R
n × R
1
+, i = 1, 2, whose intersection is a smooth surface
γ̂ = {(x, t); (t = ψ1) ∩ (t = ψ2)}
belonging to the third surface Γ
(3)
t = {(x, t); t = ψ3(x)}. Further, we
assume that the surface Γ
(3)
t is a continuation of the surfaces Γ(i) in the
following sense. We let n
(i)
⊥ denote the curves on the surfaces Γ
(i)
t and
assume that each point (x̂, t̂) on the surface γ̂ is assigned the graph con-
sisting of the trajectories n
(1)
⊥ and n
(2)
⊥ entering (x̂, t̂) and the trajectory
n
(3)
⊥ leaving this point (i.e., the trajectories n
(i)
⊥ fiber the surface Γ(i)).
We also assume that the surface Γ∪ = Γ(1) ∪Γ(2) ∪Γ(3) consists of points
belonging to these graphs. Next, we assume that u(x, t) is a piecewise
smooth vector field whose trajectories come to Γ∪.
Definition 2.2. Let
u(x, t) = u0(x, t) +
3
∑
i=1
H(t− ψi)u1i(x, t),
where ψ is the same function as previously, and u0, u1i ∈ C(Rn × R
1
+).
The function ρ(x, t) determined by relation
ρ(x, t) = R(x, t) +
3
∑
i=1
ei(x)δ(t− ψi(x)),
V. G. Danilov 51
where R(x, t) ∈ C1(Rn × R
1
+) \ {⋃ Γ
(i)
t }, is called a generalized δ-shock
wave type solution to (2.2) on the graph Γ∪ if the integral identity
∞
∫
0
∫
Rn
(Rζt + (uR,∇ζ) + aRζ) dx dt
+
3
∑
i=1
∫
Γ
(i)
t
ei
|∇ψi|
d
dn
(i)
⊥
ζ(x, t) dx = 0 (2.4)
holds for all test functions ζ(x, t) ∈ D(Rn × R
1
+), ζ(x, 0) = 0, d
dn
(i)
⊥
=
( ∇ψi
|∇ψi|
,∇
)
+ |∇ψi| ∂∂t .
Now we consider the case codim Γt > 1.
First, we note that the second integral in (2.2) can be written as
∫
Γt
e
|∇ψ|
d
dn⊥
ζ(x, t) dx =
∫
Γt
e
(( ∇ψ
|∇ψ|2 ,∇
)
+
∂
∂t
)
ζ(x, t) dx.
We note that if the surface Γt is determined by the equation S(x, t) = 0
rather than by a simpler equation presented at the beginning of this
section, then
~Vn = − St
|∇S| ·
∇S
|∇S| = − St
|∇S|2∇S
and, of course, the new vector field d
dn⊥
= (~Vn,∇) + ∂
∂t remains tangent
to Γt.
Therefore, in this more general case, using this new vector ~Vn, we can
again rewrite the integral identity from Definition 1.1 as
∞
∫
0
∫
Rn
(
Rζt + (uR,∇ζ) + aRζ
)
dxdt
+
∫
Γt
e
(
(~Vn,∇) +
∂
∂t
)
ζ(x, t) dx = 0. (2.5)
This form of integral identity can easily be generalized to the case in
which Γt is a smooth surface in R
n+1 of codimension > 1.
In this case, instead of ~Vn, we can use a vector ~v such that it is
transversal to Γt and the field (~v,∇) + ∂
∂t is tangent to Γt. We note
that the vector ~v is uniquely determined by this condition, which can
52 Global Mandelbrot transformation...
be treated as “the calculation of the velocity value on the discontinuity”
from the viewpoint of [5] and [7].
Moreover, in this case, the expression for ρ does not contain the Heav-
iside function, and it is assumed that the trajectories of the field u are
smooth, nonsingular outside Γt, and transversal to Γt at each point of Γt.
In this case, the function ρ has the form
ρ = R(x, t) + eδ(Γt),
where R(x, t) ∈ C1(Rn+1 \ Γt), e ∈ C1(Γt), and the delta function is
determined as
〈δ(Γt), ζ(x, t)〉 =
∫
Γt
ζω,
where ω is the Leray form on Γt. If Γt = {S1(x, t) = 0∩· · ·∩Sk(x, t) = 0},
k ∈ [1, n], then ω is determined by the relation, see [6, p. 274],
dt dx1 · · · d xn = dS1 · · · dSkω.
In this case, we assume that the functions Sk are sufficiently smooth
(for example, C2(Rn × R
1
+)) and their differentials on Γt are linearly
independent.
Moreover, we can assume that the inequality
J =
D(S1, . . . , Sn)
D(t, x1, . . . , xn−1)
6= 0
holds. This inequality is an analog of St 6= 0 at the beginning of this
section and allows us to write ω in the form
ω = J−1dxk · · · dxn.
The integral identity, an analog of (2.5), has the form
∞
∫
0
∫
Rn
(
Rζt + (uR,∇ζ) + aRζ
)
dxdt+
∫
Γt
e
(
(v,∇) +
∂
∂t
)
ζ(x, t)ω = 0.
Integrating the last relation by parts, we obtain equations for determining
the functions e and R similarly to (2.3).
Now we assume that the singular support of the velocity field is the
stratified manifold
⋃
Γit with smooth strata Γit of codimensions ni ≥ 1.
We also assume that the velocity field trajectories are transversal to
⋃
Γt and are entering trajectories.
V. G. Danilov 53
Then the general solution of Eq. (1.1) has the form
ρ = R(x, t) +
∑
eiδ(Γit),
where R(x, t) is a function smooth outside
⋃
Γit, ei are functions defined
on the strata Γit, and the sum is taken over all strata.
The integral identities determining such a generalized solution have
the form
∞
∫
0
∫
Rn
(Rζt + (uR,∇ζ) + aRζ) dxdt
+
∑
∫
Γit
ei
[
(
(vi,∇) +
∂
∂t
)
ζ(x, t)
]
ωi = 0. (2.6)
This implies that, outside
⋃
Γit, the function R satisfies the continuity
equation
Rt + 〈∇, uR〉 + aR = 0,
and, on the strata Γjt for nj = 1, equations of the form (2.3) hold,
which contain the values of R brought to Γjt along the trajectories. For
nl = n− k, k > 1, on the strata Γlt, we have the equations
∂
∂t
elµl + (∇, vlelµl) = 0, (2.7)
where µl is the density of the measure ωl with respect to the measure on
Γlt left-invariant with respect to the field ∂
∂t + 〈vl,∇〉. We note that it
follows from the above that the function R is determined independently of
the values of vi on the strata under the condition that the field trajectories
enter
⋃
Γit.
In conclusion, we consider the case where the coefficient a has a sin-
gular support on
⋃
Γit, i.e.,
a = f(u).
In this case, we set
aρ = ǎρ+
∑
f(vi)eiδ(Γit).
We note that such a choice of the definition of the term aρ in not unique in
this case. But, first, it is consistent with the common concept of measure
solutions and, second, it is of no importance for the construction of the
solution outside
⋃
Γit for the case in which the trajectories u enter
⋃
Γit.
54 Global Mandelbrot transformation...
In this case, identity (2.6) takes the form
∞
∫
0
∫
Rn
(
Rζt + (uR,∇ζ) + f(u)Rζ
)
dxdt
+
∑
∫
Γit
ei
[
(
(vi,∇) +
∂
∂t
+ f(vi)
)
ζ(x, t)
]
ωi = 0, (2.8)
and Eq. (2.8) can be rewritten in the form
∂
∂t
(elµl) + (∇, vlelµl) + f(vl) = 0.
3. The Maslov tunnel asymptotics
We recall that the asymptotic solutions of the Cauchy problem for
an equation with pure imaginary characteristics was first constructed by
V. P. Maslov [8]. In the present paper, we consider only the following
Cauchy problem
−h∂u
∂t
+ P
(
2
x,−h
1
∂
∂t
)
u = 0, u|t=0 = e−S0(x)/hϕ0(x), (3.1)
where P (x, ξ) is the (smooth) symbol of the Kolmogorov–Feller operator
[9], S0(x) ≥ 0 is a smooth function, ϕ0(x) ∈ C∞
0 , h → +0 is a small
parameter characterizing the frequency and the amplitude of jumps of
the corresponding random process.
It is clear [8] that, locally in t, the solution of problem (3.1) is con-
structed according to the scheme of the WKB method: the solution is
constructed in the form
u = eS(x,t)(ϕ0(x, t) + hϕ1(x, t) + · · · ),
in this case, for the functions S(x, t) and ϕi(x, t) (we consider only the
case i = 0) we obtain the following problems:
∂S
∂t
+ P
(
x,
∂S
∂x
)
= 0, S|t=0 = S0(x), (3.2)
∂ϕ0
∂t
+
(
∇P
(
x,
∂S
∂x
)
,∇ϕ0
)
+
∑
ij
∂2P
∂xi∂xj
∂2S
∂xi∂xj
ϕ0 = 0,
ϕ0|t=0 = ϕ0(x),
(3.3)
V. G. Danilov 55
As is known, the solution of problem (3.2) is constructed using the
solutions of the Hamiltonian system
ẋ = ∇ξP (x, p), x|t=0 = x0, (3.4)
ẋ = −∇xP (x, p), p|t=0 = ∇S0(x0).
This solution is smooth on the support of ϕ0(x, t) until the Jacobian
Dx/Dx0 6= 0 for x0 ∈ suppϕ0(x). We let gtH denote the translation
mapping along the trajectories of the Hamiltonian system (3.4).
Recall that the plot
Λn0 = {x = x0, p = ∇S0(x0)}
is the initial Lagrangian manifold corresponding to Eq. (3.2), and Λnt =
gthΛ
n
0 is the Lagrangian manifold corresponding to Eq. (3.2) at time t.
Let π : Λnt → R
n
x be the projection of Λnt on R
n
x, which is assumed to be
proper. The point α ∈ Λnt is said to be essential if
Ŝ(α, t) = min
β∈π−1(α)
Ŝ(β, t)
and nonessential otherwise. Here Ŝ is the action on Λnt determined by
the formula
Ŝ(α, t) =
t
∫
0
p dx−H dt,
where the integral is calculated along the trajectories of system (3.4) the
projection of whose origin is x0 = α. As is known
S(x, t) = Ŝ(π−1x, t)
at nonessential points where the projection π is bijective.
The solution of problem (3.1) is given by the Maslov tunnel canonical
operator.
To define this operator, following [8, 10] we introduce the set of es-
sential points
⋃
γit ⊂ Λnt . This set is closed because the projection π is
proper.
Suppose that the open domains Uj ⊂ Λnt form a locally finite covering
of the set
⋃
γit. If the set Uj consists of nonessential points, then we set
uj = e−Sj(x,t)/hϕ0j(x, t) (3.5)
where
ϕ0j(x, t) = ψ0j(x, t)
(
det
Dx0
Dx
)1/2
,
56 Global Mandelbrot transformation...
where ψ0j(x, t) is the solution of the equation
∂ψ0j
∂t
+ (∇P (x,∇Sj),∇ψ0j) +
1
2
tr
∂2P
∂x∂ξ
(x,∇Sj)ψ0j = 0. (3.6)
The solution uj in the domain containing essential points (at which
dπ is degenerate) is given in the following way: the canonical change of
variables is performed so that the essential points become nonessential,
then we determine a fragment of the solution in new coordinates by for-
mula (3.5) and return to the old variables, applying the “quantum” inverse
canonical transformation to the solution obtain in the new coordinates.
The Hamiltonian determining this canonical transformation has the
form
Hσ =
1
2
n
∑
i=1
σkp
2
k,
where σ1, . . . , σn = const > 0.
The canonical transformation to the new variables is given by the
translation by the time −1 along the trajectories of the Hamiltonian Hσ.
One can prove that the set of sets σ for which the change of variables
takes a essential point into a nonessential is not empty.
Next, the solution near the essential point is determined by the rela-
tion
uj = e
1
h
Ĥσ ũj , (3.7)
where ũj is given by formula (3.5) in the new variables and
Ĥσ =
1
2
n
∑
k=1
σk
(
− h
∂
∂xk
)2
.
On the intersections of singular and nonsingular charts, we must
match Sj and ψ0j . This can be done by applying the Laplace method
to the integral (whose kernel is a fundamental solution for the operator
−h ∂
∂t+Ĥσ) in the right-hand side of (3.7). In this case, since the solution
is real, the Maslov index [8] well-known in hyperbolic problems does not
appear. The complete representation of the solution of problem (3.1) is
obtained by summing functions of the type (3.5) and (3.6) over all the
domains Uj , for more detail, see [8, 10].
The asymptotics thus constructed is justified, i.e., the proximity be-
tween the exact and asymptotic solutions of the Cauchy problem (3.1) is
proved [8, 9].
Precisely as in the preceding case where the solution of the continu-
ity equation at nonessential points was independent of the values of the
solution on the singularity support (of course, the inverse influence takes
V. G. Danilov 57
place), in the case of the canonical operator, the relation between the
solutions at essential and nonessential point is also unilateral, namely,
the essential points are “bypassed” using (3.7), but the values of the func-
tions ψ̃oj contained in ũj on the singularity support, do not determine
the values at the regular points (but the converse is not true).
Now we note that the function S(x, t) such that
S(x, t)|Uj
= Sj(π
−1(α), t)
is globally determined and continuous at points of the domain π(
⋃
γit) ⊂
R
n
x. We denote this set by
⋃
Γit and assume that this is a stratified
manifold with smooth strata Γit of different codimensions. We note that,
for example, if the inequality ∇(Si(x, t)−Sj(x, t)) 6= 0 holds while we pass
from one branch Λnt ∩ ⋃
γit to another, then the set π{(S̃i − S̃j) = 0}
generates a smooth stratum of codimension 1. In the one-dimensional
case, all strata are points or curves (under the above assumptions about
the singularities are discrete).
Now we consider the equation for ψ2
0j . We denote this function by ρ
and then obtain
∂ρ
∂t
+ (∇, uρ) + aρ = 0, (3.8)
where u = ∇xiP (x,∇S) and a = tr ∂2P
∂x∂ξ (x,∇S).
If the condition
HessP (x, ξ) > 0
is satisfied, then it follows from the implicit function theorem that ∇S =
F (x, u), where F (x, u) is a smooth function and
a = f(x, u),
where f(x, z) is again a smooth function.
Thus, we have proved the following theorem.
Theorem 3.1. Suppose that the following conditions are satisfied for
t ∈ [0, T ].
(1) There exists a smooth solution of the Hamiltonian system (3.4).
(2) The singularities of the velocity field u = ∇P (x,∇S) form a strat-
ified manifold with smooth strata and HessP (x, ξ) > 0.
(3) There exists a generalized solution ρ of the Cauchy problem for
Eq. (3.8) in the sense of the integral identity (2.8).
58 Global Mandelbrot transformation...
Then at the points of Γ ⊂ Λnt , the asymptotic solution of the Cauchy
problem (3.1) has the form
u = exp(−S(x, t)/h)
√
ρ.
This theorem is a global in time analog of the corresponding Mandel-
brot statement.
References
[1] V. G. Danilov, On singularities of continuity equations // Nonlinear Analysis
(2007), doi:10.1016/j.na/2006.12.044
[2] P. G. Le Floch, An existence and uniqueness result for two nonstrictly hyperbolic
systems in Nonlinear Evolution Equations that Change Type, Springer, Berlin,
1990, pp. 126–138.
[3] G. Dal Maso, P. G. Le Floch, and F. Murat, Definition and weak stability of
nonconservative products // J. Math. Pures Appl. 74 (1995), 483–548.
[4] A. I. Volpert, The space BV and quasilinear equations // Math. USSR Sb. 2
(1967), 225–267.
[5] Hanchun Yang, Riemann problem for a class of coupled hyperbolic systems of
conservation laws // J. Diff. Equations 159 (1999), 447–484.
[6] I. M. Gelfand and G. E. Shilov, Generalized Functions. Academic Press, New
York, 1964, Vol. 1, (translated from the Russian).
[7] Wanchung Sheng and Tong Zhang, The Riemann problem for the transportation
equation in gas dynamics // Memories of AMS 137 (1999), N 64, 1–77.
[8] V. P. Maslov, Asymptotic Methods and Perturbation Theory. Nauka, Moscow,
1988.
[9] V. G. Danilov and S. M. Frolovitchev, Exact asymptotics of the density of the
transition probability for discontinuous Markov processes // Math. Nachrichten
215 (2000), N 1, 55–90.
[10] V. P. Maslov and V. E. Nazaikinskii, Tunnel canonical operator in thermodynam-
ics // Funktsional. Anal. i Prilozhen. 40 (2006) ,N 3, 12–29.
Contact information
Vladimir G.
Danilov
Moscow Technical University of
Communications and Informatics
109028, Moscow,
Russia
E-Mail: danilov@miem.edu.ru
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