Abstract
In this paper, we survey some non-blow-up results for the following generalized modified inviscid surface quasigeostrophic equation (GSQG) . This is a generalized surface quasigeostrophic equation (GSQG) with the velocity field related to the scalar by , where . We prove that there is no finite-time singularity if the level set of generalized quasigeostrophic equation does not have a hyperbolic saddle, and the angle of opening of the saddle can go to zero at most as an exponential decay. Moreover, we give some conditions that rule out the formation of sharp fronts for generalized inviscid surface quasigeostrophic equation, and we obtain some estimates on the formation of semiuniform fronts. These results greatly weaken the geometrical assumptions which rule out the collapse of a simple hyperbolic saddle in finite time.
1. Introduction
In this paper, we consider the generalized inviscid quasigeostrophic equation: where is a scalar function, representing the temperature, and is the velocity field of fluid with incompressible condition , where is a parameter. When , (1) becomes 2D Euler vorticity equation while (1) with is the surface quasigeostrophic equation. Clearly, (1) bridges the 2D Euler equation and the surface quasigeostrophic equation. Except in the case when , the global regularity issue for (1) remains open. The gradient and with for the Riesz transform defined by
Obviously, we have
This model of generalized 2D Euler/SQG equations had been proposed by Pierrehumbert et al. in 1994 in [1]; then, this model was studied by Chae et al. in [2], and they obtained a regularity criterion in terms of the norm of in the Hölder space to the generalized inviscid surface quasigeostrophic equations. Then, they researched the global regularity in [3] for a class of generalized equations that the velocity field is determined by the active scalar through where represents a family of Fourier multiplier operators. Subsequently, they obtained the global existence of a weak solution in [2] by Galerkin method and the local existence of patch-type solutions for the inviscid model. Kiselev et al. in [4, 5] studied the patch dynamics on the whole plane and on the half-plane for modified surface quasigeostrophic equations involving a parameter that appears in the power of the kernel in their Biot-Savart laws, and they established local-in-time regularity for on the whole plane and finite-time singularity for all small . It is hoped that this research sheds some light on the global regularity issue concerning the Euler equation and surface quasigeostrophic equation. However, in this paper, we pay more attention to the generalized modified quasigeostrophic equations without dissipative term, and we give some conditions of non-blow-up of the generalized inviscid surface quasigeostrophic equation.
The rest of this paper is organized as follows. Section 2 is devoted to the main results. Section 3 details the proof of these theorems and corollaries. Furthermore, throughout this paper, we use to denote the positive constants which may vary from line to line.
2. Main Results
Similar to [6], firstly, we give a definition of a hyperbolic saddle.
Definition 1. A simple hyperbolic saddle in a neighborhood of the origin is the set of curves where where are nonlinear time-dependent coordinate change with , and
Remark 2. Set (4) represents the hyperbola in -coordinate. In particular, we get the asymptote for a hyperbola when . The angle of opening of the saddle is when , while the angle of opening of the saddle is when (see also [7]).
The possible singularity is due to becoming zero in finite time. The following theorems will show that this is not possible, and can go to zero at most as an exponential decay.
Theorem 3. Suppose is a smooth solution of (1) defined for , . Assume that is a constant along the hyperbolas defined in Definition 1 for , and for each, fixed that is not a constant on any disc in , and then, exists and is not 0.
Corollary 4. Let be as in Theorem 3, let , and assume on . Then, continues to some solution of (1) on for some .
Theorem 5. Let and be as in Definition 1 but with . Assume that the seminorms of are bounded for all time . Then for all , where and are constants.
Corollary 6. Let be as in Theorem 5, let , and assume on , then
Moreover, we give a condition that rules out the formation of sharp fronts for generalized surface quasigeostrophic equations and obtains estimates on the formation of semiuniform fronts.
Theorem 7. For a generalized modified inviscid quasigeostrophic equations with a semiuniform front, if , the thickness satisfies: where the constants and depend only on the length of the front, the semiuniformity constant, the initial thickness and the norm of the initial datum .
3. Proofs of Theorems and Corollaries
3.1. Proof of Theorems 3 and 5
Suppose be a smooth solution of (1) defined for . If for that is constant along the curves , where , we denote for a function , and . Besides, we have
In the following analysis, we introduce a new set of variables to analyze the level set. The first variable is defined in Definition 1, and the is from the following identities where we write for , and is the intersection of the bisector of the angle with . We write the stream function and system (1) in terms of a time-independent change of variables
By performing the change of variables in the first equation of (1), we get
Since , we have
By , we have
Substitute (12) and (13) into (14), we get
Thus
Since and are independent of , we easily deduce that
Furthermore, the integration with respect to of (17) gives
It is obvious that . We obtain a new expression for the stream function. The expression (18) for the velocity stream function in terms of the new variable will be used in following part to obtain an estimate on the angle of the saddle.
The main strategy is to estimate the difference of the value of the stream function at a point that lies in the branch of the saddle with the value of the stream function at a point that lies in the other branch . Both point and point have the same coordinate. We need two expressions of the stream function; one was derived from the equality , and the other one was derived from the change of variables done in the above analysis. To prove both theorems, we give a lemma.
Firstly, we need the key estimate of the difference of the value of the stream function at two different points that are close to each other that is obtained by the stream function as follows: because of the fact Similar results for the 2D Euler equations have been announced in [8], and the similar phenomenon has been noticed in 2D quasigeostrophic thermal active scalar in [9, 10].
Lemma 8. Let be a solution of (1), be given by the equality , and and be defined as before, and if , then, we have where satisfies , and and are constants, and .
Proof. We evaluate at the point and with ; thus, we
If we denote . We split the integral :
where is a fixed number.
Next, we estimate every term in (22), respectively.
If we choose to be a point in the line between and , then, , and we estimate by
For the third term , we have
where we have used the fact that the norm is conserved for all time. Collecting (23), (24), and (25), we have the needed result in (20).
Then, we need to divide it into two cases to prove the main theorem. However, we only deal with the case , and for the case , we can use a similar method to prove the result, and we omit it here.
Assume . In this case, the angle of the saddle is . We take two points and lying in the same level set but in different arms. Using the identity (4), we define
Then, we take the limit approaching points and , respectively. Then, necessarily, and grow logarithmically. Then, we have the following lemma.
Lemma 9. Under the assumptions in Theorem 3, , let be given by expression (18) and defined as before, then where .
Proof. We evaluate the stream function at the points and with . According to (18), we have
Next, we take the limit when and , and it means .
By (17), we have
where we have used (10).
For a fixed , we can estimate by
where
Considering the definition of , and , we have
Now, we derive an estimate for the velocity :
We consider , and we define to be a smooth nonnegative function such that for , and for . Then, we have
For the first term, by integrating by parts, we have
For the second term, we have
Therefore, we have
Take and then, is bounded by , assuming that , where is a constant. Therefore
From the definition of , we know that when we approach the origin along the bisector with . Therefore, is at most when . This implies that
For the second term, firstly, we define
By the change of variables, we have the following expression for ,
and then by the above identify, (42) we have
Then, we utilize (31), (32), and (42) to estimate the last term in (28); it is similar to the surface quasigeostrophic equation in [7], and we omit it here.
If we take in Lemma 9, we obtain where , and is a bounded function for all .
If we denote then, (44) becomes ; therefore, (27) in Lemma 9 becomes
By , we have , thus . By the expression and , there exists constants and such that , , and , and then, we have . By Lemma 8, we have
is less than a small constant, and the constant is different from line to line. The proofs of Theorems 3 and 5 follow directly from integrating (47).
3.2. Proof of Corollaries 4 and 6
Next, we prove Corollaries 4 and 6 By the fact , we have and then, we write the integral as a limit as of integrals on . Because the two gradients applied to commute, we can choose any one of them and integrate by parts. The limit of the contribution from vanishes. In this way, we have or where .
Let , by the fact
We have and then, we deduce the two representations of . Utilizing (49), we have
Therefore, we obtain the first representation of as follows:
By (50), we used similar estimates as above, and we have an alternative expression of :
Consider with a smooth, nonnegative function of one positive variable for for .
For the second term, it is easy to obtain
Let us consider now the situation in which the direction field is smooth in the ball of centre and the radius , corresponding to the smoothly directed case. We use the representation in (54), let denote , clearly
Then, we have
By the fact , integrating by parts on the right on (59), we have
We estimate every term in (60), respectively, and then, we have
For the third term, we write , and therefore
Since then, we have
For the second term, we have where we have used the Hölder inequality and , and represents the angle in the polar coordinate through the transformation. Thus, we have
Combining (57) and (66) in (56), we have the following.
Lemma 10. Assume that is such that Then, is bounded by
Under the assumptions of Corollary 4, we know is bounded by a constant . Therefore, we can estimate on by
In particular, the function is independent of . We write the material derivative of as follows:
We estimate on by
The Definition 1 shows that . The same estimate can be obtained. Combing (69) and (71), we get on when .
Then, by (70), we have on when . However, because only depends on and and by the proposition, it follows that inequality (73) only depends on and .
We try to estimate the material derivative of on . If , then, we find such that . The inequality (73) holds for , and therefore, it holds for . Therefore, the inequality (73) holds both on and . By (70), we have the following identity
For the second term in (74), by the fact that the boundedness of on , we have
Because is independent of , the inequality (75) holds on . We compute the material derivative of and estimate it on as follows:
Combing (73), (74), and (76), we have
If , then, . By using the upper bound of , we obtain on . Therefore
Let , integrating (79) on time , we have
We have
At time . It shows that is bounded, and is bounded, and it proves the non-blow-up for solution, and this proves the Corollaries 4 and 6.
3.3. Proof of Theorem 11
Let is a solution to (1), and a level curve of can be parameterized by with in the sense that
Combining (1), (85), and (86) and the relationship , we have where denotes the inner product or dot product, i.e., ( are scalar functions). Therefore
Utilizing this formula, we can obtain an explicit equation for the change of time of the area between two fixed points and two level curves ,
Suppose that two-level curves and collapse when tends to uniformly in , i.e.
That is to say, the distance between two level sets are comparable for Denote be the thickness of the front and call the length of the interval , the length of the front.
One assumption is that for all and all By the previous results of [11], we know that if (92) holds, then, we say form a semiuniform front.
Proof of Theorem 11. By (89) and (90), we have
By virtue of (20) in Lemma 8 (where and are different point in ), we take and , and we have
That is to say
Therefore, by Gronwall’s inequality, (90) and (91), we have and this allows us to rule out the formation of sharp fronts, and this yields (8) and concludes the proof.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest with others.
Acknowledgments
This work was supported by the Fund of Fundamental Scientific Research Business Expense for Higher School of Central Government (grant no. ZY20215116). The research of M. L. Hong is partially funded by the National Science Foundation of China (grant no. 12071192).