Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators
In this paper we treat the following partial differential equation, the quasigeostrophic equation: , where is the infinitesimal generator of a convolution -semigroup of positive kernel on with Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers for . We use these estimates to obtain -decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (taking , the Laplacian), which has been studied in the literature.
In oceanography and meteorology, the quasigeostrophic equation, where represents the temperature, the velocity, and the viscosity constant, has a great importance (see for example [1, 2]). In the last years, a large number of mathematical papers are dedicated to this equation. For example, in [3, 4], A. Córdoba and D. Córdoba studied regularity and -decay for solutions. In  the well-posedness of quasigeostrophic equation was treated on the sphere, on general riemannian manifolds in  or the 2D stochastic quasigeostrophic equation on the torus in .
This equation is also denominated as advection-fractional diffusion; see for example , or it may be classified as a fractional Fokker-Planck equation . However we follow the usual terminology of quasigeostrophic equation which has appeared in our main references [1–7].
Here we replace the Laplacian operator for an arbitrary infinitesimal generator of a convolution -semigroup of positive kernel on Lebesgue spaces , with The abstract framework of -semigroups of linear bounded operators in Banach spaces was introduced by Hille and Yosida in the last fifties; see for example the monographies [10–13]. Some classical -semigroups, as Gaussian, Poisson, fractional, or the backward semigroups in classical Lebesgue spaces, fit in this approach; see for example [12, Chapter 2]. Note that in particular the Laplacian generates the Gaussian (also called heat or diffusion) semigroup [10, Chapter II, Section 2.13].
The main aim of this paper is to show the decreasing behavior for suitable solutions ofSome classical asymptotic behavior of solutions of abstract Cauchy problem,is presented in [11, Section 4.4] and for parabolic case of evolution systems in [11, Section 5.8]. Note that for in (2), we recover the classical Cauchy problem for the fractional power .
We emphasize the key role played by the Balakrishnan integral representation of fractional powers [13, p. 264] in order to get the following pointwise inequalities:for certain infinitesimal generators of convolution -semigroups on the Lebesgue space (Theorem 1). From such pointwise inequalities, and assuming convolution kernels of real symbol, one gets integral inequalities (Theorem 4 and Lemma 6), which extend [3, Lemma 1] and [4, Lemma 2.4, Lemma 2.5], respectively. For this purpose, we use Fourier transform, obtaining multiplications semigroups from convolution ones. Interesting similar pointwise inequalities have been discussed in .
The previous results allow getting a maximum principle for the solutions of (2), see Corollary 7. Moreover, one of the most important results along this paper is to estimate the decreasing behavior, for some suitable solutions and nonnegative functions , see Theorem 8. To prove that, we use some techniques which are based in . In that paper some equivalence between Super-Poincaré and Nash-type inequalities is shown for nonnegative self-adjoint operators. Some of these results were proved in the case of fractional powers of the Laplacian in [3, 4, 16].
In the last section, we apply our results to check estimations about the -decay of some solutions in concrete quasigeostrophic equations. Our main example is to consider subordinated -semigroups to Poisson or Gaussian semigroup. This approach is inspirated in . Preliminary versions of these results were included in .
Notation. Through this article with is the usual Lebesgue space and is the Banach algebra whereThe space is formed by the continuous functions such that , and ; the set is the Schwartz space and is the Gamma function.
2. Pointwise and Integral Estimates for Fractional Powers
Let be a one-parameter continuous semigroup in the Banach algebra ; i.e., for ; when for any and such that for ; see for example [12, Chapter 1]. Then the one-parameter family of linear bounded operators , defined by is a convolution -semigroup on , with Recall that the infinitesimal generator of is defined by that is, the domain of the operator is the closed and densely defined subspace where the above limit exists on , see for example [10, Definition 1.2]. Note that these -semigroups are contractive since for all We also assume that is a positive kernel. Below, there are several examples of convolution -semigroups of positive kernel: (1)The Gaussian kernel, , whose generator is the Laplacian operator ([12, Theorem 2.15]).(2)The Poisson kernel, , whose infinitesimal generator is ([12, Theorem 2.17]).(3)Subordinated semigroups in . In , new convolution -semigroups are defined by subordination principle, i.e., using the bounded algebra homomorphism , with where is an uniformly bounded continuous semigroup on ; in particular or for . Now, we take the fractionary semigroup on , with , and new type kernels are obtained by see additional details in [18, Theorem 2.1, Corollary 2.2].
In the following, denotes the fractional powers of the infinitesimal generator of these semigroups; see [13, p. 264]:for all and Our first result gives a pointwise inequality for these fractional powers. The main ingredient is to represent the -semigroup in terms of the positive kernel functions. Compare with [3, Theorem 1] and [4, Proposition 2.3] in the case of .
Theorem 1. Let be the infinitesimal generator of a -semigroup as above. Then, for all real-valued with and , the inequalityholds.
Proof. We use equality (13), almost everywhere , and to get Note thatsince if , and then If , it is trivial, and for we use the definition of the infinitesimal generator.
Given , the usual Fourier transform is given by and then Let be a convolution -semigroup of positive kernel on , with kernel Note that , with Then, it is well known that withis a contractive multiplication -semigroup. We obtain the following result as a consequence of [10, p. 28].
Proposition 2. Let be a -semigroup as above. Then there is a continuous function with for all , such that for and is the infinitesimal generator of , with and
Definition 3. We say that a convolution -semigroup of positive kernel on , , is of real symbol when the infinitesimal generator of the semigroup is a real function; i.e., .
Theorem 4. Let be a convolution -semigroup of positive kernel and real symbol on , with kernel , and infinitesimal generator satisfying and for all , If is a real function, thenfor and with positive integer.
Proof. We apply equation (14) to get with . Taking , then for the following inequality holds: On the other hand, for Therefore Note that, for , the previous equality is trivial, and, for , it is well known. Finally, by Plancherel and Parseval theorems for Fourier Transform, we obtain and then Then we conclude the proof.
In the conditions of the previous theorem, we give the following examples where also the function is identified:(1)For the Gaussian semigroup (2)For the Poisson semigroup (3)For the subordination semigroups defined in , using the Gaussian kernel and using the Poisson kernel.
Note that all these examples provide kernels and functions which depend on the norm .
3. -Decay of Solutions of Quasigeostrophic Equation
Let be the infinitesimal generator of a convolution -semigroup of positive and radius dependent kernel of real symbol on , with , and the fractional power defined by (13) for .
Let be a solution of the following:where and satisfies either or , together with the necessary conditions about regularity and decay at infinity. Existence results on for (28) with smooth initial conditions have been studied in  using a functional approach. Note that we use several notations , , through this section.
We want to study the decline in time of the spatial -norm solutions of (28), and, to do this, we will work with its derivatives, as the following lemma shows. Although the next lemma is known, we include it for the sake of completeness.
Lemma 5. Let be under the above conditions and be a solution of (28). If the function satisfies that or with for , then
Proof. Note thatOn the one hand, we suppose that satisfies that . Then where we have integrated by parts, and .
On the other hand, we suppose that with and . Similarly,
The following positivity lemma is a natural extension of [4, Lemma 2.5].
Lemma 6. Let be under the above conditions. Then for all and we have
Proof. For , a change of variables yields Then, we obtain since for all For and the above inequality is easily checked.
The previous lemma implies the following maximum principle, which completes similar approaches; see for example [4, Corollary 2.6] and [16, Theorem 1,2].
Corollary 7 (maximum principle). Let be a smooth solution of (28). Then for we have for all
Proof. It is a trivial consequence of Lemma 5 and (33).
From now on, we focus on study the decay of . Applying Theorem 4, we havefor with positive integer. For we have , then solving this differential inequality we obtain Below we see what happens to the case
Theorem 8. Assuming that the symbol is an increasing function in the radius, with , thenfor with , real-valued solution of (28), and a continuous, nonnegative and nondecreasing function.
Proof. For , we consider the bijectionwith inverse function Thus for all and one gets where denotes the usual Lebesgue measure on .
Note that is a bijection from to itself. So where is the measure of the unit sphere in We define , for , and we rewrite where is a nonnegative and decreasing function.
The operator is a nonnegative and symmetric operator, which satisfies a Super-Poincaré inequality with rate function , then by [15, Proposition 2.2] this is equivalent to a Nash-type inequality with rate function Note that the function is continuous, nonnegative and nondecreasing. So, applying this argument to , with , we obtain and therefore inequality (39) follows from (37).
4. Examples and Applications
In this last section, we check the -decay of solutions in some concrete examples of quasigeostrophic equations. This approach illustrates our results. To do that, we need to calculate the function (and also the function ) which appears in Theorem 8 for concrete examples. In [15, section 8], general properties of functions and are studied using N-functions; see also .
Let be a right continuous, monotone increasing function with(1);(2);(3) whenever ;
then, the function defined by for is called an N-function. Alternatively, the function is an N-function if and only if is continuous, even and convex with(1);(2);(3) if .
Given an N-function , we define the function for where is the right inverse of the right derivative of , . The function is an N-function called the complement of . Furthermore it is straightforward to check that the complement of is .
Now suppose that functions and are complementary N-functions. Then functions and , defined by for , and for , are also complementary N-functions.(1)We consider the Laplace operator and , see Section 2. Then where is the measure of the unit sphere in Now we have a couple of N-functions, and , with and a positive constant; see [15, Section 8]. Then , and we get for Solving this differential inequality, one obtains with and (2)For the subordinated semigroup through Poisson semigroup with , we get that with Then , with so Note that for . According to [15, Section 8], we consider , with If we take , then , and so Therefore with a positive constant. Now we apply Theorem 8 in the case of with . If we suppose that the solution of (28) is stable; i.e., , then for large enough, where we have used that , as with . We conclude that for , and large enough.
For other and , we obtain the decayment by interpolation property: if , with and , then When , we have for any integer , and if we also use that
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors have been partially supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS and Project E26-17R, D.G. Aragón, Universidad de Zaragoza, Spain.
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