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Proc Natl Acad Sci U S A. Dec 23, 2003; 100(26): 15316–15317.
Published online Dec 5, 2003. doi:  10.1073/pnas.2036515100
PMCID: PMC307564
Mathematics

A pointwise estimate for fractionary derivatives with applications to partial differential equations

Abstract

This article emphasizes the role played by a remarkable pointwise inequality satisfied by fractionary derivatives in order to obtain maximum principles and Lp-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 Lp-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, 2θΛαθ(x) ≥ Λαθ2(x), valid for fractionary derivatives in Rn, 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

equation M1

where [nabla][perpendicular]θ = (–[partial differential]θ/[partial differential]x2, [partial differential]θ/[partial differential]x1), R(θ) = (R1(θ), R2(θ)), and Rj denotes the jth-Riesz transform in R2 (see refs. 16).

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

equation M2

with the same initial data θ0, when the artificial viscosity ε tends to zero), it is proved that ||θ(·, t)||Lp, 1 ≤ p ≤ ∞, decays and, furthermore, there is a time T = T(κ, θ0) < ∞ 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 equation M3, where f is the Fourier transform of f.

Theorem 1. Let 0 ≤ α ≤ 2, x [set membership] Rn, Tn (n = 1, 2, 3...) and θ [set membership] C20(Rn), C2(Tn). Then the following inequality holds:

equation M4
[1]

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

equation M5
[2]

equation M6
[3]

where Cα, Cα > 0.

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

equation M7

The proof of the remainder case n = 1 is as follows. Given ψ(x1) an application of the previous case, n = 2, to the function equation M8 yields

equation M9

Remark 1: The family of test functions xpe–δx2, δ > 0, shows that the condition α ≤ 2 cannot be improved.

Also, the hypothesis equation M10, C2(Tn) is not necessary. Inequality Eq. 1 holds when θ(x), Λαθ(x), Λαθ2(x) are defined everywhere and are, respectively, the limits of the sequences θm(x), Λαθm(x), Λαθ2m(x), where equation M11, C2(Tn) for each m.

Applications

Lp Decay. Let it be given the following scalar equation

equation M12

where the vector u satisfies either [nabla]·u = 0 or ui = Gi(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 [set membership] C20(Rn)(C2(Tn)), it follows that

equation M13
[4]

where p = 2j and j is a positive integer.

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

equation M14

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

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

equation M15

In the periodic case, this inequality yields an exponential decay of ||θ||Lp, 1 ≤ p < ∞. For the nonperiodic case, Sobolev's embedding and interpolation produces

equation M16

where C = C(κ, α, p, ||θ0||1) is a positive constant. It then follows

equation M17

with ε = α/2(p – 1).

Remark 3: The decay for other Lp, 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

equation M18

will be called a viscosity solution with initial data θ0 [set membership] Hs(R2)(Hs(T2)), s > 1, if it is the weak limit of a sequence of solutions, as ε → 0, of the problems

equation M19
[5]

with θε(x, 0) = θ0.

Theorem 2. Let θε, ε > 0, be a solution of Eq. 5, then θε(·, t) [set membership] Hs for each t > 0 and satisfies

equation M20

uniformly on ε > 0 for all time t ≥ 0. Furthermore, for equation M21, there is a time T1 = T1(κ, ||θ0||Hs) such that ||Λs θε(t)||L2 ≤ 2||Λsθ0||L2 fort < T1.

Theorem 3. Let θ be a viscosity solution with initial data θ0 [set membership] Hs, equation M22, of the equation θt + R(θ)·[nabla][perpendicular]θ = –κΛθ (κ > 0). Then there exist two times T1T2 depending only on κ and the initial data θ0 so that:

  1. If tT1 then θ(·, t) [set membership] C1([0, T1); Hs) is a classical solution of the equation satisfying
    equation M23
  2. If tT2 then θ(·, t) [set membership] C1([T2, ∞); Hs) is also a classical solution and ||θ(·, t)||Hs is monotonically decreasing in t, bounded by ||θ0||Hs, and satisfying
    equation M24

In particular, this implies that

equation M25

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 xt, depending smoothly on t, then the equation yields

equation M26

And the decay is obtained because

equation M27

In the actual Proof the differentiability properties of Lipschitz functions are used in order to avoid the hypothesis about the existence of dxt/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 equation M28 for each L2-function f and every odd singular integral R. A typical example of that mechanism is the following chain of inequalities

equation M29

equation M30

equation M31

equation M32

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

Acknowledgments

It is a pleasure to thank C. Fefferman for his helpful comments and his strong influence in our work. The work of A.C. was partially supported by Ministerio de Ciencia y Tecnología Grant BFM2002-02269. D.C. acknowledges support from Ministerio de Ciencia y Tecnología Grant BFM2002-02042.

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.

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