In this paper we treat the following partial differential equation, the quasigeostrophic equation: ∂/∂t+u·∇f=-σ-Aαf,0≤α≤1, where (A,D(A)) is the infinitesimal generator of a convolution C0-semigroup of positive kernel on Lp(Rn), with 1≤p<∞. Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers (-A)α for 0≤α≤1. We use these estimates to obtain Lp-decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (taking A=Δ, the Laplacian), which has been studied in the literature.

MCYTSMTM2016-77710-PDGI-FEDERUniversidad de ZaragozaE26-17R1. Introduction

In oceanography and meteorology, the quasigeostrophic equation, (1)∂∂t+u·∇f=-σ-Δαf,for 0≤α≤1,where f represents the temperature, u the velocity, and σ 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 Lp-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 T2 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 [1–7].

Here we replace the Laplacian operator Δ for an arbitrary infinitesimal generator (A,D(A)) of a convolution C0-semigroup of positive kernel on Lebesgue spaces Lp(Rn), with 1≤p<∞. The abstract framework of C0-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 C0-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(2)∂∂t+u·∇f=-σ-Aαf,0≤α≤1.Some classical asymptotic behavior of solutions of abstract Cauchy problem,(3)∂∂tf=Af,f∈DA,is presented in [11, Section 4.4] and for parabolic case of evolution systems in [11, Section 5.8]. Note that for u=0 in (2), we recover the classical Cauchy problem for the fractional power -σ(-A)α.

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:(4)fx-Aαfx≥12-Aαf2xa.e,for certain infinitesimal generators of convolution C0-semigroups on the Lebesgue space Lp(Rn) (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 [14].

The previous results allow getting a maximum principle for the solutions of (2), (5)f·,tp≤f·,0p,t≥0,see Corollary 7. Moreover, one of the most important results along this paper is to estimate the decreasing behavior,(6)ddtfpp≤-σfppDfpp, for some suitable solutions f∈S(Rn) and nonnegative functions D, 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 Lp-decay of some solutions in concrete quasigeostrophic equations. Our main example is to consider subordinated C0-semigroups to Poisson or Gaussian semigroup. This approach is inspirated in [15]. Preliminary versions of these results were included in [17].

Notation. Through this article (Lp(Rn),p) with 1≤p≤∞ is the usual Lebesgue space and (L1(Rn),1,∗) is the Banach algebra where(7)f∗gx≔∫Rnfx-ygydy,x∈Rn.The space C0(Rn) is formed by the continuous functions f such that limx→∞f(x)=0, and f∞≔maxx∈Rnfx; the set S(Rn) is the Schwartz space and Γ is the Gamma function.

2. Pointwise and Integral Estimates for Fractional Powers

Let (kt)t>0⊂L1(Rn) be a one-parameter continuous semigroup in the Banach algebra L1(Rn); i.e., kt∗ks=kt+s for t,s>0; kt∗f→f when t→0 for any f∈L1(Rn) and such that kt1=1 for t>0; see for example [12, Chapter 1]. Then the one-parameter family of linear bounded operators K=(K(t))t≥0, defined by(8)Ktf≔kt∗f,f∈LpRn,t>0;K0=I, is a convolution C0-semigroup on Lp(Rn), with 1≤p<∞. Recall that the infinitesimal generator (A,D(A)) of K is defined by (9)Af=limt→0+kt∗f-ft,f∈DA, that is, the domain of the operator A is the closed and densely defined subspace where the above limit exists on Lp(Rn), see for example [10, Definition 1.2]. Note that these C0-semigroups (K(t))t≥0 are contractive since kt1=1 for all t>0. We also assume that (kt)t>0 is a positive kernel. Below, there are several examples of convolution C0-semigroups of positive kernel:

The Gaussian kernel, gt(x)=(4πt)-n/2e-x2/4t, whose generator is the Laplacian operator Δ ([12, Theorem 2.15]).

The Poisson kernel, pt(x)=Γn+1/2/πn+1/2t/(t2+x2)n+1/2, whose infinitesimal generator is --Δ ([12, Theorem 2.17]).

Subordinated semigroups in L1(Rn). In [18], new convolution C0-semigroups are defined by subordination principle, i.e., using the bounded algebra homomorphism Θa:L1(R+)→L1(R), with (10)Θaf=∫0∞ftatdt,f∈L1R+,

where a=(at)t>0 is an uniformly bounded continuous semigroup on L1(R); in particular at=gt or pt for t>0. Now, we take the fractionary semigroup on L1(R+), (11)Ist≔ts-1Γse-t,t>0,

with s>0, and new type kernels are obtained by (12)ΘaIsx=∫0∞Istatxdt=∫0∞ts-1Γse-tatxdt,x∈Rn,

see additional details in [18, Theorem 2.1, Corollary 2.2].

In the following, (-A)α denotes the fractional powers of the infinitesimal generator of these semigroups; see [13, p. 264]:(13)-Aαf=Γ-α-1∫0∞t-α-1Kt-Ifdt,for all f∈D(A) and 0<α<1. Our first result gives a pointwise inequality for these fractional powers. The main ingredient is to represent the C0-semigroup (K(t))t>0 in terms of the positive kernel functions. Compare with [3, Theorem 1] and [4, Proposition 2.3] in the case of A=Δ.

Theorem 1.

Let (A,D(A)) be the infinitesimal generator of a C0-semigroup (K(t))t≥0 as above. Then, for all f∈D(A) real-valued with f2∈D(A) and 0≤α≤1, the inequality(14)fx-Aαfx≥12-Aαf2xa.eholds.

Proof.

We use equality (13), almost everywhere x∈Rn, and 0<α<1 to get (15)fx-Aαfx=Γ-α-1∫0∞t-α-1∫Rnktx-rfrfxdr-f2xdt=Γ-α-1∫0∞t-α-1∫Rnktx-rfrfx-f2xdrdt.Note that(16)-f2x-frfx=-12fx-fr2+12f2x-f2r≤-12f2x-f2r,since Γ(-α)<0 if 0<α<1, and then (17)fx-Aαfx≥Γ-α-12∫0∞t-α-1∫Rnktx-rf2r-f2xdrdt=12-Aαf2xa.e. If α=0, it is trivial, and for α=1 we use the definition of the infinitesimal generator.

Given f∈L1(Rn), the usual Fourier transform is given by (18)f^η≔∫Rnfxe-2πiη·xdx,η∈Rn, and then f^∈C0(Rn). Let K=(K(t))t≥0 be a convolution C0-semigroup of positive kernel on Lp(Rn), with kernel (kt)t>0. Note that kt^∈C0(Rn), with kt^∞≤kt1=1. Then, it is well known that TK≪(T(t))t≥0 with(19)Tt: C0Rn→C0Rng↦kt^g,t>0,is a contractive multiplication C0-semigroup. We obtain the following result as a consequence of [10, p. 28].

Proposition 2.

Let K=(K(t))t≥0 be a C0-semigroup as above. Then there is a q:Rn→C continuous function with Re(q(x))≤0 for all x∈Rn, such that kt^=etq for t>0 and (B,D(B)) is the infinitesimal generator of TK, with B=qI and (20)DB=f∈C0Rn∣qf∈C0Rn.

Definition 3.

We say that a convolution C0-semigroup of positive kernel on Lp(Rn), K=(K(t))t≥0, is of real symbol when the infinitesimal generator of the semigroup TK is a real function; i.e., q:Rn→(-∞,0].

Theorem 4.

Let K=(K(t))t≥0 be a convolution C0-semigroup of positive kernel and real symbol on Lp(Rn), with kernel (kt)t>0, and infinitesimal generator (A,D(A)) satisfying S(Rn)⊂D(A) and for all h∈S(Rn), qh^∈L2(Rn). If f∈S(Rn) is a real function, then(21)∫Rnfxp-2fx-Aαfxdx≥1p∫Rn-Aα/2fp/2x2dx,for 0≤α≤1 and p=2j with j positive integer.

Proof.

We apply equation (14) to get (22)∫Rnfxp-2fx-Aαfxdx≥12∫Rnfxp-2-Aαf2xdx≥12l∫Rnfxp-2l-Aαf2lxdx, with l∈N0. Taking l=j-1, then for 0≤α≤1 the following inequality holds: (23)∫Rnfxp-2fx-Aαfxdx≥2p∫Rnfxp/2-Aαfp/2xdx. On the other hand, for 0<α<1(24)-Aαf^η=∫Rne-2πiη·xΓ-α-1∫0∞t-α-1kt∗fx-fxdtdx=Γ-α-1∫0∞t-α-1∫Rne-2πiη·xkt∗fx-fxdxdt=Γ-α-1∫0∞t-α-1kt∗f^η-f^ηdt.Therefore (25)-Aαf^η=-Bαf^η=-qηαf^η∈L2Rn∩C0Rn. Note that, for α=0, the previous equality is trivial, and, for α=1, it is well known. Finally, by Plancherel and Parseval theorems for Fourier Transform, we obtain (26)∫Rn-Aα/2fp/2x2dx=∫Rnfp/2x-Aαfp/2xdx, and then (27)∫Rnfxp-2fx-Aαfxdx≥1p∫Rn-Aα/2fp/2x2dx. Then we conclude the proof.

In the conditions of the previous theorem, we give the following examples where also the function q is identified:

For the Gaussian semigroup q(x)=-4π2x2.

For the Poisson semigroup q(x)=-2πx.

For the subordination semigroups defined in [18], q(x)=-log(1+4π2x2) using the Gaussian kernel and q(x)=-log(1+2πx) using the Poisson kernel.

Note that all these examples provide kernels and functions q which depend on the norm x.

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</mml:math></inline-formula>-Decay of Solutions of Quasigeostrophic Equation

Let (A,D(A)) be the infinitesimal generator of a convolution C0-semigroup of positive and radius dependent kernel of real symbol on Lp(Rn), with 1≤p<∞, and (-A)α the fractional power defined by (13) for 0<α<1.

Let f be a solution of the following:(28)∂∂t+u·∇f=-σ-Aαf,where 0≤α≤1 and u satisfies either ∇·u=0 or ui=Gi(f), together with the necessary conditions about regularity and decay at infinity. Existence results on Lp for (28) with smooth initial conditions have been studied in [16] using a functional approach. Note that we use several notations f, f(x,t), f(·,t) through this section.

We want to study the decline in time of the spatial Lp-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 (A,D(A)) be under the above conditions and f be a solution of (28). If the function u satisfies that ∇·u=0 or ui=Gi(f) with Gi∈S(Rn) for 1≤i≤n, then(29)ddtfpp=-σp∫Rnfp-2f-Aαfdx.

Proof.

Note that(30)ddtfpp=p∫Rnfp-2f∂f∂tdx=p∫Rnfp-2f-u·∇f-σ-Aαfdx.On the one hand, we suppose that u satisfies that ∇·u=0. Then (31)∫Rnfp-2fu·∇fdx=∫Rn∑j=1nfp-2f∂f∂xjujdx=∑j=1n∫Rn-1∫Rfp-1∂f∂xjujdxjdx^=-∑j=1n∫Rn-1∫Rfpp∂uj∂xjdxjdx^=-∫Rnfpp∇·udx=0, where we have integrated by parts, and dx^=dx1dx2…dxj-1dxj+1…dxn.

On the other hand, we suppose that ui=Gi(f) with Gi∈S(Rn) and 1≤i≤n. Similarly, (32)∫Rnfp-2fu·∇fdx=∫Rn∑j=1nfp-2fGjf∂f∂xjdx=∑j=1n∫Rn-1fp-2fGjf22-∞∞dx^=0.

The following positivity lemma is a natural extension of [4, Lemma 2.5].

Lemma 6.

Let (A,D(A)) be under the above conditions. Then for all f∈D(A) and 0≤α≤1 we have(33)∫Rnfp-2f-Aαfdx≥0.

Proof.

For 0<α<1, a change of variables yields (34)∫Rnfp-2f-Aαfdx=∫0∞t-α-1Γ-α∫Rn∫Rnfxp-2fxktx-rfr-fxdrdxdt=-∫0∞t-α-1Γ-α∫Rn∫Rnfrp-2frktx-rfr-fxdrdxdt. Then, we obtain(35)2Γ-α∫Rnfp-2f-Aαfdx=∫0∞1tα+1∫Rn∫Rnfxp-2fx-frp-2frktx-rfr-fxdrdxdt≥0, since (fxp-2f(x)-frp-2f(r))(f(r)-f(x))≤0 for all x,r∈Rn. For α=0 and α=1 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 f∈D(A) be a smooth solution of (28). Then for 1≤p<∞ we have (36)f·,tp≤f·,0p,for all t≥0.

Proof.

It is a trivial consequence of Lemma 5 and (33).

From now on, we focus on study the decay of d/dtfpp. Applying Theorem 4, we have(37)ddtfpp≤-σ∫Rn-Aα/2fp/22dx,for p=2j with j positive integer. For α=0 we have d/dtfpp≤-σfpp, then solving this differential inequality we obtain (38)f·,tpp≤e-σtf·,0pp.Below we see what happens to the case 0<α≤1.

Theorem 8.

Assuming that the symbol -q is an increasing function in the radius, with limx→∞g(x)=∞, then(39)ddtfpp≤-σfppDfpp,for p=2j with j∈N, f∈S(Rn) real-valued solution of (28), and D a continuous, nonnegative and nondecreasing function.

Proof.

For 0<α≤1, we consider the bijection(40)0,+∞→0,+∞u↦uαwith inverse function u1/α. Thus for all t>0 and h∈S(Rn) one gets (41)h22=h^22=∫x∈Rn:1≤t-qxαh^2xdx+∫x∈Rn:1>t-qxαh^2xdx≤t∫Rn-qxαh^2xdx+h^∞2mnx∈Rn:1t>-qxα≤t-qαh^,h^+h12mnx∈Rn:1t1/α>-qx,

where mn denotes the usual Lebesgue measure on Rn.

Note that q(x)=q(x) is a bijection from R+ to itself. So (42)mnx∈Rn:1t1/α>-qx=∫x<-q-11/t1/αdx=-q-11t1/αnwn where wn is the measure of the unit sphere in Rn. We define β(t)≔((-q)-1(1/t1/α))nwn, for t>0, and we rewrite (43)h22≤t-Aαh,h+h12βt, where β is a nonnegative and decreasing function.

The operator (-A)α is a nonnegative and symmetric operator, which satisfies a Super-Poincaré inequality with rate function β, then by [15, Proposition 2.2] this is equivalent to a Nash-type inequality (44)h22Dh22≤-Aαh,h, with rate function (45)Ds=supt>0t-tβ1/ts,s>0. Note that the function D is continuous, nonnegative and nondecreasing. So, applying this argument to fp/2, with p=2j, we obtain (46)-Aα/2fp/222≥fp/222Dfp/222, and therefore inequality (39) follows from (37).

4. Examples and Applications

In this last section, we check the Lp-decay of solutions in some concrete examples of quasigeostrophic equations. This approach illustrates our results. To do that, we need to calculate the function D (and also the function β) which appears in Theorem 8 for concrete examples. In [15, section 8], general properties of functions β and D are studied using N-functions; see also [19].

Let r:[0,∞)→[0,∞) be a right continuous, monotone increasing function with

r(0)=0;

limt→∞r(t)=∞;

r(t)>0 whenever t>0;

then, the function defined by R(x)≔∫0xr(t)dt for x∈R is called an N-function. Alternatively, the function R:R→[0,∞) is an N-function if and only if R is continuous, even and convex with

limx→0(Rx/x)=0;

limx→∞Rx/x=∞;

R(x)>0 if x>0.

Given an N-function R, we define the function G(x)≔∫0xg(t)dt for x>0 where g is the right inverse of the right derivative of R, r. The function G is an N-function called the complement of R. Furthermore it is straightforward to check that the complement of G is R.

Now suppose that functions β and D are complementary N-functions. Then functions h and h∗, defined by h(t)≔tβ(1/t) for t>0, and h∗(x)≔xD(x) for x>0, are also complementary N-functions.

We consider the Laplace operator and q(x)=-4π2x2, see Section 2. Then (47)-q-1t=t1/22π,and βt=wnt-n/2α2πn,t>0,

where wn is the measure of the unit sphere in Rn. Now we have a couple of N-functions, h(t)=wnt1+n/2α/2πn and h∗(x)=cnxq, with 1/q+1/1+n/2α=1 and cn a positive constant; see [15, Section 8]. Then D(x)=cnx2α/n, and we get (48)ddtfpp≤-Cnfpp1+2α/n,

for p=2j. Solving this differential inequality, one obtains (49)f·,tpp≤f·,0pp1+εCntf·,0ppε1/ε,

with ε=2α/n and p=2j.

For the subordinated semigroup through Poisson semigroup with q(x)=-log(1+2πx), we get that (50)-q-1t=et-12π,and ht=tβ1t=cntet1/α-1n,t>0,

with cn=wn/2πn. Then h(t)=∫0tu(s)ds, with (51)ut=cnet1/α-1n+nαet1/αet1/α-1n-1t1/α,t>0,

so h∗(x)=∫0xu-1(t)dt. Note that (52)ut≤cnet1/α-1n+nαe2t1/αet1/α-1n-1≤cn+nαen+1t1/α-1≤cn+nαn+1αt1-α/αen+1αt1/α≕gt,

for t>0.

According to [15, Section 8], we consider h4(t)=etp-1, with p>1. If we take p=1/α, then h4′(t)=1/αt(1-α)/αet1/α, and so (53)g-1t=cn+nαα-1h4′-1n+1αt,t>0.

Now we apply Theorem 8 in the case of p=2j with j∈N. If we suppose that the solution of (28) is stable; i.e., limt→∞f(·,t)pp=0, then (55)ddtfpp≲-Cn,αfpp1/1-α,

for t large enough, where we have used that h4∗(x)~cqxq, as x→0+ with 1/q+α=1. We conclude that (56)f·,tpp≤f·,0pp1+εCn,αtf·,0ppε1/ε,

for ε=α/(1-α), and t large enough.

For other p≠2j and 1<p<∞, we obtain the decayment by interpolation property: if 1≤p1<p<p2<∞, with 1/p=(1-θ)/p1+θ/p2 and 0<θ<1, then fp≤fp11-θfp2θ. When p>2, we have 2j<p<2j+1 for any integer j≥1, and if 1<p<2 we also use that f(·,t)1≤f(·,0)1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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.

SchertzerD.TchiguirinskaiaI.LovejoyS.TuckA. F.Quasi-geostrophic turbulence and generalized scale invariance, a theoretical replyVannesteJ.Enhanced dissipation for quasi-geostrophic motion over small-scale topographyCórdobaA.CórdobaD.A pointwise estimate for fractionary derivatives with applications to partial differential equationsCórdobaA.CórdobaD.A maximum principle applied to quasi-geostrophic equationsAlonso-OránD.CórdobaA.MartínezA. D.Global well-posedness of critical surface quasigeostrophic equation on the sphereAlonso-OránD.CórdobaA.MartínezA. D.Integral representation for fractional Laplace-Beltrami operatorsRöcknerM.ZhuR.ZhuX.Sub and supercritical stochastic quasi-geostrophic equationSilvestreL.Hölder estimates for advection fractional-diffusion equationsTristaniI.Fractional Fokker-Planck equationEngelK.-J.NagelR.PazyA.SinclairA. M.YosidaK.CaffarelliL. A.SireY.On some pointwise inequalities involving nonlocal operatorsGentilI.MaheuxP.Super-Poincaré and Nash-type inequalities for subordinated semigroupsde la LlaveR.ValdinociE.Lp-bounds for quasi-geostrophic equations via functional analysisAbadiasL.Campos-OrozcoJ. S.GaléJ. E.Special functions as subordinated semigroups on the real lineRaoM. M.RenZ. D.Applications Of Orlicz Spaces