Uniqueness for SQG patch solutions
Abstract
This paper is about the evolution of a temperature front governed by the surface quasi-geostrophic equation. The existence part of that program within the scale of Sobolev spaces was obtained by the third author (2008). Here we revisit that proof introducing some new tools and points of view which allow us to conclude the also needed uniqueness result.
1. Introduction
Among the more important partial differential equations of fluid dynamics we have the three dimensional Euler equation, modelling the evolution of an incompressible inviscid fluid, and the surface quasi-geostrophic (SQG) which describes the dynamics of atmospheric temperature Reference 19. SQG also has the extra mathematical interest of capturing the complexity of the 3D Euler equation but in a two dimensional scenario, as was described in the classical work Reference 8.
This model reads
where is the temperature of the 2D fluid with The velocity . is related to the temperature through the Riezs transforms given by
Within the equation there is an underlying particle dynamic which preserves the value of implying that the norms , , remain constants under the evolution. ,
In this paper we consider the patch problem, on which the temperature takes two constant values in two complementary domains and the solution of SQG has to be understood in a weak sense, namely:
for every That is, the temperature reads .
where is a simply connected domain. It gives rise to a contour equation for the free boundary
which is moving with the fluid and whose exact formulation can be found in Reference 10. It is then clear that the evolution of the patch is equivalent to that of its free boundary Therefore an important question for this problem is the propagation in time of the regularity of the interface . or to the contrary the existence of finite time blow-up phenomena.
This problem was first considered by Resnick in his thesis Reference 20. Local-in-time existence and uniqueness was proven by Rodrigo Reference 21 for initial data using the Nash-Moser inverse function theorem. In Reference 10 the third author proves local-in-time existence for the problem in Sobolev spaces, using energy estimates and properties of a particular parameterization of the contour. Namely, one such that the modulus of the tangent vector to the curve does not depend on the space variable, depending only on time Reference 16 and giving us extra cancellations which allows to integrate the system.
In the distributional sense, the gradient of the temperature is given by
for a given parameterization of the contour and
Then the Biot-Savart formula helps us to get the velocity field, outside the boundary, in terms of the geometry of the contour, that is,
where is the Riesz potential of order which on the Fourier side is multiplication by , The above integral diverges when . approaches the boundary but only on its tangential component, while its normal component is well defined. This fact is crucial to assign a normal velocity field to the boundary governing its evolution. Since the contribution of the tangential component amounts to a reparameterization of the boundary curve, we are free to add such a component satisfying both purposes: to be bounded and having a tangent vector with constant length. For a given parameterization approaching the boundary in both domains we obtain ,
And we get the task of finding a good parameterization and a function so that
and the two purposes mentioned above are achieved.
Having the length of the vector as a function in the variable only provides the following two identities:
The first one gives extra cancellations while the second allows us to perform convenient integration by parts.
Although we cannot give justice to the many interesting contributions due to the different authors quoted in our references, let us say that, at the beginning, there was a conjecture about the formation of singularities in the evolution of a vortex patch for Euler equations in dimension two Reference 2. It was disproved by Chemin in a remarkable work Reference 7 using paradifferential calculus, and later Bertozzi-Constantin Reference 1 obtained a different proof taking advantage of an extra cancellation satisfied by singular integrals having even kernels.
Between the patch problem for 2D Euler and SQG there is a continuous set of interpolated equations given by
The case is the most regular, 2D Euler, while for one gets SQG. The patch problem for those equations was first studied in Reference 9, where Córdoba, Fontelos, Mancho, and Rodrigo introduced a very interesting scenario for which they could show numerical evidence of singularity formation: two patches with different temperature approach each other in such a way that they collide at a point where the curvature blows-up. Let us mention that recently it has been shown analytically Reference 11 that if the curvature is controlled, then pointwise collisions cannot happen in the patch problem for SQG. In Reference 22Reference 23 a different finite time singularity scenario is shown where numerics point at a self-similar blow-up behaviour for SQG patches.
The system above can also be considered in more singular cases than SQG, replacing the last identity by the following one:
where here whose Fourier symbol is , See .Reference 6 for results on this equation with patch solutions.
A classical result in fluid dynamics is the existence for all time of vortex patches for the Euler equation which are rotating ellipses Reference 2. The patch problem for the system Equation 5 and SQG present a more complex dynamic, as ellipses are not rotational solutions and some convex interfaces lose this property in finite time Reference 5. See Reference 12 for a study of the growth of the patch support. Recently, in a remarkable series of papers and with an ingenious use of the Crandall-Rabinowitz mountain pass lemma, the authors have extended those global-in-time existence results to a more general class of geometrical shapes for the vortex patch problem Reference 14Reference 15, the -systemEquation 5 Reference 13 and also to the SQG equation Reference 3Reference 4.
There are two articles Reference 17Reference 18 where the patch problem for the is considered in a half plane with Dirichlet’s condition. The system is proved to be well-posed for -system in the more singular scenario where the patch intersects the fixed boundary. In this framework, singularity formation is shown when two patches of different temperature approach each other.
In this paper we will take advantage of a special parameterization of the boundary in the following terms:
We say that a bounded simply connected domain is for if there exists a parameterization of the boundary
such that Specifically, a domain . given by
is said to be equal to if there exists a change of variable
such that Furthermore, a time dependent simply connected domain . belongs to if there exist parameterizations of the boundaries
such that Throughout the paper we shall also deal with time dependent simply connected domains in the space . with , Sobolev spaces for meaning that its evolving boundary , belongs to that time dependent space.
Another main character of this play is the so-called arc-chord condition which help to control the absence of self-intersections of the boundary curve. This is done through the following quantity:
with
whose norm has to be controlled in the evolution.
As was mentioned before, patch solutions for the SQG equation are understood in a weak sense. Any such solution with a free boundary given by a smooth parameterization has to satisfy the equation below
where we have taken for the sake of simplicity. On the other hand, any smooth parameterization satisfying Equation 6 provides a weak SQG solution with the temperature given by Equation 2, Equation 3 (see Reference 10 for more details).
It is easy to check that the equation above is a reparameterization invariance object, and that the following formula, introduced in Reference 20 and Reference 21, has a well-defined tangential velocity and identical normal component:
The local-in-time existence result was given in Reference 10 for initial data satisfying Equation 4 and evolving by
We state the result here for completeness.
The main purpose of this paper is to show uniqueness for the patch problem for SQG which was until now an open problem. The following theorem provides this result:
This is an important part of the paper and it is proved in its section 2. In particular we show that any weak solutions of Equation 1 identified by a patch Equation 2, for a given parameterization Equation 3 with a certain regularity, can be reparameterized satisfying Equation 4. This property is preserved in time and, together with a new reparameterized curve, help us to fix the tangential velocity for a contour that evolves by Equation 8, Equation 9 giving the patch solution. Then, one just needs to get uniqueness for the system Equation 8, Equation 9. Next we check the evolution of the Sobolev norm of the difference among two different curves evolving by Equation 8, Equation 9. We close the estimate revisiting the previous existence results and introducing new cancellation and tools to find uniqueness by Gronwall’s lemma. However, in this process several different points of view with respect to the previous literature are introduced.
In the following we are going to show how it is possible to go from Equation 8, Equation 9 to equation Equation 7 through a convenient change of variable. This procedure is also valid to go from Equation 8, Equation 9 to an SQG patch contour equation with a different and more convenient tangential term.
We denote by a solution of Equation 8, Equation 9 and let be given by
or equivalently
where
is a reparameterization in for any positive time. Here is a solution of the linear system
The existence and uniqueness for that system is given in the following proposition, for whose formulation we introduce the space:
The proof of the proposition is given in section 3. The space is needed because we can only assume that for (see the proof of Proposition 1.3). Observe that the logarithmic modification of Sobolev norms is not a problem in the proof of the existence theorem given in Reference 10, because only control of the norm of is needed, which is far from the norm. In the energy estimates which provide local existence, one needs to consider the integral
whose most singular term coming form is given by
Integration by parts yields
and using identity Equation 4 one gets the bound
with and constants depending on (it is easy to observe that this extra cancellation cannot be used in the equation).
Next we shall show that is a solution of Equation 7. Here we consider regular enough ( with so that it is a bona fide reparameterization satisfying )Equation 10.
The chain rule implies
On the other hand, the equation for the evolution provides
and therefore
The fact that is a solution of Equation 11 together with identities Equation 12, Equation 13 allow us to get
Introducing the change of variable in the integral above and taking we obtain as a solution of Equation 7 replacing by , by and by Therefore, . as a consequence of the Leibniz rule for derivatives of composite functions. An interesting feature in this process is the logarithm loss of derivative which affects the solutions of Equation 7; nevertheless, we will show later how to take care of that.
Once at this point one can see clearly how this reparameterization process helps to solve the following system:
for any having the same regularity as We just have to repeat the argument but with the equation .
where the function acts as a source term, and as long as and have the same regularity, the argument works. We then arrive at Equation 14 with This shows that the systems .Equation 14 or Equation 7 come from the system Equation 8, Equation 9 by a change of variable.
Theorem 1.1 together with Proposition 1.3 yield the existence of solutions for the system Equation 7. Then Theorem 1.2 implies uniqueness:
The uniqueness part of this theorem will be discussed in section 4. Its proof will not assume property Equation 4 and it will be done controlling the evolution of the norm of the difference between any two given solutions.
An important linear operator in the study of patch solutions for SQG is given by
for Since -periodic. is a translations invariance (where we have extended periodically), the operator is a Fourier multiplier given by
Uniqueness for the 2D Euler vortex patch problem was obtained in the classical Yudovich work Reference 24. The results presented in that paper hold in a more general setting but it is also valid for any 2D Euler weak solution with vorticity in For the . weak solutions given by patches have been shown to be unique in -system,Reference 18. The uniqueness result in the present paper corresponds to the more singular and physically relevant case: but the arguments can be extended for , In those cases the equations for the reparameterization are more regular than .Equation 11 and there is no logarithm derivative loss in the change of variable process. Solutions for one of the contour evolution equations were shown to be unique in Reference 10 for .
2. Uniqueness for the SQG patch problem
This section is devoted to showing the proof of uniqueness of SQG weak solutions given by patches: Theorem 1.2. As a consequence of its proof, the solutions found in Reference 10 are unique:
3. Existence of an appropriate parameterization and commutator estimate
First let us define the operators used within the proofs, namely and a derivative and potential operator, respectively, as the following Fourier multipliers: ,
for Clearly we have that . belongs to if
Next we show a commutator estimate needed in the existence and uniqueness proofs.
4. Uniqueness for the system Equation 7
This section is devoted to showing uniqueness for the system Equation 7. The argument shown below is straight, dealing with the system Equation 7 without any change of parameterization. As before, to simplify notation we shall write , and when there is no danger of confusion.
We consider two solutions for the system Equation 7:
given by and in the space with the same initial data. During the time of existence one finds and in Here . denotes a constant which may be different from inequality to inequality but only depends on , , and , .
Let us consider the function One finds .
where
Next we symmetrize and integrate by parts to get
We have the splitting: where ,
Then a simple exchange of variables yields We have: .
hence
It remains an estimate for We rewrite .
and decompose where ,
and
As before, we control and in the following manner:
Adding both estimates we obtain the bound for which together with ,Equation 39 yield
Next we show that
We have
for and, therefore, inequality with gives Equation 41. Introducing that estimate in Equation 40 we obtain
for Since . we can conclude that the maximal solution of this inequality satisfies ,
for Therefore, choosing . and taking the limit as we prove uniqueness.
Acknowledgments
The first author was partially supported by the grant MTM2014-56350-P (Spain). The first and second authors were partially supported by the ICMAT Severo Ochoa project SEV-2015-556. The second and third authors were partially supported by the grant MTM2014-59488-P (Spain). The third author was partially supported by the Ramón y Cajal program RyC-2010-07094, the grant P12-FQM-2466 from Junta de Andalucía (Spain), and the ERC Starting Grant 639227.