On blowups of vorticity for the homogeneous Euler equation

Blowups of vorticity for the three- and two- dimensional homogeneous Euler equations are studied. Two regimes of approaching a blowup points, respectively, with variable or fixed time are analysed. It is shown that in the $n$-dimensional ($n=2,3$) generic case the blowups of degrees $1,..,n$ at the variable time regime and of degrees $1/2,..,(n+1)/(n+2)$ at the fixed time regime may exist. Particular situations when the vorticity blows while the direction of the vorticity vector is concentrated in one or two directions are realisable.

In the papers [3] and [16] it was observed that in the case of compressible fluid the behavior of vorticity for the Euler equation is intimately connected with that of homogeneous Euler equation (HEE) ut + u • ∇u = 0 . (1.1) without the constraint ∇ • u = 0.In the papers [3,4] an explicit integral-type formula for the vorticity ω = ∇ × u for the equation (1.1) has been presented.Another type of formula for the vorticity has been found in [13,14].The blowup of vorticity as t → tc > 0 has been analysed in [14,16] (see also [4] and [17]).Homogeneous Euler equation (1.1) is the most simplified version of the basic equations of the hydrodynamics when one can neglect all effects of pressure, viscosity etc.. Nevertheless it has number of applications in physics and represent itself an excellent touchstone for an analysis of blowups of vorticity.
In this paper we present some results concerning the blowups of vorticity for the three-and two-dimensional homogeneous Euler equation (1.1).Our analysis is based in part on the previous study of the structure and hierarchies of blowups of derivatives for the n-dimensional HEE [10,11].
We consider the behavior of vorticity in two different regimes of approaching the blowup points at the blowup hypersurface.The first regime is to approach such a point along the t axis, i.e. t → t b while the coordinates u in the hodograph space remain fixed.It is shown that, in the generic case, i.e. for generic initial data for the 3D HEE (1.1), the vorticity in this regime may have singularities of three different degrees Such blowups occur on the intersection of m branches of the blowup hypersurface Γ.The existence of blowups of type (1.2) with m = 1, 2 has been observed earlier in [14].In the second regime the time t b is fixed while the coordinates u are variyng.In this regime of approaching the blowup point for 3D HEE (1.1) generically it may exists 4 levels of blowups of the vorticity ω with the behavior where ∼ |δx| → 0. Blowups (1.3) occur on the subspaces Γm of the blowup hypersurface Γ and dimΓm = 4 − m, m = 1, 2, 3, 4.
It may happens also that the components of the vorticity ω behave differently on certain subspaces of Γ.In particular, at the first level m = 1 there may exist one-dimensional subspace Γ1 at which the component ω3 blows as −1/2 when → 0 while the components ω1 and ω2 remain bounded.In such a case the direction of the vorticity ω is a unit vector oriented along one axes, namely ω = 0, 0, 1 . (1.4) The calculations are performed both in the special coordinates introduced in [11] as well as in cartesian coordinates x and u.
For the 2D HEE (1.1) the vorticity blows-up as in (1.2) and (1.3) with m taking the values m = 1, 2 for (1.2) and m = 1, 2, 3 for (1.3), respectively.Three particular solutions of the 2D HEE with different blowup behavior are considered.
It is noted that we analyze the behavior of vorticity at certain points on the blowup hypersurface Γ and at the time t b which can be negative or positive.The realisability of blowups of different orders at positive time remains an open problem.
Similar results for the n-dimensional homogeneous Euler equation are briefly discussed too.The paper is organized as follows.Section 2 contains a brief exposition of the results of the paper [11] for the 3D HEE.Blowups of vorticity in the first regime t → t b are analysed in section 3. Blowups of vorticity for the 3D HEE in the regime with fixed t are studied in sections 4 and 5. Similar results for the 2D HEE are presented in section 6.Three particular solutions of the 2D HEE with different blowup behavior are described in detail in Section 7. The n-dimensional n ≥ 4 case is discussed in Section 8. Conclusion 9 contains some indications on possible future developments.

Blowups of derivatives
Here for conveniency we report some results concerning the blowup of derivatives for the 3-dimensional homogeneous Euler equation obtained in the paper [11].We also slightly change the notations in order to make the corresponding formulas more convenient for the further calculations.
The starting point of the analysis are the hodograph equations [23,3,7,10] where fi(u) are arbitrary functions locally inverse to the initial data ui(t = 0, x).Any solution u(x, t) of the system (2.1) is a solution of the 3D system HEE (1.1).
The matrix M with the elements plays a central role in the analysis of blowups of derivatives and possible gradient catastrophes.In particular, The blowups occur on the 3D hypersurface Γ defined by the equation where a2(u), a1(u) are certain functions of u and a0(u) = det(M (t = 0, u)) = 0 for generic initial data.
The blowup hypersurface Γ is the union of the branches tα = φα(u) corresponding to real roots of the cubic equation (2.4).In the three dimensional case the number of branches can be one or three [11].
In the generic case the rank r of the matrix M may assume two values r = 2 and r = 1.Equivalently it means that there exists 3 − r vectors R (α) (u b ) and (2.5) The existence of such vectors suggests the introduction of new dependent and independent variables v1, v2, v3 and y1, y2, y3 defined by the relations [11] δu where the vectors R(β) are r vectors complementary to the set of 3 − r vectors R (α) and vectors P (α) , P (β) are defined by the relation where L(β) are r vectors complementary to the set of 3 − r vectors L (α) .One also has where = δij.The use of variational consequences of the hodograph equations (2.1) shows that derivatives ∂vα ∂y β (u b ) behave differently in different subsectors of the independent and dependent variables [10,11].For instance, for r = 2, on the first level of blows-ups, the derivatives ∂v1 explode on the hypersurface Γ (2.4) while the derivatives remain bounded.These blowups may happen both at positive and negative time.
It is noted that all vectors given above and the behavior of derivatives ∂vα ∂y β vary with the variation of the point u b belonging to the hypersurface Γ (2.4).
On the first level of blows-ups the derivatives explode as −1/2 , ∼ |δy| → 0 and the behavior of derivatives at fixed time t b presented in (2.9) and (2.10) can be resumed in the formula where and νij, i, j = 1, 2, 3 are connected with the values of ∂f i ∂u j (u b ) and ∂ 2 f i ∂u j ∂u k (u b ) evaluated at the point u b ∈ Γ1 (see [11]).We emphasize that the formulae (2.12) represent the relations between the infinitesimal variations of the variables yi and vi around a point u b ∈ Γ at fixed time t b .Blowup time t b can be positive or negative.Blowup at t b > 0 is refereed as gradient catastrophe.In this paper, as in [11], we will not discuss conditions which guarantee that t b > 0.
It is also noted the domain Du of variations of u constructed via equation (2.1) and, consequently, the domain of variations of variables u parameterizing the blowup hypersurface Γ (2.4), coincides with the domain Du 0 of variations of the initial values u0, since u(x, t) = u0(x − ut).

Blowup of vorticity
The formula (2.3) provide us with the explicit and useful expression for the vorticity vector in the original Cartesian coordinates in terms of the components ui, i = 1, 2, 3 of the velocity.Namely, where M is the adjugate matrix.We consider first the case rank(M (t b , u b )) = 2. Let us fix the point u b on the blow-up hypersurface Γ (2.4) and take the corresponding real t b , i.e. the real root of the cubic equation(2.4) which always exists for the 3D HEE [10].The formula (3.1) implies that ( [10]) where and . Generically for r = 2 M jk (t b , u b ) = 0 and D1(t b , u b ) = 0. Hence, in the generic case, in the first regime the vorticity blows up on the full hypersurface Γ as where σi ≡ 3 j,k=1 ijk M kj (t b , u b )/D1(t b , u b ) for i = 1, 2, 3. Existence of the higher order singularities is correlated with the structure of the blowup hypersurface Γ.If it has a single branch (single real root of the equation (2.4)) then M (t b , u b ) cannot be zero.Hence, due to (3.2) and (3.3) in this case only the blowup of type (3.4) occurs.
Situation is different when Γ has three real branches, i.e. all roots of the equation (2.4) are real.In this case one has the formulae (3.2) and (3.3) and three different values of t bα , α = 1, 2, 3 for the same value u b .Moreover, the condition i.e. the condition that det(M (t b , u b )) has a double zero at t b , is now admissible.Let the condition be satisfied at one branch.It defines the two-dimensional submanifold D (2) and at the corresponding t bα the vorticity blows-up as Moreover, the condition (3.6) (cf.(3.3)) means that the root t bα is a double root, i.e. coincides with another root t bβ .So, the branches α and β of the blowup hypersurface Γ intersect along the two-dimensional surface Γ2 corresponding to values of u b ∈ D (2) u and on Γ2 the vorticity blows up as in (3.7).Hence, in the particular case (3.6) the vorticity ω blows up ad (t − t b ) −2 on the intersection of two branches of Γ and blows up as (t − t b ) −1 on the third branch.
Finally if, in addition to (3.6), the condition is satisfied, but 3 j,k=1 ijk M kj (t b , u b ) = 0, with i = 1, 2, 3 then the vorticity ω blows up as The situation (3.9) happens on the curve Γ3 in Du defined by the conditions (3.6) and (3.8).Since such conditions means that the root t bα is a triple root, the behavior (3.9) occurs at the intersection of all three branches of the blowup surface Γ.
The existence of the blowups of the types (3.4), (3.7), and (3.9) becomes rather obvious if one rewrites the formula (2.4) as det (M (t, u)) It is noted that one can treat the conditions (3.6) and (3.8) in a different manner, namely, to consider them as the equations for the functions f1(u), f2(u), f3(u).Within such a viewpoint, equation (3.6) defines those functions fi(u), i = 1, 2, 3 for which two branches of the hypersurface Γ identically coincide.All three branches of Γ coincide in the particular case of initial data such that the functions fi(u), i = 1, 2, 3 are solutions of the pair of equations (3.6) and (3.8).
The formulae (3.4) and (3.7) reproduce the results previously obtained in [14] with the use of the Lagrangian analogue of the formula (3.1).The behavior of type (3.9) was not present in [14] due to the particular geometry of the vortex lines considered there.
An analysis of the behavior of vorticity and its integral characteristics has been performed also in [4] with the use of an explicit integral representation of the Lagrangian type derived in [3].
The components ωi behave according to (3.4), (3.7), and (3.9) in the general case when all σi = 0.In this case the direction of the vorticity vector (see e.g.[5]) is regular with components Let us assume now that one of σi vanishes, e.g.σ3, i.e.
This condition defines the two-dimensional subspace D2 ⊂ Du in the hodograph space.At the points u ∈ D2 one has σ3 = 0 and, hence instead of (3.4) the vorticity vector direction blows up as as where σ i ≡ 3 j,k=1 ijk M kj (t b , u b ) for i = 1, 2, 3. Consequently, the vector ω is of the form Generically, for m = 1 such situation may occur on the two-dimensional subsurface of the blow-up hypersurface Γ.For m = 2 it may happens along the curve belonging to the two-dimensioanl intersection of two-branches of Γ.For m = 3 it may occur at the point belonging to the curve of intersection of the three branches of Γ.
In the very particular case of two vanishing components of σi, e.g.σ1 = σ2 = 0, one has and ω = (0, 0, 1) . (3.17) Generically such behavior may exists only for m = 1, 2. For m = 1 it may happens along a curve on Γ while for m = 2 it may occur at the point belonging to the intersection of two branches of Γ.The behaviour of vorticity described above corresponds to the case of rank r = 2 for the matrix M evaluated on the blowup hypersurface Γ.It occurs on the whole blowup hypersurface [11].In contrast, the matrix M (t b , u b ) may have rank 1 only on a set of points Γ0 on Γ [11].Moreover for r = 1 the adjugate matrix M vanishes identically: On the other hand generically M ij Γ 0 are different from zero.So, in such a situation the components of vorticity remain bounded when t is approaching t b which correspond to a point u b belonging to Γ0.

Blowups of vorticity at fixed time
The formulae (3.4), (3.7), and (3.9) describe the behavior of the vorticity in the situation when time t approach the blowup time t b along the t axis with fixed coordinate u b .
The approach presented in [11] and briefly reproduced in the section 2 looks more appropriate for the analysis of the blowups of vorticity in the regime when time t is fixed while the coordinates u are subject to variations.
The formulas presented in the section 2 (see also [11]) indicate that non-cartesian coordinates yi and vi, i = 1, 2, 3 are rather convenient for the analysis of blowups of the derivatives.In order to use such coordinates for the analysis of blowups of vorticity, one has to consider its coordinate-independent definition as the differential two-form (see e.g.[1,22]) where θ = u • dx.
We will use such definition in the form to study the behavior of vorticity at the point u b of the blowup hypersurface Γ.
Using the formulae (2.6), one gets where Then, due to the relation (2.11), at the blowup point u b one obtains where The components of the vorticity vector ω in these coordinates are defined as usual as At the first level of blowup and rank r = 2 the matrix C is of the form (2.12).Consequently, the element of ω αβ , written in terms of the vorticity components ωi, behave as where as → 0 and So, generically, i.e. when all Sα = 0, the vorticity ω blows-up as −1/2 , → 0 at the point u b of the three-dimensional blowup hypersurface Γ.In this case the direction of the vorticity vector ω is regular with the components However, particular situations are also admissible.Indeed, if there exist a point u b ∈ Γ such that S3(u b ) = 0 then at this point the components ω1 and ω2 of the vorticity blowup while the component ω3 remain finite.The condition S3(u b ) = 0 has co-dimension one.So, such situation is realisable, in principle, on the two-dimensional sub-surface of the blowup hypersurface Γ and ω is of the form Further, there may exist the points belonging to a certain curve on Γ at which At these points the components ω1 and ω2 remain bounded and only one component ω3 of the vorticity blows up.Hence, the vorticity direction vector (3.11) assumes a particular form Such a situation when the vorticity vector ω becomes very large in modulus, but concentrated in one direction looks rather special and of interest.
It may even happens at a certain point u b ∈ Γ that In such a case the vorticity ω remains bounded in the point of the first level blowups of derivatives.Finally, in order to analyse the blowup of vorticity in the Cartesian coordinates it is sufficient to perform the change of coordinates y → x in the r.h.s. of (4.5).Performing the transformation (2.8) in (4.5), one obtains As a result, the components ωi = 3 j,k=1 ijk ∂u k ∂x j of the vorticity vector ω = ∇ × u blows-up on the whole hypersurface Γ, namely ωi = −1/2 Si(ub) + Ti(ub) , i = 1, 2, 3 , → 0 (4.17 where Si and Ti are bounded functions obtained by a change of variables from (4.10).
The same result can be obtained directly, using the formulae (2.6), (2.11) and (2.8).Namely, one gets and, then one obtains the formula (4.17).Again, it may happens that along certain curves Γ1 belonging to Γ, one has At the points on this curve, the components ω1 and ω2 remain bounded while the component ω3 → ∞ and ω = (0, 0, 1).Such a situation, when the vorticity vector ω becomes very large in modulus but concentrated in one direction, resembles somehow certain well-known physical phenomena.

Blowups at rank 1 and higher levels
In the case of rank r = 1, which occurs at a set of points u b ∈ Γ the matrix C is of the form (cf. [11]) The components of the vorticity vector ω again are of the form (4.10) or (4.17).
However in this case one cannot impose any constraint of the type S3 = 0 or (4.13), if one considers the situation with generic function fi(u) of initial data.Such constraints may be admissible for particular special initial data.
Blowups of second, third and fourth level for r = 2 occur on certain subspaces of the three-dimensional blowup hypersurface Γ [11].
One of the subsections of the second level of blowups (in the rank 2 case) is characterized by the following behavior of derivatives [11] which corresponds to a matrix C given by where ηij are certain coefficients depending on ∂f i ∂u j (u b ) and ∂ 2 f i ∂u j ∂u k (u b ) evaluated at the point u b .Consequently, the components ωi of the vorticity have the following behavior at the blowup point of the second level where Yi, Si, and Ti are certain bounded functions of u b ∈ Γ.In this case the direction of vorticity vector (3.11) is where So, in contrast to the first level (4.10) the components of the vorticity vector generically blows up in a different manner.Such realization occurs in the two-dimensional subspace of the blowup hypersurface Γ [11].So, one can impose at most two constraints.
Imposing two constraints, one may have essentially two different situations.Indeed if all components of vorticity blow up in the same manner, namely, (5.12) and the vorticity direction vector is generic one.On the other hand, if it happens that then the components of vorticity behave quite differently since ω ∼ (O(1), −1/2 , −2/3 ) , → 0. (5.14) In this case the vorticity direction vector ω is oriented along the third axis, namely ω = 0, 0, 1 . (5.15) Such situation is realisable in principle at the points of intersection of the curves defined by (5.8) and (5.6).
One observes similar behaviors of vorticity in other subsectors of the second level of blowups.
(5.17) and ω = 0, 1, 0 . (5.18) Finally, the fourth level may occur at a point on Γ and this point (see also [11]) Again, one has formula (5.4) with the substitution −2/3 → −4/5 in the first term in the r.h.s. and generically no constraints are allowed.

Vorticity for two-dimensional HEE
For the two-dimensional HEE an analog of the formula (3.1) for the vorticity ω3 = ∂u 2 ∂x 1 − ∂u 1 ∂x 2 is given by where M0 ≡ M (t = 0, u) is the matrix with components (M0)ij = ∂f i ∂u j , i, j = 1, 2. The quadratic equation t 2 + tr(M0)t + det (M0) = 0, defining the blowup surface Γ [10], may have, obviously, either two real roots or no one, depending on the sign of the discriminant So, in contrast to the three-dimensional HEE, in two dimensions there are solutions with blowups free vorticity (cf.[10]).
It is natural to consider subdomains D + u ⊂ Du, and D − u ⊂ Du defined as follows and the curve D 0 is the boundary between D + u and D − u .In the case Du = D − u , one has the blowup free situation.In the rest of this section we will assume that the subdomain D + u is not empty and hence the blowup surface has two branches Γ+ and Γ−.
Let u b a point at D + u and t b be the corresponding value of time t on the first or the second branches of Γ.In the first regime, i.e. when t → t b with fixed u b , one has the vorticity ω3 blows up as This happens at each point of the blowup surface Γ.
If instead the vorticity ω3 blows up as Such a behavior (6.9) occurs the curve defined by the condition (6.8).
It is the condition of coincidence for the values of the branches Γ±, i.e. t b+ = t b− .Hence, the blow-up of the type (6.9) occurs along the curve of intersection of two branches of the blow-up surface Γ.The corresponding curve (6.8) in the hodograph space can be the border curve between two subdomains D + u or D − u when D + u = Du or D − u = Du respectively.Similar to the three-dimensional case one can view the conditions (6.8) as the equation which defines those functions f1(u) and f2(u) for which two branches of Γ coincide.
In order to analyze the behavior of the vorticity ω3 at fixed time t b , similar to (2.6), one introduces the variables y and v (see also [11]) At the first level of blowups one has the following behavior of derivatives [11] So, one has the relation with the matrix In the two-dimensional case the vorticity is the differential two-form where and S(u b ) and T (u b ) are certain combinations of ν αβ and R α • P β (see in analogy the three-dimensional case the (4.4) and (4.5) relations).
In the cartesian coordinates the vorticity ∂u 2 ∂x 1 − ∂u 1 ∂x 2 also is of the form (6.16).Along the curve defined by the condition S(u b ) = 0 (6.17 the vorticity is bounded.Blowups of second level occurs on the curve contained in Γ and on this curve (see [11]) and, consequently, the vorticity blows-up as Finally, at the third level which may occur at a point on Γ, one has ∂v 1 ∂y 1 ∼ −3/4 and, hence, the vorticity blows-up as ω12 = −3/4 .

Examples in 2D.
Here we will present three characteristic examples for the two-dimensional HEE.

Blowup free solutions
Let the functions f1 and f2 be of the form where the real function W (u1, u2) obeys the Laplace equation It is easy to see that in this case for any function W except a linear one.So, the corresponding solutions u1 and u2 of the 2D HEE have no blowups.
The vorticity (6.1) is given by and it is blowup free too.
The particular choice or f1 = u 2 α , f2 = − u 1 α corresponds to initial velocities u1 = αx2 and u2 = −αx1 where α is an arbitrary real constant.Such initial condition gives and ω3 = 2α It is the rotational type vortex solution of the 2D HEE with the initial strenght 2α and α −1 as the characteristic decaying time.
It is worth to note that the subclass of solutions of the 2D HEE corresponding to the choice (7.1) has a simple description in terms of complex coordinates [10] Indeed, in these variables the conditions (7.1) and (7.2) are given by and where W(V ) is an arbitrary analytic function (note that (7.1) implies that W is real-valued), then For such function F the hodograph equation assume the form Solutions of the hodograph equation (7.13) obeys the equation In the complex variables the vorticity (7.4) is given by For the solution (7.6) F = −iV /α.For the generic analytic function F (V ) the corresponding solution V (Z, t) of the equation (7.14) and its vorticity are blowup free.In the trivial particular case F = βV , where β is an arbitrary real constant, the solution V (Z, t) = Z t+β of equation (7.14) and its derivative exhibit the blowup at t = −β while the vorticity ω3 = 0.In this case the 2D HEE is decomposed into two one-dimensional Burgers-Hopf equations.
The fact that for the generic analytic solutions of the 2D HEE the derivatives are blowups free has been noted in [10] (Section 5).In different contexts the equation (7.14) has been considered earlier in [12,8,24].

Nongeneric blowup
Let us choose The corresponding initial data are In this case the matrix M is and the blowup surface Γ is defined by the equation The discriminant ∆(u1, u2) is ∆(u1, u2) ≡ 4u 4 1 + 28u The subdomain D − u is located around the origin u1 = u2 = 0 as shown in figure 1.The blowup surface Γ has two    In the first regime of approaching of generic blowup point (u1, u2) ∈ D + u the vorticity behaves as Approaching the points which belongs to the curve of intersection of two branches t+ and t−, the vorticity blows up as In this case the curve ∆(u1, u2) = 0 is the boundary line between the subdomains D + u and D − u .

Gaussian initial data
Finally we consider solution of the HEE with the initial data Table 1: The local inverses of the initial data (7.27).
Each pair of equations (7.28) define a solution u ab (x, t) in the corresponding subdomain.So, solution of the 2D HEE with the initial data (7.27) is a union

.29)
In other words Moreover the domain Du is the square 0 < u1(x, t), u2(x, t) ≤ 1.Using the standard formulae u(x, t) = u0(ξ1, ξ2) with ξi = xi − uit, i = 1, 2, one can view the piecewise solution (7.30) as Then four corresponding matrices M are of the form and the corresponding branches of the blowup surface are defined by the equation The values of the vorticity ω3 for the branches (a, b) are given by The discriminant ∆ of the equation (7.33) is positive for all values of a and b since  w.r.t. to space variables, numerically computed using Mathematica.The behavior is in agreement with the analytical predictions (7.42).Since ∆ ab (u) = 0 for all u ∈ Du, two branches (7.36) do not intersect.So, the blowup of the type ω ∼ (tc − t) −2 is absent in this case.
8 Blowups for n-dimensional case.
An extension of the results presented in this paper to the n-dimensional HEE is quite straightforward.Indeed, the components ωij of the vorticity two-form (4.1) in Cartesian coordinates are given by ωij(t, u) = (M −1 )ji(t, u) − (M −1 )ij(t, u) = Mji(t, u) − Mij(t, u) det(M (t, u)) , i, j = 1, . . ., n.Similar to the results described in [11] blowups of the vorticity exhibit rather rich fine structure.
The formulae (8.3) and (8.4) imply certain behavior of the characteristics of vorticity in different dimensions discussed in [1].
One obtains analogous results for the stress tensor

Conclusions
The results presented in this note are in part the consequences of those obtained in the paper [11].As in [11] we are dealing with the most simplified version of the Navier-Stokes equation, namely with HEE (1.1) and do not discuss the possibility of blowups of vorticity of type (1.3) for positive values of time.
All that indicates at least two possible direction of further study.The first is the verification of the realisability of hierarchy of blowups (1.3) for positive times that is of most interest in physical applications.
An extension of such type of analysis for more physical systems would be the second direction.In particular, it may be applicable to those hydrodynamical systems which are obtainable as the constraints of the multidimensional homogeneous Euler equation [9].

2 Figure 1 :
Figure 1: In the gray D + u region the discriminant ∆(u 1 , u 2 ) (7.20) positive and therefore blowups are possible.In the complementary region D − u the discriminant ∆(u 1 , u 2 ) is negative and therefore no blowups are possible.

Figure 3 :
Figure 3: The time evolution of the vorticity depending on u with initial data given by (7.27).From left to right the times are t = 0, t = 0.85t c , t = 0.999t c where t c = 0.642593 is the catastrophe time.Remark the change in the vertical scale in the last plot.

Figure 4 :
Figure 4: The time evolution of the vorticity depending on x with initial data given by (7.27).From left to right the times are t = 0, t = 0.85t c , t = 0.999t c where t c = 0.642593 is the catastrophe time.The dashed vertical line indicates the catastrophe direction of the vorticity in the catastrophe place x c .
(2) .It is easy to see that for both branches t+ ≥ t− > 0 (see figure(2)).The time of the gradient catastrophe