## Abstract

This article emphasizes the role played by a remarkable pointwise inequality satisfied by fractionary derivatives in order to obtain maximum principles and *L*^{p}-decay of solutions of several interesting partial differential equations. In particular, there are applications to quasigeostrophic flows, in two space variables with critical viscosity, that model the Eckman pumping [see Baroud, Ch. N., Plapp, B. B., She, Z. S. & Swinney, H. L. (2002) *Phys. Rev. Lett.* 88, 114501 and Constantin, P. (2002) *Phys. Rev. Lett.* 89, 184501].

The decay in time of the spatial *L*^{p}-norm, 1 ≤ *p* ≤ ∞, is an important objective in order to understand the behavior of solutions of partial differential equations. The purpose of this article is to analyze the following pointwise inequality, 2θΛ^{α}θ(*x*) ≥ Λ^{α}θ^{2}(*x*), valid for fractionary derivatives in *R*^{n}, n ≥ 1, 0 ≤ α ≤ 2, together with its applications to several maximum principle and decay estimates. In particular, it is applied to the quasigeostrophic equation with critical viscosity

where ^{}θ = (–θ/*x*_{2}, θ/*x*_{1}), *R*(θ) = (*R*_{1}(θ), *R*_{2}(θ)), and *R*_{j} denotes the *j*^{th}-Riesz transform in *R*^{2} (see refs. 1–6).

Given a weak solution θ(*x, t*) (obtained as limit of solutions of the equations

with the same initial data θ_{0}, when the artificial viscosity ε tends to zero), it is proved that θ(·, *t*)_{L}^{p}, 1 ≤ *p* ≤ ∞, decays and, furthermore, there is a time *T* = *T*(κ, θ_{0}) < ∞ after which θ becomes regular.

In this article, we describe the main ideas of the proofs, together with some of the heuristic arguments. The complete details will appear elsewhere.

## Pointwise Estimate

The nonlocal operator Λ = (–Δ)^{1/2} is defined with the Fourier transform by , where is the Fourier transform of *f*.

**Theorem 1.** *Let* 0 ≤ α ≤ 2, *x* *R*^{n}, T^{n} (*n* = 1, 2, 3...) *and* θ *C*^{2}_{0}(*R*^{n}), *C*^{2}(*T*^{n}). *Then the following inequality holds:*

*Proof:* It is easy to check that the inequality is satisfied when α = 0 and α = 2. For 0 < α < 2 and *n* ≥ 2, there are the formulas

where *C*_{α}, _{α} > 0.

With Eq. **2** (and Eq. **3** in the periodic case) inequality Eq. **1** is obtained easily:

The proof of the remainder case *n* = 1 is as follows. Given ψ(*x*_{1}) an application of the previous case, *n* = 2, to the function yields

*Remark 1:* The family of test functions *x*^{p}e^{–δ}^{x}^{2}, δ > 0, shows that the condition α ≤ 2 cannot be improved.

Also, the hypothesis , *C*^{2}(*T*^{n}) is not necessary. Inequality Eq. **1** holds when θ(*x*), Λ^{α}θ(*x*), Λ^{α}θ^{2}(*x*) are defined everywhere and are, respectively, the limits of the sequences θ_{m}(*x*), Λ^{α}θ_{m}(*x*), Λ^{α}θ^{2}_{m}(*x*), where , *C*^{2}(*T*^{n}) for each *m*.

## Applications

**L**^{p} Decay. Let it be given the following scalar equation

where the vector *u* satisfies either ·*u* = 0 or *u*_{i} = *G*_{i}(θ), together with the appropriate hypothesis about regularity and decay at infinity, which will be specified each time, in order to allow the integration by parts needed in the proofs.

**Lemma 1.** *If* 0 ≤ α ≤ 2 *and* θ *C*^{2}_{0}(*R*^{n})(*C*^{2}(*T*^{n})), *it follows that*

*where p* = 2^{j} and j is a positive integer.

*Proof:* An iterated application of inequality Eq. **1** yields:

taking *k* = *j* – 1 and using Parseval's identity with the Fourier transform inequality Eq. **4** is obtained.

*Remark 2:* When *p* = 2^{j} (*j* ≥ 1) *Lemma 1* implies the following improved estimate:

In the periodic case, this inequality yields an exponential decay of θ_{L}^{p}, 1 ≤ *p* < ∞. For the nonperiodic case, Sobolev's embedding and interpolation produces

where *C* = *C*(κ, α, *p*, θ_{0}_{1}) is a positive constant. It then follows

with ε = α/2(*p* – 1).

*Remark 3:* The decay for other *L*^{p}, 1 < *p* < ∞, is obtained easily by interpolation. However, the *L*^{∞} decay needs further arguments that will be presented in the next section.

## Viscosity Solutions of the Quasigeostrophic Equation

A weak solution of

will be called a viscosity solution with initial data θ_{0} *H*^{s}(*R*^{2})(*H*^{s}(*T*^{2})), *s* > 1, if it is the weak limit of a sequence of solutions, as ε → 0, of the problems

with θ^{ε}(*x*, 0) = θ_{0}.

**Theorem 3.** *Let* θ *be a viscosity solution with initial data* θ_{0} *H*^{s}, , *of the equation* θ_{t} + *R*(θ)·^{}θ = –κΛθ (κ > 0). *Then there exist two times T*_{1} ≤ *T*_{2} *depending only on* κ *and the initial data* θ_{0} *so that:*

*If t* ≤

*T*_{1} *then* θ(·,

*t*)

*C*^{1}([0,

*T*_{1});

*H*^{s})

*is a classical solution of the equation satisfying* *If t* ≥

*T*_{2} *then* θ(·,

*t*)

*C*^{1}([

*T*_{2}, ∞);

*H*^{s})

*is also a classical solution and* θ(·,

*t*)

_{H}^{s} is monotonically decreasing in t, bounded by θ

_{0}_{H}^{s},

*and satisfying*

*In particular, this implies that*

*Sketch of the Proofs:* For the *L*^{∞}-decay there is the following heuristic argument. Assuming that θ(·, *t*) get its maximum value at the point *x*_{t}, depending smoothly on *t*, then the equation yields

And the decay is obtained because

In the actual *Proof* the differentiability properties of Lipschitz functions are used in order to avoid the hypothesis about the existence of *dx*_{t}/*dt*.

The *Proof* of *Theorem 3* is based on both the *L*^{∞}-decay and a bootstrap mechanism associated with the evolution of different Sobolev norms. A crucial ingredient is the fact that *fR*(*f*) belongs to Hardy's space for each *L*^{2}-function *f* and every odd singular integral *R*. A typical example of that mechanism is the following chain of inequalities

valid for some universal constant *C*, uniformly with respect to the artificial viscosity ε.

## References

1. Chae, D. & Lee, J. (2003) Commun. Math. Phys. 233**,** 297–311.

2. Constantin, P., Cordoba, D. & Wu, J. (2001) Indiana Univ. Math. J. 50**,** 97–107.

3. Constantin, P. & Wu, J. (1999) SIAM J. Math. Anal. 30**,** 937–948.

4. Resnick, S. (1995) Ph.D. thesis (University of Chicago, Chicago).

5. Wu, J. (2002) Comm. Partial Differ. Eq. 27**,** 1161–1181.

6. Wu, J. (1997) Indiana Univ. Math. J. 46**,** 1113–1124.