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 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^αθ(x) ≥ Λ^αθ²(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

where ∇^⊥θ = (–∂θ/∂x₂, ∂θ/∂x₁), R(θ) = (R₁(θ), R₂(θ)), and R_j denotes the j^th-Riesz transform in R² (see refs. 1–6).

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

with the same initial data θ₀, 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, 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 , where f̂ is the Fourier transform of f.

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

[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

[2]

[3]

where C_α, C̃_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(x₁) an application of the previous case, n = 2, to the function yields

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

Also, the hypothesis , C²(Tⁿ) is not necessary. Inequality Eq. 1 holds when θ(x), Λ^αθ(x), Λ^αθ²(x) are defined everywhere and are, respectively, the limits of the sequences θ_m(x), Λ^αθ_m(x), Λ^αθ²_m(x), where , C²(Tⁿ) 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 ∈ C²₀(Rⁿ)(C²(Tⁿ)), it follows that

[4]

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, ∥θ₀∥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 θ₀ ∈ H^s(R²)(H^s(T²)), s > 1, if it is the weak limit of a sequence of solutions, as ε → 0, of the problems

[5]

with θ^ε(x, 0) = θ₀.

Theorem 2. Let θ^ε, ε > 0, be a solution of Eq. 5, then θ^ε(·, t) ∈ H^s for each t > 0 and satisfies

uniformly on ε > 0 for all time t ≥ 0. Furthermore, for , there is a time T₁ = T₁(κ, ∥θ₀∥_Hs) such that ∥Λ^s θ^ε(t)∥_L2 ≤ 2∥Λ^sθ₀∥_L2 for ≤ t < T₁.

Theorem 3. Let θ be a viscosity solution with initial data θ₀ ∈ H^s, , of the equation θ_t + R(θ)·∇^⊥θ = –κΛθ (κ > 0). Then there exist two times T₁ ≤ T₂ depending only on κ and the initial data θ₀ so that:

1. If t ≤ T₁ then θ(·, t) ∈ C¹([0, T₁); H^s) is a classical solution of the equation satisfying

   
2. If t ≥ T₂ then θ(·, t) ∈ C¹([T₂, ∞); H^s) is also a classical solution and ∥θ(·, t)∥_H^s is monotonically decreasing in t, bounded by ∥θ₀∥_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 L²-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.