# On a Fractional Nonlinear Hyperbolic Equation Arising from Relative Theory

Authors: Zujin Zhang; Xiaofeng Wang; Zheng-an Yao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We obtain the existence of a weak solution to a fractional nonlinear hyperbolic equation arising from relative theory by the Galerkin method. Its uniqueness is also discussed. Furthermore, we show the regularity of the obtained solution. In our proof, we use harmonic analysis techniques and compactness arguments.


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## Body

## 1. Introduction

This paper is concerned with the following fractional partial differential equations in𝕋n=ℝn/ℤn:
(1)∂2u∂t2-Λ2αu+|u|ρu=f,(2)u(x,0)=u0(x),(3)∂u∂t(x,0)=u1(x),
where Λ=(-Δ)1/2 is the square root of the Laplacian operator, α∈(0,1), ρ∈[0,∞) are two real parameters, and f(x,t) is given. Equations (1)–(3) play an important role in nuclear force and relativistic theory.The fractional diffusion operatorΛ2α is nonlocal except when α∈ℕ, which means that Λ2αu(x) depends not only on u(y) for y near x, but on u(y) for all y∈ℝn. Fractional differential equations, arising from mathematical physics such as viscoelasticity, electrochemistry, control theory, porous media, and electromagnetism, now attract the interests of many mathematicians; see [1–4] and references therein. In the past ten years, the quasi-geostrophic equation with fractional dissipation has been extensively studied; see Constantin et al. [5–9] and references therein. The relativistic equation shares some similar difficulties with the quasi-geostrophic equation. However, the equations studied in this paper are more complicated in that the fractional diffusion operator and the nonlinear term in (1)–(3) bring new difficulties in passing to the limits of the approximate solutions, and hence, new devices must be introduced to overcome these obstacles.Whenα=1, (1)–(3) become the standard equations, which were intensively studied in the past century. The readers are referred to[10–12] for more details.Interestingly enough, the parabolic version of (1)–(3) with convection corresponds to the Navier-Stokes equations with damping; see [13, 14].We now collect the notations in this paper. The square root of the negative Laplacian-Δ, Λ is given by (in terms of Fourier series)
(4)(Λf^)k=|k|f^k,k∈ℤn,
where f^k is the Fourier coefficients of f:
(5)f^k=1(2π)n∫𝕋n‍f(x)e-ik·xdx,k∈ℤn.
More generally, Λ2αf for α∈ℝ can be defined as
(6)(Λ2αf^)k=|k|2αf^k,k∈ℤn.We will also invoke the notion of homogeneous Sobolev spaceH˙s(𝕋n)(s∈ℝ), which comprises all tempered distributions f on ℝn such that
(7)∥f∥H˙s(𝕋n)=(∑k∈ℤn∖{0}|k|2s|u^k|2)1/2.
We also recall the meaning of by the weak (weakly*) convergence in Lp(0,T;Lq(𝕋n)). Since the dual of Lr(𝕋n)(1≤r<∞) is Lr/(r-1)(𝕋n) and the space C∞(𝕋n) is dense in Lr(𝕋n) (noticing the periodic boundary conditions), we have
(8)fk⇀finLp(0,T;Lq(𝕋3))
with 1≤p,q<+∞ if and only if
(9)∫0T‍∫𝕋n‍fkϕdxdt⟶∫0T‍∫𝕋n‍fϕdxdt,∀ϕ∈C0∞(0,T;C∞(𝕋n)),(10)fk⇀finLp(0,T;Lq(𝕋3))
with p=+∞ or q=+∞ if and only if (9) holds.We now close this introduction by outlining the rest of this paper. In Section2, we prove the existence of a weak solution to (1)–(3); see Theorem 4. The uniqueness of such weak solutions is discussed in Section 3; see Theorem 5. Finally, a regularity result is obtained in Section 4; see Theorem 6.


## 2. Existence of a Weak Solution

First, let us recall the following two fundamental lemmas.Lemma 1 (see [15]).
LetX be a Banach space, f∈Lp(0,T;X), and ∂f/∂t∈Lp(0,T;X)(1≤p≤∞), then f∈C([0,T];X).Lemma 2 (see [15]).
Let𝒟 be a bounded domain in ℝn×ℝ+, and {gk}k=1∞,g belong to Lq(𝒟)(1<q<∞) with
(11)∥gk∥Lq(𝒟)≤C,gk⟶ga.e.𝒟.
Then gk⇀g in Lq(𝒟) weakly.Let us now give the weak formulation of (1)–(3).Definition 3.
LetT>0, u0∈H˙α(𝕋n)∩Lρ+2(𝕋n),u1∈L2(𝕋n), and f∈L2(0,T;L2(𝕋n)). A measurable function u(x,t) is said to be a weak solution on [0,T] to (1)–(3) if the following conditions hold:   (1)
u∈L∞(0,T;H˙α(𝕋n)∩Lρ+2(𝕋n)) and ∂u/∂t∈L∞(0,T;L2(𝕋n));(2)
(1) holds in the sense of distributions; that is,
(12)∫0T∫𝕋n∂u∂t·∂ϕ∂tdxdt-∫0T∫𝕋nΛαu·Λαϕdxdt+∫0T∫𝕋n|u|ρuϕdxdt=∫𝕋nu1ϕtdx+∫0T∫𝕋nfϕdxdt,
for eachϕ∈C∞(𝕋n×(0,T)); (3)
u(x,0)=u0(x) a.e. in 𝕋n;(4)
(∂u/∂t)(x,0)=u1(x) a.e. in 𝕋n.To see how item (4) in Definition3 makes sense, we rewrite (1) as
(13)∂2u∂t2=f+Λ2αu-|u|ρu.
Noticing that [16]
(14)Λ2αis continuous fromH˙α(𝕋n)toH˙-α(𝕋n),
we have
(15)Λ2αu∈L∞(0,T;H˙-α(𝕋n)).
Also, due to the fact that
(16)∥|u|ρu∥L(ρ+2)/(ρ+1)(𝕋n)=∥u∥Lρ+2(𝕋n)ρ+1,
we deduce from (13) that
(17)∂2u∂t2∈L2(0,T;L2(𝕋n))+L∞(0,T;H˙-α(𝕋n)+L(ρ+2)/(ρ+1)(𝕋n)).
Thus
(18)∂2u∂t2∈L2(0,T;H˙-α(𝕋n)+L(ρ+2)/(ρ+1)(𝕋n)).
Hence, by item (1) of Definition 3, (18), and Lemma 1, we gather that
(19)∂u∂t∈C([0,T];H˙-α(𝕋n)+L(ρ+2)/(ρ+1)(𝕋n)).
From this, we see that item (4) of Definition 3 makes sense, as claimed.Now, we state our existence results in the following theorem.Theorem 4.
LetT>0,
(20)u0∈H˙α(𝕋n)⋂Lρ+2(𝕋n),(21)u1∈L2(𝕋n),(22)f∈L2(0,T;L2(𝕋n));
then there exists at least one weak solution u on [0,T] to (1)–(3), taking u0,u1 as initial data.Proof.
We use Galerkin method to establish the existence of such a solution.
Step
1 (construction of approximate solution). Let {wi}i=1∞ be a dense and total basis in H˙α(𝕋n)∩Lρ+2(𝕋n), and consider the approximate solution which has the form
(23)um(t)=∑i=1m‍cim(t)wi,
where cim satisfy the following ordinary differential system:
(24)(∂2um∂t2,wj)-(Λ2αum,wj)+(|um|ρum,wj)=(f,wj),1≤j≤m,(25)um(0)=u0m=∑i=1m‍αimwi⟶u0inH˙α(𝕋n)⋂Lρ+2(𝕋n),asm⟶∞,(26)∂um∂t(0)=u1m=∑i=1m‍βimwi⟶u1inL2(𝕋n),asm⟶∞.
Here and hereafter, we denote (u,v)=∫𝕋n‍uvdx for u,v∈L2(𝕋n).
The system (24)–(26) is nonsingular because {wi} are linear independent. Thus, we may apply standard theory of ordinary differential equations to obtain the existence of a local solution to (24)–(26) on [0,tm], for some tm>0. We will then, in the next step, establish some a priori estimates of the obtained solutions which will ensure that tm=T.
Step
2 (a priori estimates). Taking the inner product of (24) with cim′(t) in L2(ℝn), we obtain
(27)12ddt[∥∂um∂t(t)∥L2(𝕋n)2+(Λαum(t),Λαum(t))]+1ρ+2ddt∫𝕋n‍|um(x,t)|ρ+2dx=(f(t),∂um∂t(t)).
Integrating over [0,t] and invoking Hölder inequality then yield
(28)12[∥∂um∂t(t)∥L2(𝕋n)2+∥Λαum(t)∥L2(𝕋n)2]+1ρ+2∥um(t)∥Lρ+2(𝕋n)ρ+2≤12[∥∂um∂t(0)∥L2(𝕋n)2+∥Λαum(0)2∥L2(𝕋n)]+1ρ+2∥um(0)∥Lρ+2(𝕋n)ρ+2+∫0t∥f(τ)∥L2(𝕋n)∥∂um∂τ(τ)∥L2(𝕋n)dτ,+∫0t‍∥f(τ)∥L2(𝕋n)hhhhhhhhh∀t∈[0,T].
Due to (20), (25), and (26), we have that ∥(∂um/∂t)(0)∥L2(𝕋n)2,∥um(0)∥2,and∥um(0)∥Lρ+2(𝕋n)ρ+2 are uniformly bounded. Thus, (28) becomes
(29)12[∥∂um∂t(t)∥L2(𝕋n)2+∥Λαum(t)∥L2(𝕋n)2]+1ρ+2∥um(t)∥Lρ+2(𝕋n)ρ+2≤C+12∫0t‍∥f(τ)∥L2(𝕋n)2dτ+12∫0t‍∥∂um∂τ(τ)∥L2(𝕋n)2dτ,∀t∈[0,T].
Noticing that f∈L2(0,T;L2(𝕋n)), we have
(30)12[∥∂um∂t(t)∥L2(𝕋n)2+∥Λαum(t)∥L2(𝕋n)2]+1ρ+2∥um(t)∥Lρ+2(𝕋n)ρ+2≤12∫0t‍∥∂um∂τ(τ)∥L2(𝕋n)2dτ+C,∀t∈[0,T].
Applying Gronwall's inequality then yields
(31)∥∂um∂t(t)∥L2(𝕋n)≤C,
and hence,(32)∥um(t)∥Lρ+2(𝕋n)≤C,∥Λαum(t)∥L2(𝕋n)≤C.
Step
3 (passage to limit m→∞).
By (31) and (32), we have, up to a subsequence, still denoted by um, that
(33)um⇀uweakly*inL∞(0,T;H˙α(𝕋n)⋂Lρ+2(𝕋n)),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhasm⟶∞,∂um∂t⇀∂u∂tweakly*inL∞(0,T;L2(𝕋n)),asm⟶∞,
and also that um are uniformly bounded in L2(0,T;H˙α(𝕋n)). Thus by Lemma 1, we find that
(34)um⟶ustronglyinL2(0,T;L2(𝕋n)),a.e.𝕋n,asm⟶∞.
Hence, there exists a function w(x,t) such that
(35)|um|ρum⟶w(x,t)weakly*inL∞×(0,T;L(ρ+2)/(ρ+1)(𝕋n)),asm⟶∞.
By Lemma 2 and the fact that
(36)|um|ρum⟶|u|ρu,a.e. in𝕋n,
we know then that
(37)w=|u|ρu.
Fixing j, we now pass to the limit m→∞ in (24) to deduce that
(38)(∂2u∂t2(t),wj)+(Λ2αu(t),wj)+(|u(t)|ρu(t),wj)=(f(t),wj).
A simple density argument then shows that
(39)(∂2u∂t2(t),v)+(Λ2αu(t),v)+(|u(t)|ρu(t),v)=(f(t),v),
for all v∈H˙α(𝕋n)∩Lρ+2(𝕋n).
Up to now, we have proved items (1) and (2) in Definition3. Let us turn our attention to items (3) and (4) in Definition 3.
By (33) and Lemma 1, we know um(0)⇀u(0) weakly in L2(𝕋n), and from (25), um(0)→u0 in H˙α(𝕋n)∩Lρ+2(𝕋n); thus item (3) of Definition 3 is verified.
By (24) and (39), we see that
(40)(∂2um∂t2(0),wj)⟶(∂2u∂t2(0),wj)weakly*,
and by Lemma 1, we have
(41)(∂um∂t(0),wj)⟶(∂u∂t(0),wj).
On the other hand, (26) implies that
(42)(∂um∂t(0),wj)⟶(u1,wj).
Hence
(43)(∂u∂t(0),wj)=(u1,wj),∀j∈ℕ.
This verifies item (4) in Definition 3.


## 3. Uniqueness of Weak Solutions

In this section, we will discuss the uniqueness of weak solutions of (1)–(3). We only obtain partial results in case 0≤ρ≤2α/(n-2α). More precisely, we have the following theorem.Theorem 5.
Assuming as in Theorem4, then there exists an unique weak solution of (1)–(3), in case 0≤ρ≤2α/(n-2α).Proof.
Letu, v be two weak solutions for (1)–(3) given in Theorem 4 with the same datum. Then w=u-v satisfies
(44)∂2w∂t2+Λ2αw=|v|ρv-|u|ρu,(45)w(0)=0,∂w∂t(0)=0.
Also, we have
(46)w∈L∞(0,T;H˙α⋂Lρ+2(𝕋n)),∂w∂t∈L∞(0,T;L2(𝕋n)).
Taking the inner product of (44) with ∂w/∂t in L2(𝕋n), we obtain
(47)12ddt[∥∂w∂t∥L2(𝕋n)2+∥w∥H˙α(𝕋n)2]=∫𝕋n‍(|v|ρv-|u|ρu)∂w∂tdx.
Invoking Hölder and Sobolev inequalities, we obtain(48)∫𝕋n‍(|v|ρv-|u|ρu)∂w∂tdx≤(ρ+1)∫𝕋n‍sup(|u|ρ,|v|ρ)|w||∂w∂t|dx≤C(∥|u|ρ∥Ln/α(𝕋n)+∥|v|ρ∥Ln/α(𝕋n))∥w∥Lq(𝕋n)∥∂w∂t∥L2(𝕋n)≤C(∥u∥H˙α(𝕋n)ρ+∥v∥H˙α(𝕋n)ρ)∥w∥Lq(𝕋n)∥∂w∂t∥L2(𝕋n)≤C(∥u∥H˙α(𝕋n)ρ+∥v∥H˙α(𝕋n)ρ)∥w∥H˙α(𝕋n)∥∂w∂t∥L2(𝕋n),
where (1/q)+(α/n)+(1/2)=1, and we use 0≤ρ≤2α/(n-2α).
Thus, (47) becomes
(49)∥∂w∂t∥L2(𝕋n)2+∥w∥H˙α(𝕋n)2≤C∫0t‍(∥∂w∂t∥L2(𝕋n)2+∥w∥H˙α(𝕋n)2)dτ,
and we get w=0, u=v as desired.


## 4. Regularity of the Weak Solution

Now we discuss the regularity of solutions for (1)–(3). If the initial value and force are more regular, then so is the solution.Theorem 6.
Let(50)u0∈H˙2α(𝕋n)⋂H˙α(𝕋n)⋂Lρ+2(𝕋n),u1∈H˙α(𝕋n)⋂L2(𝕋n),f∈L∞(0,T;L2(𝕋n)),∂f∂t∈L2(0,T;L2(𝕋n)),
and 0≤ρ≤2α/(n-2α). Then there exists a unique weak solution for (1)–(3). Furthermore,
(51)u∈L∞(0,T;H˙2α(𝕋n)),∂u∂t∈L∞(0,T;H˙α(𝕋n)),∂2u∂t2∈L∞(0,T;L2(𝕋n)).Proof.
We just establish the a priori bounds, since the verification follows directly from passing to the limit for Galerkin approximate solutions.
Step  1 (bounds for initial data). Observing that
(52)0≤ρ≤nn-2α⟹n2(ρ+1)≥-α+n2,
we see by Sobolev inequality that
(53)∥|u0|ρu0∥L2(𝕋n)≤∥u0∥L2(ρ+1)(𝕋n)ρ+1≤∥u0∥H˙α(𝕋n)ρ+1.
By formula (13), we have
(54)∂2u∂t2(0)∈L2(𝕋n).
Step  2 (bounds for ∂u/∂t, ∂2u/∂2t). Differentiating (1) with respect to t, we find that
(55)∂3u∂t3-Λ2α∂u∂t+(ρ+1)|u|ρ∂u∂t=∂f∂t.
Taking the inner product of (55) with ∂2u/∂t2, we obtain by integration by parts that
(56)12ddt[∥∂2u∂t2∥L2(𝕋n)2+∥Λα∂u∂t∥L2(𝕋n)2]≤(∂f∂t,∂2u∂t2)-(ρ+1)(|u|ρ∂u∂t,∂2u∂t2)≡I1+I2.
The first term I1 can be easily dominated by using Hölder inequality as
(57)I1≤∥∂f∂t∥L2(𝕋n)∥∂2u∂t2∥L2(𝕋n)≤12∥∂f∂t∥L2(𝕋n)2+12∥∂2u∂t2∥L2(𝕋n)2.
To tackle I2, we invoke Hölder and Sobolev inequalities to deduce that
(58)I2≤C∥|u|ρ∥Lp(𝕋n)∥∂u∂t∥Lq(𝕋n)∥∂2u∂t2∥L2(𝕋n)≤C∥u∥Lρp(𝕋n)ρ∥∂u∂t∥Lq(𝕋n)∥∂2u∂t2∥L2(𝕋n)≤C∥Λαu∥L2(𝕋n)ρ∥Λα∂u∂t∥L2(𝕋n)∥∂u∂t∥Lq(𝕋n)≤C(∥Λα∂u∂t∥L2(𝕋n)2+∥∂u∂t∥Lq(𝕋n)2),
where p, q are chosen so that
(59)nρp=-α+n2=nq,
and thus
(60)1p+1q+12≤1.
Gathering (57) and (58) into (56), it follows that
(61)ddt[∥∂2u∂t2∥L2(𝕋n)2+∥Λα∂u∂t∥L2(𝕋n)2]≤∥∂f∂t∥L2(𝕋n)2+C(∥Λα∂u∂t∥L2(𝕋n)2+∥∂u∂t∥Lq(𝕋n)2).
Gronwall inequality then implies that
(62)∥∂2u∂t2∥L2(𝕋n)2+∥Λα∂u∂t∥L2(𝕋n)2≤eCt[∥∂2u∂t2(0)∥L2(𝕋n)2+∥Λαu1∥L2(𝕋n)2]+eCt∫0t∥∂f∂s(s)∥L2(𝕋n)2ds<∞.
Step  3 (bounds for u). Rewriting (1) as
(63)Λ2αu=∂2u∂t2+|u|ρu-f,
we have u∈L∞(0,T;H˙2α(𝕋n)), in view of similar inequalities satisfied by |u|ρu as (53).
This completes the proof of Theorem6.



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