Note on the weak–strong uniqueness criterion for the β-QG in Morrey–Campanato space
Introduction
We consider the following two-dimensional β-generalized quasi-geostrophic equations (see e.g. [24], [25]): where θ(x, t) is a scalar function representing the temperature, u(x, t) is the velocity field of the fluid with and determined by the Riesz transforms of the potential temperature θ: where is the Zygmund operator and are the usual Riesz transforms in . Here the Riesz potential operator is defined through the Fourier transform [20]: where denotes the Fourier transform of f. Here 0 < α ≤ 2 and 1 ≤ β < 2 are two fixed parameters.
The β-generalized surface quasi-geostrophic Eqs. (1.1) and (1.2) was introduced by Kiselev [11] which is deeply related to the important model in geophysical fluid dynamics used in meteorology and oceanography, see [11], [25] and the references therein.
When the surface quasi-geostrophic Eqs. (1.1) and (1.2) has been intensively investigated due to both its mathematical importance and its potential for applications as models of atmospheric and ocean fluid flow. For more details about its physical background, we can refer to Constantin et al. [2], Pedlosky [19] and references therein. The surface quasi-geostrophic equation with subcritical (1 < α ≤ 2) or critical dissipation have been shown to possess global classical solutions whenever the initial data are sufficiently smooth. However, the global existence of smooth and uniqueness issue remains open for the supercritical case (0 < α < 1). Various regularity (or blow-up) criteria have been produced to shed light on this difficult global regularity problem (see e.g. [1], [3], [4], [12] and the references therein). This work is partially motivated by the recent progress on the 2D incompressible generalized MHD system, we refer to [7], [28] and the references therein. The above works were subsequently extended to the case 1 ≤ β < 2 by [3], [13], [17], [21], [22], [23], [24], [26], [27].
However, the uniqueness of such weak solution for the 2D quasi-geostrophic equation is still open. When 1 < α ≤ 2, Constantin and Wu [4] showed the following weak–strong uniqueness results: if a weak solution lies in the regular class then there is at most one solution to the quasi-geostrophic equation with the initial data . That is to say, all the weak solutions with the same initial data θ0 coincide with the strong solution. Later on, Dong and Chen [5] more or less established the weak–strong uniqueness for the critical and supercritical dissipative quasi-geostrophic equations in the regularity class
Recently, Dong and Chen [6] have refined the above conditions to the following uniqueness criterion in the framework of Besov spaces: the weak–strong solutions are unique in the class In [16], Marchand has proved the weak–strong uniqueness for the critical case () when one of the two weak solutions θ satisfies and the norm is small enough, where BMO is the space of bounded mean oscillations. Recently, for 0 < α < 2, Liu et al. [15] have refined the above result in the following sense
It should be mentioned that the Serrin’s (1.3) condition implies a connection of regularity criteria of weak solutions and a scaling invariance property, that is, solves (1.1) and (1.2) if and only if solves (1.3) and the scaling invariance holds true for all λ > 0 if and only if r and p satisfy the Serrin’s condition (1.3). Due to its similarity with 3D incompressible Navier–Stokes/Euler equations, we consider the case as the subcritical case, the case as the critical case, and the case as the supercritical case. While the regularity and uniqueness of global weak solutions of the Navier–Stokes/Euler equations remains an outstanding open problem in mathematical physics, the global well-posedness of Eqs. (1.1) and (1.2) seems to be in a satisfactory situation in the subcritical and critical cases. Recently, the global existence of smooth solutions of Eqs. (1.1) and (1.2) with suitable choices of α and β has been studied by many authors (cf. [22]), the uniqueness of weak solution in the critical case and supercritical case is a rather challenging problem in mathematical fluid mechanics.
The aim of this paper to improve and extend the above results in the following sense that if θ is a weak solution toEqs. (1.1) and (1.2) and if then, θ is unique in the class of weak solutions. Here stands for the homogeneous Morrey–Campanato space. We point out here that the Morrey–Campanato spaces have been studied by Lemarié–Rieusset and co-workers [14]. They are useful tools for stating minimal regularity requirements on the coefficients of partial differential operators for proving regularity or uniqueness of solutions. Since the following embedding relation with 1 < q < p < ∞ holds, our uniqueness criterion can be understood as an extension of the uniqueness result of Zhao–Liu [25]. It is a natural way to extend the space widely and improve the previous results. However, the spaces and are different and no inclusion relation between them.
Section snippets
Morrey–Campanato spaces and main result
For the statement of our main result we introduce a function space, used previously in [8], [9], [14]. This kind of spaces play an important role in studying the regularity of solutions to partial differential equations.
Definition 2.1 For the homogeneous Morrey–Campanato space is defined by:
where B(x, R) denotes the closed ball in with center x with radius R.
Those spaces were introduced by Morrey [18]. One can find that
Acknowledgments
This project was supported by King Saud University, Deanship of Scientific Research, College of Sciences Research Center. The authors would like to express gratitude to reviewers for careful reading of the manuscript, many valuable comments and suggestions for its improvement.
References (28)
Weak–strong uniqueness criteria for the critical quasi-geostrophic equation
Phys. D
(2008)- et al.
Global wellposedness for a modified critical dissipative quasi-geostrophic equation
J. Differ. Equat.
(2012) On the solutions of quasi-linear elliptic partial differential equations
Trans. Amer. Math. Soc
(1938)- K. Yamazaki, A remark on the global well-posedness of a modified critical quasi-geostrophic equation,...
Regularity criteria of supercritical beta-generalized quasi-geostrophic equation in terms of partial derivatives
Electron. J. Differ. Equat.
(2013)- et al.
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation
Ann. Math.
(2010) - et al.
Formation of strong fronts in the 2d quasi-geostrophic thermal active scalar
Nonlinearity
(1994) - et al.
Global regularity for a modified critical dissipative quasi-geostrophic equation
Indiana Univ. Math. J.
(2011) - et al.
Behavior of solutions of 2d quasi-geostrophic equations
SIAM J. Math. Anal.
(1999) - et al.
Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation
Nonlinearity
(2006)
On the weak-strong uniqueness of the dissipative surface quasi-geostrophic equation
Nonlinearity
Global cauchy problem of 2d generalized MHD equations
Monatsh. Math.
Regularity criteria for the 3d magneto-micropolar fluid equations in the morrey-campanato space
Nonlinear Differ. Equat. Appl.
Remark on a regularity criterion in terms of pressure for the navier-stokes equations
Quart. Appl. Math.
Cited by (3)
Remarks on global regularity for the 3D MHD system with damping
2018, Applied Mathematics and ComputationGlobal existence and asymptotic behavior for the 3D generalized Hall-MHD system
2017, Nonlinear Analysis, Theory, Methods and ApplicationsA regularity criterion for the three-dimensional micropolar fluid system in homogeneous Besov spaces
2016, Electronic Journal of Qualitative Theory of Differential Equations