Blowup with vorticity control for a 2D model of the Boussinesq equations
We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a simplified vorticity stretching term and Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable, controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data.
The two-dimensional Boussinesq equations for the vorticity and density
with velocity field given by
are an important system of partial differential equations arising in atmospheric sciences and geophysics, where the system models an incompressible fluid of varying, temperature dependent density subject to gravity. On the other hand, the system (1) exhibits some mathematical features of great interest. It is well-known that the three-dimensional axisymmetric Euler equations in a cylinder are almost identical to (1) at least at points away from the cylinder axis . That is, (1) contains an analogue of the 3D Euler vorticity stretching mechanism in the right-hand side term of the first equation of (1).
The question whether solutions of the 3D Euler equations blow up in finite time from smooth data with finite energy is a profound, as of yet unanswered question . Recent progress was made by by T. Hou and G. Luo using numerical simulations. In  the authors compute approximate solutions for which the magnitude of the vorticity vector appears to become infinite close to an intersection of the domain boundary with a symmetry plane at . Their blowup scenario is referred to as the hyperbolic flow scenario. We refer to Figure 1 for an illustration of the flow geometry.
A promising idea that leads to a better understanding of the singularity formation was introduced by A. Kiselev and V. Šverák in . There, the hyperbolic flow scenario for the 2D Euler equation
with velocity field (2) was considered. More precisely, the authors studied the flow close to the intersection of a symmetry axis of the solution with a domain boundary. The setup created a stagnant point of the flow, towards which the flow is directed.
For the two-dimensional Boussinesq equation, there is an additional vorticity stretching term on the right-hand side of the vorticity transport equation. It is possible that methods based on the construction in , exploiting the hyperbolic flow scenario, can be used to prove finite time blowup for the two-dimensional Boussinesq equations. Owing to the possible growth in vorticity and various other issues, it seems that there are many obstacles along the path.
It is reasonable to approach the problem by studying first a simplified system of equations, which captures several essential aspects of the hyperbolic flow scenario, while keeping the nonlocal and nonlinear features of the original equations. The model equations we consider here are inspired by the two-dimensional Boussinesq equations but contain a vorticity streching term of a different form.
The system reads as follows:
Note that the vorticity stretching term on the right-hand side of the equation for arises by replacing the slope by .
The functions are defined on , and we consider solutions whose support initially is bounded away from the vertical axis . We will also consider a simplified velocity field of the form ,
where is defined by
and is the sector
with arbitrary large, fixed .
1.1. Motivation and discussion
In this part, we motivate the introduction of the system consisting of (4), (5) and (6). First we would like to remark that the simplified velocity field (5), (6) is motivated by . There the authors consider the hyperbolic flow scenario for the two-dimensional Euler equation in vorticity form on a disc. A stagnant point of the flow is created at the intersection point of a symmetry axis and the boundary of the disc. The proof in  is accomplished by tracking the evolution of a “projectile”, a region on which . The projectile is carried towards the stagnant point, and due to the nonlinear interaction, the projectile is fast enough to create double-exponential growth of .
As mentioned before, it is conceivable that the hyperbolic scenario might be used to prove blowup for the Boussinesq system. By choosing an initial profile for that is monotone increasing for , one would hope that remains mostly positive up to the time of blowup. The compression of the hyperbolic flow should lead to a growth in which translates into a growth of via the vorticity stretching. This in turn should lead to further, amplified growth in . In order to prove finite-time blowup however, one has to carefully quantify the growth rates of these quantities. The problem is compounded by the nonlocal nature of the Biot-Savart law.
For this reason we have chosen to study a simplified system inspired by the Boussinesq system, which retains a similar nonlocal and nonlinear structure. In the following, we would like to further remark on these issues:
A central result of  is the representation
for the velocity field. Here, the integrals are the main contribution to the velocity field in the sense that they keep the hyperbolic flow scenario going, and and are error terms which can be bounded by .
Thus, for the two-dimensional Euler equations the error terms can be controlled owing to the conservation of the -norm of the vorticity. For the Boussinesq system, this is not possible due to the presence of the vorticity stretching term. This means that for the original Biot-Savart law it is not straightforward to guarantee the continued persistence of the hyperbolic flow scenario. During a phase of strong vorticity growth, the error contributions might become dominant.
Our velocity field (5) is motivated by the structure of the main terms in (8), although not identical. By writing the velocity field in the form (5), we create a stagnant point of the flow at , and for nonnegative we obtain the compression and expansion along the two coordinate directions typical for the hyperbolic flow scenario. We note that the role of the symmetry axis in our setup is played by the vertical coordinate axis, whereas the horizontal axis corresponds to the boundary of the disc from .
In summary, our velocity field (5) allows us to set up the hyperbolic scenario in a natural fashion. Control of vorticity growth will play an important role in our proof, and we believe that the techniques we develop here will allow us at a later stage to successfully control the errors, and to revert back to the original Biot-Savart-Law.
Let us now remark on the simplified vorticity streching mechanism in (4). The most obvious advantage is that the sign of is definite, if has a definite sign. This is not true for the original vorticity streching , and at this stage the control of the sign of proves to be challenging. The simplified vorticity streching creates a more direct connection between the compression of the fluid and the feedback into vorticity growth. However, we do not expect that the fine structure of the singularity of the model problem (4) corresponds to that of the Boussinesq problem. We refer to the forthcoming paper  for a more thorough discussion of this aspect of the problem.
1.2. Simplified picture of blowup mechanics.
Our technique emphasizes the idea to control the vorticity over regions of space and up to the time of blowup. The importance of this idea is readily illustrated by the following considerations (we follow the presentation in  by P. Constantin). Suppose we wish to study vorticity growth in the context of the 3D Euler equation
For the magnitude of vorticity, we have the equation
where the vorticity stretching factor is given by a singular integral operator acting on the vorticity field . The heuristic idea to achieve blowup therefore would be to look for situations where
However, since depends in a nonlocal way on , it is not clear for which vorticity distributions we should expect (11) to hold for a long enough time to produce infinite growth of . Indeed, geometric properties of the velocity field can lead to absence of blowup [5, 6].
Our overall strategy carried out in the context of our simplified Boussinesq system will therefore be to study particular flow scenarios, such as the hyperbolic flow scenario at a domain boundary, and to find ways to control the distribution of in space. The geometric properties of the velocity field in this situation are not in contradiction to a blowup (see [12, 13]). The control of over some significant areas in space will create leverage to achieve finite time blowup.
Let us now give a simplified picture of the blowup for (4), where the control is achieved via barrier functions. To this end, we consider only the behavior of on the horizontal axis, i.e. . The barrier functions that we construct have the follwing form for :
with positive powers satisfying . (12) holds for
where is a strictly decreasing function which will be defined below (for an illustration, see Figure 3). Slightly different control conditions hold for due to technical reasons.
The key to the proof will be to show that if was initially enclosed between the lower and the upper barrier, it will remain so for as long as it remains smooth. Note that we start from initial data with support contained in .
While the barrier functions are constant in time, the barrier domain (13) is dynamic, and one of our main tasks is to find a suitable evolution equation for . To achieve this we take the behavior of for into account, where further suitable control conditions are in place. It turns out that will satisfy
which is caused by the compression of hyperbolic flow. This implies that reaches zero in finite time. Informally, we can say that the lower bound of the barrier pushes the values of to . While only the lower bound is needed to show blowup, the upper bound is crucial to stabilize the scenario, ensure continued control over and to derive the lower bound. Of course, the smooth solution can break down before becomes zero. In this case, we use a criterion of Beale-Kato-Majda type (see Theorem 1) to show that the vorticity becomes infinite at the blowup time.
1.3. Related results.
The construction in this paper is inspired a by a corresponding result by V. Hoang and M. Radosz for a one-dimensional system . The one-dimensional case was in turn strongly inspired by the work of S. Denisov [7, 8, 9], especially his idea of starting from a singular steady solution. For our system (4), no explicit singular steady state solution is known, but for the one-dimensional analogue
a steady singular solution is
being a suitable constant. A very natural question arises: what happens if we smooth out the steady solution profile to obtain a which is compactly inside and use as initial data for the evolution? If the singular steady state is “stable” in a suitable sense, one might conjecture that the smooth solution approaches the singular steady solution, causing blowup in finite time.
The stationary solution also motivates the form of the barrier functions used in the blowup proof. It is crucial that , indicating that the power in (14) plays a special role.
The investigation of equations with simplified Biot-Savart laws of the type considered here was begun in , where a simplified version of a one-dimensional model given in  was investigated (see also ).
There seem to be few results that deal with finite time blowup in the context of two-dimensional active scalar equations or systems of equations. D. Chae, P. Constantin and J. Wu  present an example of a two-dimensional active scalar equation with nonlocal flux with finite-time blowup. In , A. Kiselev, L. Ryzhik, Y. Yao and A. Zlatoš prove finite time blowup for patch solutions of the modified SQG equations.
Independently from our work, A. Kiselev and C. Tan  prove finite time blowup for the system (4) using another simplified Biot-Savart law. Their approach is completely different from ours (barrier functions) and their model has different properties. For instance, their velocity field in incompressible, whereas ours is not. On the other hand, the “2D Euler equation” derived by setting and using their Biot-Savart law exhibits only exponential gradient growth. The version of the 2D Euler equation with our Biot-Savart law, has solutions with double exponential gradient growth. This can be shown similar to the results in .
The recent preprint  explores a class of Lagrangian models for the 3D Euler equations, which are not directly related to our hyperbolic flow scenario.
1.4. Plan of the paper.
2. Problem statement and main results
We consider classical solutions
where denotes the class of functions such that
where has compact support such that .
Observe that functions in in general do not vanish on the horizontal axis. Throughout the paper, we write and for any matrix norm on real -matrices. For matrix-valued functions , we shall also write
The following local existence and Beale-Kato-Majda continuation result holds:
Given , there exists a and a unique solution of (4) satisfying (15) with initial data . Moreover, for solutions with nonnegative initial data , the following continuation criterion holds: The existence of a such that
implies that the solution can be continued with the same spatial smoothness (15) to a slightly larger time interval , .
Our main result states the existence of finite-time blowup and gives a description of a class of initial data that blows up in finite time.
Let be positive parameters satisfying
Let be the region
where is the parameter in (7).
We call initial data suitably prepared, if are non-negative and the following conditions are satisfied:
In addition, assume and everywhere.
We refer to figure 4 for an illustration of . Our main blowup result reads:
For all , we have the following statement: There exist positive parameters with conditions such that for any suitably prepared initial data the corresponding smooth solution blows up in finite time, i.e.
where is the lifespan of the solution.
Moreover, there exists and a smooth, positive, strictly decreasing function with and such that
holds for .
The essence of the conditions (18) is that the initial vorticity is enclosed between the two singular power functions and for , where is small. For , goes smoothly to zero. From the proof of Theorem 2 it is seen that the gap between upper and lower bound can be arbitrarily wide as long as is chosen sufficiently small. This allows for a wide class of initial profiles.
Theorem 2 shows that the blowup is stable in , i.e. small changes in the initial vorticity still lead to blowup. We could have easily also allowed variations in the values of , e.g. we could have required for .
can be computed in terms of known quantities, see (40). We remark that is an upper bound on the lifespan of the solution, and not the precise time of regularity breakdown, i.e. may not be zero. If , then (20) holds for , and the lower bound in Theorem 2 would imply . We know (19), however, independently of (20) (see the proof of Theorem 2).
The slope in the definition of is for ease of presentation only. We could have chosen for any . The same applies to the definition of and in Section 4.
3. Local Existence of Solutions
3.1. Particle trajectory method.
Following the particle trajectory method (see also ), we first derive equations for the flow map
The velocity field is given by
By integrating the first equation of (4) along a trajectory and observing that is transported, we get
We obtain the following equation for :
The local existence proof is now performed in a rather standard fashion, by finding a suitable metric space of flow maps, on which is a contraction. For the reader’s convenience, a detailed proof can be found in the Appendix.
3.2. Continuation of solutions.
We address the continuation of smooth solutions. As in , it can be shown that
is sufficient to continue the solution to a slightly larger time interval (recall that denotes a suitable matrix norm).
Similar calculations as in the proof of Lemma 10 show that can be bounded by a finite constant provided and there exist such that
for all .
In the first step, we show a bound on . Since the initial data is nonnegative, all particles move to the left. Let denote any particle trajectory such that . Then at all times , and by integrating the vorticity equation from (4) we obtain the a-priori bound
Together with (16), we obtain
We now seek a function and a constant such that
for all . We may choose such that the support of lies to the left of . The support of will remain to the left of for all times since to the right of .
Let be such that
and suppose (27) holds on some time interval . Then
for and .
4. Construction of Singular Solutions
In the following, we consider smooth solutions that satisfy a number of control conditions dependent on a set of parameters. We seek to extend the validity of the control conditions up to the time of blowup, provided certain conditions were valid at and provided the parameters in Definition 1 are suitably chosen.
Let be the function defined by
where will be chosen below. We also define the following region
where will be chosen below (see Figure 4).
Observation. Since , remains non-negative for all times. This follows directly from the first equation of (4). As a consequence, we note that
So the particles all move to the left and upwards. Moreover,
as long as the smooth solution is defined.
We call controlled on if the following holds:
for all .
Suppose is controlled on the time interval . Then
In the above,
and is a suitable universal constant.
We note first that is contained in for each . For such ,
can be written as
yielding the representation (32).
For the upper bound, we use the upper bounds from the control condition and find
For we have
whereas for we have
We now fix the choice of :
Let be the solution of
Note that does not correspond to any particle trajectory.
As already mentioned, the main idea of the proof is to show that the solution is controlled on the time-dependent control region up to the blowup time. In the following, we will use the notation for particle trajectories
with initial position . In particular, we need information on the initial positions of the particles that are inside the control region at any time . The following Lemmas achieve this. In order to avoid constant repetitions, we state the following
holds for all and for some . Consider all particles with
(i.e. those that start at time above the graph of ). In the course of the evolution, these particles cannot enter the region at a point with and at some time .
Let be a trajectory as above that crosses at some time (see Figure 4 (c)). We will derive a contradiction.
and observe that . Assume crossing happens, i.e. at a point such that . We write , and in the following all expressions are evaluated at time . We skip and suppress the nonessential arguments . First note
A computation gives and from (36), the assumption , and the definition of follows
because . The above estimation implies
Now write , and . (37) becomes