Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators

. In this paper we treat the following partial differential equation, the quasigeostrophic equation: (


Introduction
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 [5] the well-posedness of quasigeostrophic equation was treated on the sphere, on general riemannian manifolds in [6] or the 2D stochastic quasigeostrophic equation on the torus T 2 in [7].This equation is also denominated as advection-fractional diffusion; see for example [8], or it may be classified as a fractional Fokker-Planck equation [9].However we follow the usual terminology of quasigeostrophic equation which has appeared in our main references [1][2][3][4][5][6][7].
Here we replace the Laplacian operator Δ for an arbitrary infinitesimal generator (, ()) of a convolution  0semigroup of positive kernel on Lebesgue spaces   (R  ), with 1 ≤  < ∞.The abstract framework of  0 -semigroups of linear bounded operators in Banach spaces was introduced by Hille and Yosida in the last fifties; see for example the monographies [10][11][12][13].Some classical  0 -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 Δ 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 of Some 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  = 0 in (2), we recover the classical Cauchy problem for the fractional power −(−)  . 2

Journal of Function Spaces
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  0semigroups on the Lebesgue space   (R  ) (Theorem 1).From such pointwise inequalities, and assuming convolution kernels of real symbol, one gets integral inequalities (Theorem 4 and Lemma 6), which extend [ for some suitable solutions  ∈ S(R  ) and nonnegative functions , see Theorem 8. To prove that, we use some techniques which are based in [15].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  0 -semigroups to Poisson or Gaussian semigroup.This approach is inspirated in [15].Preliminary versions of these results were included in [17].
In the following, (−)  denotes the fractional powers of the infinitesimal generator of these semigroups; see [13, p. 264]: for all  ∈ () and 0 <  < 1.Our first result gives a pointwise inequality for these fractional powers.The main ingredient is to represent the  0 -semigroup (()) >0 in terms of the positive kernel functions.Compare with [3, Theorem 1] and [4, Proposition 2.3] in the case of  = Δ.
Proof.We use equality (13), almost everywhere  ∈ R  , and Note that since Γ(−) < 0 if 0 <  < 1, and then If  = 0, it is trivial, and for  = 1 we use the definition of the infinitesimal generator.
Given  ∈  1 (R  ), the usual Fourier transform is given by and then f ∈  0 (R  ).Let K = (()) ≥0 be a convolution  0semigroup of positive kernel on   (R  ), with kernel is a contractive multiplication  0 -semigroup.We obtain the following result as a consequence of [10, p. 28].
Proof.We apply equation ( 14) to get with  ∈ N 0 .Taking  =  − 1, then for 0 ≤  ≤ 1 the following inequality holds: On the other hand, for 0 <  < 1 Therefore Note that, for  = 0, the previous equality is trivial, and, for  = 1, 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 () = −4 2 || 2 .
Note that all these examples provide kernels and functions  which depend on the norm ||.
Let  be a solution of the following: where 0 ≤  ≤ 1 and  satisfies either ∇ ⋅  = 0 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 [16] 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) On the one hand, we suppose that  satisfies that ∇ ⋅  = 0.
On the other hand, we suppose that   =   () with   ∈ S(R  ) and 1 ≤  ≤ .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 0 ≤  ≤ 1 we have Proof.For 0 <  < 1, a change of variables yields Then, we obtain since For  = 0 and  = 1 the above inequality is easily checked.
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 have for  = 2  with  ∈ N,  ∈ S(R  ) real-valued solution of (28), and  a continuous, nonnegative and nondecreasing function.
The operator (−)  is a nonnegative and symmetric operator, which satisfies a Super-Poincaré inequality with rate function , then by [15,Proposition 2.2] and therefore inequality (39) follows from (37).

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 [19].
Given an N-function , we define the function () fl ∫  0 ()  for  > 0 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 ℎ and ℎ * , defined by ℎ() fl (1/) for  > 0, and ℎ * () fl () for  > 0, are also complementary N-functions.