Two-Front Solutions of the SQG Equation and its Generalizations
We derive regularized contour-dynamics equations for two-front solutions of the surface quasi-geostrophic (SQG), generalized surface quasi-geostrophic (GSQG), and Euler equations when the fronts are a graph, as well as scalar reductions of these equations, including ones that describe one-front solutions in the presence of a rigid, flat boundary. We prove local-in-time existence of solutions of the front equations in different parameter regimes for the Euler, SQG, and GSQG equations.
- 1 Introduction
- 2 Preliminaries
- 3 Regularized two-front GSQG systems
- 4 Linearized stability
- 5 A priori estimates for the two-front GSQG () systems
- 6 A priori estimates for the two-front SQG ( systems
- 7 A priori estimates for the two-front GSQG () systems
In this paper, we derive contour dynamics equations for the motion of two fronts in a class of piecewise constant solutions of the surface quasi-geostrophic (SQG) and generalized surface quasi-geostropic (GSQG) equations; these two-front solutions are described in more detail in Section 1.1.3 below. We also prove local existence and uniqueness theorems for the resulting front-equations.
The GSQG equations are a family of active scalar equations in two spatial dimensions, depending on a parameter , which arise naturally from fluid dynamics. They consist of a transport equation for a scalar function that is transported by a divergence-free velocity field which depends non-locally on :
Here, is the spatial variable, is the perpendicular gradient, and is a nonlocal operator defined by the Riesz potential in for [58, 59] or the logarithmic Newtonian potential for . Alternatively, we can introduce a stream function , and write
When , equation (1.1) is the vorticity-stream function formulation of the two-dimensional, incompressible Euler equation for an inviscid fluid, and the scalar is the vorticity . It has long been established that the 2D Euler equation has global smooth solutions [38, 64]. Further results on the 2D Euler equation can be found in [47, 49] and the references therein.
When , equation (1.1) is the (inviscid) SQG equation. This equation describes the motion of quasi-geostrophic flows confined near a surface [34, 45, 46, 52], and is usually referred to as the potential temperature or the surface buoyancy. From an analytical point of view, the SQG equation has many similar features to the 3D incompressible Euler equation [13, 14]. In particular, the scalar has the same dimensions as the velocity field that transports it.
The SQG equation has global weak solutions in -spaces () [48, 54], and convex integration shows that low-regularity weak solutions need not be unique . A class of nontrivial global smooth solutions is constructed in , but — as for the 3D incompressible Euler equations — the question of whether general smooth solutions of the SQG equation remain smooth for all time or form singularities in finite time is open.
The other cases in the family, with or , correspond to a natural generalization of the Euler and SQG equations. Local existence of smooth solutions of these equations is proved in , but the global existence of smooth solutions with general initial data is not known for any .
1.1. Patch and front solutions
Equation (1.1) has a class of piecewise constant solutions of the form
where is a positive integer, are constants, are domains with smooth boundaries such that can be described as a family of disjoint simple curves and
and denotes the indicator function of . The transport equation (1.1) preserves the form of these weak solutions, and to study their evolution, we only need to understand the dynamics of the boundaries of the regions .
Depending on the number of regions and the boundedness of each region, we distinguish the following three different types of solutions (see Figure 1.1). In this paper, we will be concerned with the third type of two-front solutions.
Equation (1.2) is a patch solution if it satisfies the following assumptions:
, but for each ;
for each , the region is bounded with smooth, simple, closed boundary that is diffeomorphic to the circle ;
the region is unbounded.
Under these assumptions, has compact support and contour dynamics equations for the motion of the patches are straightforward to derive. Patch solutions of the 2D Euler equation are usually called vortex patches. The 2D Euler equation has global weak solutions with vorticity in [47, 65], and smooth vortex patch boundaries remain smooth and non-self-intersecting for all times [3, 11, 12]. Some special types of nontrivial global-in-time smooth vortex patch solutions are constructed in [6, 8, 20, 35, 36, 37].
Local well-posedness of the contour dynamics equations for SQG and GSQG patches is proved in [10, 15, 29, 30]. The question of whether finite-time singularities can form in smooth boundaries of SQG or GSQG patches remains open, but it is proved in  that splash singularities cannot form, and some particular classes of nontrivial global solutions for SQG and GSQG patches have been shown to exist [7, 19, 32, 33, 36].
The local existence of smooth GSQG patches in the presence of a rigid boundary is shown in [30, 44] for a range of , and the formation of finite-time singularities is proved for a range of close to . By contrast, vortex patches in this setting (with ) have global regularity .
Numerical solutions for vortex patches show that they form extraordinarily thin, high-curvature filaments [22, 23], although their boundaries remain smooth globally in time. On the other hand, numerical solutions for SQG patches suggest that complex, self-similar singularities can form in the boundary of a single patch  and provide evidence that two separated SQG patches can touch in finite time .
Equation (1.2) is a front solution if it satisfies the following assumptions:
are distinct constants;
both and are unbounded and they share a boundary which is a simple, smooth curve diffeomorphic to .
When , the kernel of the (generalized) Biot-Savart law, which recovers the velocity field from the scalar , decays too slowly at infinity for the formal front equations to converge. This differentiates the patch problems and the front problems, since there are no convergence issues at infinity in the case of patches with compactly supported . Regularized equations for a single front that is a graph located at are derived in .
The front problem for vorticity discontinuities in the Euler equation is studied in [4, 53]. Local existence and uniqueness for spatially periodic SQG fronts is proved for solutions in  and analytic solutions in . Almost SQG sharp fronts are studied in [16, 24, 26, 27]. Smooth solutions for spatially periodic GSQG fronts with also exist locally in time .
In the non-periodic setting, smooth solutions to the GSQG front equations on with are shown to exist globally in time for small initial data in . When , a regularization procedure is needed in the derivation of the front equations to account for the divergence of the naive contour dynamics equations at infinity . Local well-posedness of a cubically nonlinear approximation of the SQG front equations is proved in , and the global well-posedness of an initial-value problem on for the full SQG front equation is proved in  for small, smooth initial data .
Equation (1.2) is a two-front solution if it satisfies the following assumptions:
with and ;
there is a diffeomorphism , satisfying , , and .
This case is the one we study here. We derive equations for the locations of the two fronts by use of the regularization method introduced in  and prove well-posedness results for the resulting systems. From now on, we write , , , with the same subscript changes applying to , , . We also define the jumps in across the fronts, scaled by a convenient factor given in (3.2), by
Numerical solutions of the contour dynamics equations for spatially-periodic two-front solutions of the Euler equation and a study of the approximation of vortex sheets by a thin vortex layer are given in .
1.2. Main results
We consider two-front solutions whose fronts are graphs located at
where denote the perturbations from the flat fronts , , and . We also write
As we will see, there are different features for , , and , which are a consequence of a loss of local integrability in the restriction of the potential for to the front for , leading to an infinite tangential velocity on the front, and a loss of global integrability for . The nonlinear terms in the front equations also behave differently, losing derivatives if , and having good, hyperbolic-type energy estimates if (see Table 1.1).
|Global Integrability||Tangential Velocity||Derivative Loss|
|1||No111The two-front system is globally integrable only if .||Unbounded||Yes|
|(1,2]||No111The two-front system is globally integrable only if .||Bounded||No|
The equations describing the dynamics of the fronts are given by (3.8) for Euler, (3.9) for SQG, (3.10) for GSQG with , and (3.11) for GSQG with . Symmetric and anti-symmetric scalar reductions of these equations are given in (3.12) and (3.13), respectively.
1.2.1. Local well-posedness
We briefly summarize our local well-posedness results for the front equations. As explained further in Section 2, we use to denote a Weyl paraproduct with symbol .
Theorem 1.1 ().
Let be an integer. Suppose that satisfy
Then there exists , depending only on , , , , and , such that the initial value problem of the system (3.10) has a unique solution with .
Theorem 1.2 ().
The smallness conditions on the paraproducts in the preceding theorems arise from the fact that the nonlinear terms in the front equations lose derivatives, so we need to use the linear terms to control them, which can only be done if the solutions are sufficiently small. The case is simpler than , since the nonlinear terms do not lose derivatives.
Theorem 1.3 ().
Let be an integer. Suppose that . Then there exists , depending only on , , such that the initial value problem of the system (3.11) has a unique solution with .
1.3. Outline of the paper
In Section 2, we provide the definitions and some properties of the Weyl paradifferential calculus and modified Bessel function of the second kind. In Section 3, we derive the two-front equations. In Section 4, we analyze the linearized stability of the unperturbed, flat two-front solutions, which is a particular example of a GSQG shear flow, and in Sections 5–7, we prove the local existence and uniqueness theorems for the front equations.
JKH was supported by the NSF under grant number DMS-1616988. JS would like to thank Javier Gómez-Serrano for discussions in the “MathFluids” Workshop held in Mathematical Institute of University of Seville, Seville, Spain, June 12–15, 2018.
2.1. Para-differential calculus
In this section, we state several lemmas for the Fourier multiplier operators
We denote the Fourier transform of by , where is given by
For , we denote by the space of Schwartz distributions with , where
Throughout this paper, we use to mean there is a constant such that , and to mean there is a constant such that . We use to mean that and . The notation denotes a term satisfying whenever there exists such that , and denote a term satisfying pointwise.
Let be a smooth function supported in the interval and equal to on . If is a Schwartz distribution on and is a symbol, then we define the Weyl para-product by
where denotes the partial Fourier transform of with respect to . For , we define a normed symbol space by
If and , with , then and
In particular, if is real-valued, then is a self-adjoint, bounded linear operator on .
Let and , then
where is the Poisson bracket between and , and is a remainder term satisfies estimates
As a consequence, we have
where also satisfies estimates (2.2).
Finally, we state an expansion of acting on para-products whose proof can be found in .
If , and , then
where means that the differential operator acts on the function for fixed , and similarly for and .
2.2. Modified Bessel function of the second kind
The modified Bessel functions of the second kind are defined for by
and for . When and , we can also write as
In (2.3), and throughout this paper, denotes the Gamma function.
The following lemma collects the properties of modified Bessel functions of the second kind that we need; proofs can be found in .
The modified Bessel functions of the second kind have following properties:
For each , is a real-valued, analytic, strictly decreasing function on .
For each fixed , .
When is fixed and ,
When is fixed and ,
For an integer , let be defined as
Then attains its maximum. Let the maximum be attained at , then
Proof of (v).
It follows from (ii) that we only need to consider . We use identities (see )
When , for fixed , , thus is decreasing in , and its maximum is attained at with
When , it is clear that is smooth in with
then the maximum is attained at its critical points. Therefore, must satisfy
3. Regularized two-front GSQG systems
3.1. Contour dynamics
Following the approach introduced by Zabusky et. al.  for the Euler equation, we will derive contour dynamics equations for two-front solutions of the Euler, SQG, and GSQG equations.
Using the GSQG equation (1.1) and Green’s theorem, we find that the velocity field is
where the Green’s function for the operator on is given by with
Here, the jumps are defined in (1.3), is the upward unit normal to , , and is arc-length on .
The integrals in (3.1) converges when or for . In other cases, we will use a regularization procedure adopted in  to make sense of the equations. In all cases, we first cut-off the integration region to a -interval about some point and then consider the limit . We assume that both the top and bottom fronts asymptotically converge to and as , respectively, and that the two curves are one-to-one and do not intersect each other.
Let the top and bottom fronts have parametric equations and , where
Since is transported by the velocity field, the fronts move with normal velocity
so the cut-off equations for and are
where and are arbitrary functions corresponding to time-dependent reparametrizations of the fronts.
If the fronts are given by graphs that are perturbations of and , then the top front is located at and the bottom front at , and we can solve for and to get
We then obtain a coupled system for and
3.2. Cut-off regularization
Notice that for fixed , and when is given by (3.2) we have for ,
We only write out the computation for the first equation; the computation for the second equation is similar. The linear term
can be written as 
and as , where is the Fourier multiplier operator with symbol
Here, is the Euler–Mascheroni constant .
As for the second nonlinear term, we have
where is a divergent part (or zero if converges) and is a convergent part
and as , where is the Fourier multiplier with symbol
for any , which gives
We further denote by the limit
The cut-off system (3.4) can then be written as
In the limit , the possibly problematic terms in these equations are and . The only case when these two terms converge to finite limits are when or . Otherwise, we regularize the equations by choosing suitable Galilean transformations. Indeed, if we choose
and make a Galilean transformation , then the system becomes
The asymptotic behavior of and as is given by
Therefore, converges as , and we define
Putting everything together and letting , we get the regularized system in conservative form
One can also take the derivatives inside the integral to obtain the non-conservative form