On the ne structure and hierarchy of gradient catastrophes for multidimensional homogeneous Euler equation B.G.Konopelchenko1and G.Ortenzi2

2025-05-02 0 0 1.97MB 22 页 10玖币
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On the fine structure and hierarchy of gradient catastrophes for
multidimensional homogeneous Euler equation
B.G.Konopelchenko 1and G.Ortenzi 2∗
1Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Universit`a del Salento, 73100 Lecce, Italy
2Dipartimento di Matematica e Applicazioni, Universit`a di Milano-Bicocca, via Cozzi 55, 20126 Milano, Italy
2INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
October 11, 2022
Abstract
Blow-ups of derivatives and gradient catastrophes for the n-dimensional homogeneous Euler equation are discussed. It is
shown that, in the case of generic initial data, the blow-ups exhibit a fine structure in accordance of the admissible ranks
of certain matrix generated by the initial data. Blow-ups form a hierarchy composed by n+ 1 levels with the strongest
singularity of derivatives given by ∂ui/∂xk∼ |δx|−(n+1)/(n+2) along certain critical directions. It is demonstrated that
in the multi-dimensional case there are certain bounded linear superposition of blow-up derivatives. Particular results for
the potential motion are presented too. Hodograph equations are basic tools of the analysis.
1 Introduction
Multidimensional quasi-linear partial differential equations and systems have been studied intensively during last decades.
The homogeneous n-dimensional Euler equation
ut+u· ∇u= 0 ,u:Rn→Rn(1.1)
is one of the most important representative of this class of equations. It is the basic equation in the theory of continuous
media in the case when effects of dissipation, pressure etc. are negligible (see e.g. [14, 13, 20]). Equation (1.1) arises in
various branches of physics from hydrodynamics to astrophysics [14, 13, 20, 21, 17].
A remarkable feature of the homogeneous Euler equation (HEE) is that it is solvable by the multidimensional version
of the classical hodograph method [21, 17, 4, 5, 6]. Namely, solutions of the HEE are provided by the vector hodograph
equation
xi−uit−fi(u) = 0 , i = 1,...,n (1.2)
where fi(u) are the local inverse of the initial data ui(x,0) for equation (1.1). The hodograph equation (1.2) is also a
powerful tool for the analysis of the singularities of solutions for HEE, in particular, of blow-ups of derivatives and gradient
catastrophes [21, 17, 4, 11, 10]. It was demonstrated that features and properties of blow-ups and gradient catastrophes
(GC) for the multidimensional HEE are quite different from those in the text-book one-dimensional examples [14, 20].
Existence or nonexistence of blow-ups in different dimensions, boundedness of certain linear superpositions of derivatives,
the case of potential flows have been discussed briefly recently in paper [10].
In this paper the detailed analysis of the blow-ups and GC for HEE (1.1) is presented. It is shown that the blow ups
of derivatives, which occur on the hypersurface Γ defined by the equation
det(M(t, u)) = 0 ,where Mil =tδil +∂fi
∂ul
, i, l = 1,...,n, (1.3)
form a hierarchy. Degree of singularities of derivatives and dimensions of subspaces in Γ at which they occur are different
for different members (levels) of such hierarchy.
Blow-up of the first level have a fine structure due to the different admissible ranks r1of the degenerate matrix M.
It is shown that there are r1-dimensional subspaces in the spaces of variation δukand δxisuch that the corresponding
derivatives ∂uk
∂xiare bound. The rest of the derivatives blow-up on Γ as
∂uk
∂xi∼−1/2, →0,(1.4)
∗Corresponding author. E-mail: giovanni.ortenzi@unimib.it, Phone: +39(0)264485725
1
arXiv:2210.03939v1 [nlin.SI] 8 Oct 2022
where is the distance from the catastrophe point in the most singular direction. Admissible values of the rank r1and the
dimensions dim Γr1of the corresponding subspaces in Γ are calculated for generic functions f(u), i.e. for generic initial
data u(x,0). For instance, for dimension n= 3, the admissible ranks r1are r1= 2,1 and correspondingly dim Γr1=2 = 3
and dim Γr1=1 = 0, while for n= 8, one has r1= 7,6,5 and dim Γr1= 8,5,0. On the other side, in the particular case of
potential flows, dim Γr1assumes others values. This case is analysed in the paper too.
It is shown that in the generic case the hierarchy of blow-ups has n+ 1 levels. On the m-th level there are also the
subsections of smaller dimensions where superpositions of derivatives are bounded while the blowing-up derivatives behave
as ∂uk
∂xi∼−m/(m+1) , →0.(1.5)
So, the most singular behavior of derivatives for HEE in the generic case is given by
∂uk
∂xi∼−(n+1)/(n+2) , →0.(1.6)
Non-generic and Poincar´e cases for initial data are discussed too. Characteristic features of blow-ups of derivatives for
the two- and three-dimensional HEE are considered in details. Some concrete cases are analyzed. The results presented
in this paper confirm an expected great difference between the properties of the one-dimensional and multi-dimensional
homogeneous Euler equations.
The paper is organized as follows. In Section 2 blow-ups of derivatives of the first level are analyzed. Admissible
ranks of the matrix M(t, u) and corresponding dim Γr1are calculated in Section 3. Non-generic and Poincar´e cases are
considered in Section 4. The situation with potential motion is discussed in section 5. Blow-ups of higher level and
hierarchy are described in Section 6 for the maximal rank case and in Section 7 for the lower rank case. Two-dimensional
and three-dimensional cases are described in Sections 8 and 9, respectively. An explicit 2D example has been carried out
in Section 10.
2 Generic blow-ups of derivatives: first level
The hodograph equations (1.2) will be our basic tool in the study of blow-ups of derivatives and gradient catastrophes for
the HEE (1.1).
Differentiating (1.2) with respect to xk, one obtains [6, 11, 10]
n
X
l=1
Mil
∂ul
∂xk
=δik ,with Mil =tδil +∂fi
∂ul
, i, k, l = 1,...,n, (2.1)
where δik is the Kronecker symbol. This relation implies that the derivatives ∂ul
∂xkand also ∂ul
∂t become unbounded at the
hypersurface Γ defined by the equation
det(M(t, u)) = 0 .(2.2)
The degeneracy of the matrix Mis the central point in the analysis of blow-ups. General properties of the hypersurface
(2.2) and some of their implications have been discussed in [10].
For deeper study of blow-ups one needs to consider the infinitesimal version of the hodograph mapping (1.2) around a
point u0on the blow-up hypersurface Γ (2.2) at fixed time t, i.e. with the expansion
δxi=tδui+X
k
∂fi
∂uk
(u0)δuk+1
2X
k,l
∂2fi
∂uk∂ul
(u0)δukδul+1
6X
k,l,m
∂3fi
∂uk∂ul∂um
(u0)δukδulδum+. . .
=X
k
Mik(u0)δuk+1
2X
k,l
∂2fi
∂uk∂ul
(u0)δukδul+1
6X
k,l,m
∂3fi
∂uk∂ul∂um
(u0)δukδulδum+. . . , I = 1,...,n.
(2.3)
For variable tone has the expansions (2.3) with the substitution δx→δ(x−u0t) in the l.h.s. after the subtraction of
the contribution from the infinitesimal Galilean transformation.
An important feature of the multi-dimensional case is the invariance of HEE (1.1) under the group SO(n) of rotations
in Rn. It is easy to see that in order to ensure that the variables (u1,...,un) are the component of the n-dimensional
vector uit is sufficient to require that the functions f(u) = (f1(u),...,fn(u)) in (1.2) are transformed as the component
of a vector. In such a case Mik are components of a tensor and the hodograph equations (1.2), condition (2.2) and (2.3)
are invariant to the group of rotations SO(n) too. Consequently, it would be preferable to formulate the basic properties
of blow-ups in a covariant form.
Another important peculiarity of the multi-dimensional HEE is that the terms linear in δudo not disappear in the
expansion (2.3) in contrast to the one-dimensional case. The coefficients Mik(u0) do not vanish on Γ, however they are
very special. They are elements of the degenerate n×nmatrix M, which is a key element for the subsequent analysis.
2
Let us assume that the rank of the matrix Mat the point u0∈Γ is equal to r1. It means that (see e.g. [7]) there exist
n−r1vectors R(α)(u0) and L(α)(u0) with α= 1,...,n−r1such that
n
X
k=1
Mik(u0)R(α)
k= 0 , i = 1, . . . , n , α = 1,...,n−r1,(2.4)
and n
X
i=1
L(α)
iMik(u0) = 0 , k = 1, . . . , n , α = 1,...,n−r1.(2.5)
It is well known (see e.g. [7]) that the requirements that the matrix Mhas rank r1and that there are n−r1linearly
independent vectors R(α)and L(α)satisfying the conditions (2.4) and (2.5) are equivalent. An advantage of conditions
(2.4) and (2.5) is that they are invariant under the rotation in Rn.
The properties (2.4) and (2.5) of the matrix M(u) have immediate consequences for the expansion (2.3). First, the
relations (2.4) imply that for the variations δuof the form
δ+uk=
n−r1
X
α=1
R(α)
kδaα,(2.6)
where δaαare arbitrary infinitesimals, one has
n
X
k=1
Mik(u)δ+uk= 0 , i = 1,...,n. (2.7)
This fact suggests us to introduce r1linearly independent vectors ˜
R(β),β= 1,...,r1complementary to the vectors R(α),
α= 1,...,n−r1such that arbitrary variation δucan be decomposed as
δu=
n−r1
X
α=1
R(α)δaα+
r1
X
β=1
˜
R(β)δbβ,(2.8)
where δbβare arbitrary infinitesimals. Substituting (2.8) into (2.3) one gets
δx=
r1
X
β=1
Qβ
(01)δbβ+
r1
X
β1,β2=1
Qβ1β2
(02) δbβ1δbβ2+
n−r1
X
α1,α2=1
Qα1α2
(20) δaα1δaα2+
n−r1
X
α=1
r1
X
β=1
Qαβ
(11)δaαδbβ+. . . , (2.9)
where
Qβ
(01) i≡
n
X
k=1
Mik(u0)˜
R(β)
k,
Qα1α2
(20) i≡1
2
n
X
k,l=1
∂2fi
∂uk∂ul
(u0)R(α1)
kR(α2)
l,
Qβ1β2
(02) i≡1
2
n
X
k,l=1
∂2fi
∂uk∂ul
(u0)˜
R(β1)
k˜
R(β2)
l,
Qαβ
(11) i≡1
2
n
X
k,l=1
∂2fi
∂uk∂ul
(u0)R(α)
k˜
R(β)
l,
. . .
Qα1...αrβ1...βs
(rs)i≡1
(r+s)!
n
X
k1,...,kr,
l1...,ls=1
∂r+sfi
∂uk1. . . ∂ukr∂ul1. . . ∂uls
(u0)R(α1)
k1. . . R(αr)
kr
˜
R(β1)
l1. . . ˜
R(βs)
ls, r +s≥2
i= 1, . . . , n , α, α1, α2,··· = 1,...,n−r1, β, β1, β2,··· = 1,...,r1.
(2.10)
One observes that the variables δaαand δbβbehave differently at the blow-up points.
Further, the existence of the vectors L(α),α= 1,...,n−r1with the property (2.5) suggests to introduce new variables
δyα=L(α)·δx, δ˜yγ=˜
L(γ)·δx, α = 1,...,n−r1, γ = 1,...,r1(2.11)
where (L(1),...,L(n−r1),˜
L(1),...,˜
L(r1)) form a basis of Rn.
3
Note that the transformation from the variables δxito the variables δyα, δ ˜yγis invertible.
Taking the scalar product of (2.9) with L(α)and ˜
L(γ), one obtains respectively the expansions
δyα=
r1
X
β1,β2=1 Q(α)β1β2
(02) δbβ1δbβ2+
n−r1
X
α1,α2=1 Q(α)α1α2
(20) δaα1δaα2+
n−r1
X
α=1
r1
X
β=1 Q(α)αβ
(11) δaαδbβ+. . . ,
α= 1,...,n−r1
(2.12)
and
δ˜yγ=
r1
X
β=1
˜
Q(γ)β
(01) δbβ+
r1
X
β1,β2=1
˜
Q(γ)β1β2
(02) δbβ1δbβ2+
n−r1
X
α1,α2=1
˜
Q(γ)α1α2
(20) δaα1δaα2+
+
n−r1
X
α=1
r1
X
β=1
˜
Q(γ)αβ
(11) δaαδbβ+. . . , γ = 1,...,r1
(2.13)
where the coefficients in (2.12) and (2.13) are defined as
Q(α)α1...αrβ1...βs
(rs)≡Lα·Qα1...αrβ1...βs
(rs),
˜
Q(γ)α1...αrβ1...βs
(rs)≡˜
Lγ·Qα1...αrβ1...βs
(rs).(2.14)
It is noted that the coefficients in these expansions depend on the point u0∈Γ and generically Q(α)β
01 =˜
L(α)·Bβ
(0) 6= 0.
It was briefly mentioned in paper [10] that the relations (2.9) and (2.12) are essential for the analysis of blow-ups of
derivatives for HEE (1.1). As we will see, the expansions (2.9), (2.12) and (2.13) completely define the structures and
characteristic properties of the blow-up of the first and higher levels as well as the corresponding hierarchy of blow-ups.
Blow-ups of derivatives of the first level correspond to the situation when all families of bilinear forms in (2.9), (2.12)
and (2.13) and all their possible linear superpositions do not vanish. Let us denote the (n−r1)-dimensional subspace of
variation of uaround the point u0given by the formula (2.6) as H+
(1)u0and as H−
(1)u0its complement spanned by the
vectors ˜
R(β)
δ−u=
r1
X
β=1
˜
R(β)δbβ.(2.15)
Due to the invertibility of the transformations (2.11), the generic variation of xaround the point x0can also be
represented in the form
δx=δ+x+δ−x(2.16)
where
δ+x=
n−r1
X
α=1
Pαδyα, δ−x=
r1
X
β=1
˜
Pαδ˜yα(2.17)
and n−r1
X
α=1
PiαL(α)
k+
r1
X
β=1
˜
Piβ ˜
L(β)
k=δik .(2.18)
We will denote the subspace of variations of xigiven by the first of (2.17) as H+
(1)x0and that given by the second of (2.17)
as H−
(1)x0.
Now, it is easy to see that the expansions (2.12) and (2.13) define the behavior of derivations with the variations of δu
and δxbelonging to different subspaces. Indeed, the expansion (2.13) implies that if
δu∈H−
(1)u0, δx∈H−
(1)x0,(2.19)
one has
δ˜yγ∼˜
Q(γ)β
(01) δbβ,(2.20)
and, consequently, ∂bβ
∂˜yγ∼O(1) .(2.21)
Analogously, if
δu∈H+
(1)u0, δx∈H±
(1)x0,(2.22)
the expansions (2.12) and (2.13) imply that
δaα∼ |δx|1/2(2.23)
4
and so ∂aα
∂xk∼ |δx|−1/2,|δx| → 0.(2.24)
Finally if
δu∈H−
(1)u0, δx∈H+
(1)x0,(2.25)
one has
δbβ∼ |δy|1/2,(2.26)
and, hence, ∂bβ
∂yα∼ |δy|−1/2,|δy| → 0.(2.27)
It is noted that
∂
∂yα
=Pα·∂
∂x=
n
X
i=1
Piα ·∂
∂xi
, α = 1,...,n−r1(2.28)
and
∂
∂˜yα
=˜
Pγ·∂
∂x=
n
X
i=1
˜
Piγ ·∂
∂xi
, γ = 1,...,r1(2.29)
where the matrices Piα and ˜
Piγ are defined by (2.17). Summarizing one has the following table for the behavior of
derivatives ∂ui
∂xkas δx→0
δu
δxH+
(1)u0H−
(1)u0
H+
(1)x0∞ ∞
H−
(1)x0∞O(1)
It is noted that the derivatives ∂ui
∂xkdo not blow-up in r1-dimensional subspace of variations δ−u,δ−x. The rank r1of
the matrix M(u0) and dimensions of corresponding subspaces may vary from point to point on the blow-up hypersurface
Γ. Subspaces H+
(1)u0,H−
(1)u0,H+
(1)x0, andH−
(1)x0are elements of the orbits generated by the group SO(n) of rotations. The
dimensions of all elements of corresponding orbits are, obviously, the same.
3 On the admissible ranks of the matrix M(u0)
The matrix M(u), playing the central role in the whole construction, has a rather special form (2.1). It is parametrized
by nfunctions fi(u), i= 1,...,n of nvariables u=u1,...,un. Functions fiare essentially local inverse to the initial
data u(x, t = 0) for the equation (1.1).
In the analysis of the admissible constraints for matrix Mone should distinguishes two cases. First, usually considered
case, corresponds to the so-called generic initial data u0and, consequently, to the generic functions f(u). In such a case
the elements of the matrix Mdepend on n+ 1 variables t, u1,...,unfor the generic functions f1,...,fn.
The second case corresponds to the situation when one considers not the fixed initial data but a family of initial data.
In this case one can view the function fias depending not only on nvariables u1,...,un, but also on a certain number
(possibly infinite) of parameters λi,i > 0. For the discussion of such situation in the one-dimensional case see e.g. [10].
As we will see, the situation with possible ranks of the matrix Mis quite different in these two cases.
Let us begin with the generic case. The matrix Mdepends on n+ 1 variables t, u. The requirement that M(u0) has
the rank r1imposes (n−r1)2constraints of order (r1+ 1) (see e.g. [7]).
Hence, the dimension of the subspace at which the matrix M(u0) has rank r1is given by
dim Γr1=n+ 1 −(n−r1)2.(3.1)
Since dim Γr1cannot be negative, in the generic case, one has the following constraint on the rank r1
(n−r1)2≤n+ 1 ,(3.2)
i.e. ln−√n+ 1m≤r1< n , (3.3)
5
摘要：

Onthe nestructureandhierarchyofgradientcatastrophesformultidimensionalhomogeneousEulerequationB.G.Konopelchenko1andG.Ortenzi2*1DipartimentodiMatematicaeFisica\EnnioDeGiorgi",UniversitadelSalento,73100Lecce,Italy2DipartimentodiMatematicaeApplicazioni,UniversitadiMilano-Bicocca,viaCozzi55,20126Milan...

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