# Weak Solutions to a Landau–Lifshitz Equation with ap-Laplacian Operator as an Exchange Field

Authors: Xiuping Men; Chahid Ayouch; M. Tilioua; M. R. Sidi Ammi; Anwar Zeb; Sehra; Salih Djilali
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

We are dealing with the classical Landau–Lifshitz equation with an unusual exchange field expressed in terms of ap-Laplacian operator for p greater than 2. We obtain weak solutions to the model by using penalization, compactness arguments, and monotonicity method.


---

## Body

## 1. Introduction

In continuum physics, the Landau–Lifshitz (LL) equation describes the precession of the magnetization field inside ferromagnets. This equation is commonly used in several forms to modelize effects of magnetic field on magnetic materials. In particular, it can be used to describe the temporal behavior of magnetic elements under the magnetic field. In a ferromagnetic material, the magnetization vector can vary in space but keeps constant amplitude. The LL equation predicts the rotation of magnetization in response to different applied torques. Its general form writes(1)∂tm=m∧Δm−βm∧m∧Δm.For more details on the LL equation, we refer for example to [1]. The LL equation has attracted much attention in recent years and many results are already obtained. Since the pioneering work [2, 3] where, it is generally proved that weak solutions are unique while exist globally, several theoretical works on LL (sometimes coupled to the Maxwell equations) have been conducted. For example in [2, 3], the existence and nonuniqueness of weak solution to Landau–Lifshitz equation is established. The work [4] addressed global existence of weak solution to a modern structure of the Landau–Lifshitz–Gilbert equation. The modification lies in occurrence of the effective field of a nonlinear term describing vertical spin stiffness and discuss the limit of got results as a vertical spin stiffness parameter converges to zero. The model of LLG considered in [5] contains a transport-type term in the effective field and global existence of weak solution is investigated. In the same context, a Landau–Lifshitz–Slonczewski equation is considered and global weak and classical solutions have been established in [6]. In [7], existence, regularity, and local uniqueness of the solutions to the Maxwell–Landau–Lifshitz system in three dimensions is proved. In the paper [8], a model of a ferromagnetic material governed by a nonlinear Landau–Lifshitz equation coupled with Maxwell equations is considered. Existence of weak solutions is proved and the ω-limit set of all trajectories is identified to the solutions of the stationary model. Another LLG model is considered [9] with inertial effects and the global existence of the proposed model is established. Here, the modification is taken in the presence of second order time derivative of magnetization in the effective field. In [10], global existence of the solutions for Landau–Lifshitz equation of the ferromagnetic spin chain from a m-dimensional manifold M into the unit sphere S2 of ℝ3 is established and some links between harmonic maps and the solutions of the Landau–Lifshitz equation are studied. All of these proofs are based on some type of penalization and regularization. Finally, we note that significant progress has been made in developing techniques for constructing weak solutions to the generic LLG equation. The convergence of several techniques to weak solutions was demonstrated. For example, in the convergence theory of numerical schemes, a considerable step forward has been made, see for example [11–23].In the present paper, we investigate the global existence of weak solutions for the following LL equation involving thep-Laplacian operator.(2)∂tm=m∧Δpm−βm∧m∧Δpm,inQ=0,T×Ω,m=0on0,T×∂Ω,m0,x=m0xinΩ,where T>0 and Ω is a bounded regular subset of ℝd. The vector m=mt,x denotes the three-dimensional magnetization vector. The notation ∧ shows the vector cross product in ℝ3. The positive constant β parameterizes damping parameter and Δpm stands for the p-Laplacian of m which is defined by(3)Δpm=div∇mp−2∇m,forp>1.This operator can be extended to a monotone, bounded, hemicontinuous, and coercive operator between the spaceW01,pΩ and its dual as follows:(4)−Δp:W01,pΩ↦W−1,p′Ω,−Δpm,ψp=∫Ω∇mp−2∇m⋅∇ψdx,1<p<∞,for all ψ∈W01,pΩ and 1/p+1/p′=1.In (2), the initial condition m0 is assumed to satisfy the saturation constraint m0=1 a.e in Ω.Many researchers have studied several forms of the exchange field. We refer for example to the paper [24] where a model incorporating a singular kernel is proposed for the exchange field. Another form for the exchange field is considered in [25] where the exchange field is expressed by means of a function G of ∇m with specific hypotheses on G. For a third consideration for the exchange field we refer to the paper [26]. The main novelty of the present paper with respect to other related papers (dealing with the particular case p=2) is that p>2. Since our problem involves a p-Laplacian, related studies on the LL equation are not known yet, as far as the authors know. Thus, this paper aims to establish an existence result for (2) where p>2 by essentially using penalization, compactness arguments, and monotonicity method. The main difficulty in the proof of existence is to establish the convergence almost everywhere of the gradient ∇mε where ε goes to 0 in order to pass to the limit in the nonlinear term ∫0t∫Ω∇mεp−2∇mε⋅mε∧∇φdxds (Section 3.2).The current paper is structured as follows. In section2, we present some notations and discuss our main result. Section 3 is dedicated to the proof of the main result. Finally, a conclusion is drawn is the last section.


## 2. Notations and the Main Result

Throughout, in current research, fork∈1,∞,p≥2, LkΩ=LkΩ3, W1,pΩ=W1,pΩ3 are the usual Lebesgue and Sobolev spaces. The norm in LkΩ is denoted by ⋅LkΩ, for simplicity, we utilize the notation ⋅k sometimes.We begin by introducing the notion of weak solutions to the problem (2).Definition 1.
Letm0∈W01,pΩ such that m0=1 a.e, we say that the vector field m is a weak solution of (2) if(i)
For allT>0, m∈L∞0,T,W01,pΩ, ∂tm∈L2Q and m=1 a.e;(ii)
For allφ∈Lp0,T,W01,pΩ, there holds(5)∫Q∂tm⋅φdxdt+β∫Qm∧∂tm⋅φdxdt=1+β2∫Q∇mp−2∇m∧m⋅∇φdxdt,(iii)
The following energy estimate holds(6)1p∫Ω∇mpdx+β1+β2∫Q∂tm2dxdt≤1p∫Ω∇m0pdx.
The purpose of this paper is to prove the following main result.Theorem 1.
LetΩ⊂ℝd (with d≥1) a bounded regular set, m0∈W01,pΩ such that m0=1 a.e, and p satisfies the following conditions:(7)ℋ:p≥4dd+4,whend>4,p>2,whend≤4.Then there exists a weak solution of (2) such that m∈Cα0,T,L2Ω for all α∈0,1/2.The proof of Theorem1 will be given in the next section by using the penalization, compactness arguments, and monotonicity method. Let us recall some auxiliary lemmas required later.Lemma 1 (See [26, 27]).
Consider a reflexive Banach spaceX and its dual space of X′. Suppose that an operator A:X⟶X′ satisfies(i)
A is monotone on X, i.e., Au−Av,u−v≥0 for all u,v∈X.(ii)
A is semi-continuous, i.e., the map t↦Au+tv,w, is continuous on 0,1 for all u,v,w∈X.(iii)
Ifun⇀u weakly in X, un⇀b weakly in X′ as n⟶+∞ and lim supn⟶+∞Aun,un≤b,u.Thenb=Au.Lemma 2 (See [26]).
Letγ,δ∈ℝd(8)γp−2γ−δp−2δ⋅γ−δ≥2p−2γ−δp,ifp≥2,(9)γp−2γ−δp−2δ⋅γ−δγp+δp2−p/p≥p−1γ−δp,if1<p<2.
Note that this inequality is also true ifγ and δ are two tensors. In our case we apply it for γ=∇u and δ=∇v where u,v∈ℝ3, to show the monotony of the operator −Δp. We end this section with this useful theorem for differential systems.Theorem 2.
LetF:t0−δ,t0+δ×K⟶ℝn where K=u∈ℝn,u−u0<α for some δ,α>0 satisfies the following Caratheodory conditions:(i)
t↦Fit,u is measurable for all i=1,…,n and for all u∈K,(ii)
u↦Fit,u is continuous for almost all t∈t0−δ,t0+δ,(iii)
there exists an integrable functionG:t0−δ,t0+δ⟶ℝ such that(10)Fit,u≤Gt,∀t,u∈t0−δ,t0+δ×K⟶ℝn,∀i=1,…,n.Then, there existsδ′∈0,δ and a continues function u:t0−δ′,t0+δ′⟶ℝn such that.(iv)
du/dt exists for almost all t∈t0−δ′,t0+δ′.(v)
u solves the problem(11)du′dt=Ft,u,t∈t0−δ,t0+δ,ut0=u0∈ℝn.


## 3. Proof of the Main Result

### 3.1. Existence for the Penalized Problem

Our aim is to prove Theorem1. We begin by constructing weak solutions to a penalized problem associated with (2) where the constraint m=1 is relaxed, i.e., for ε>0 we introduce(12)11+β2β∂tmε−mε∧∂tmε=Δpmε−1εmε2−1mεinQ,mε=0on0,T×∂Ω,mε0,.=m0inΩ..We have the following result.Proposition 1.
Letm0∈W01,pΩ such that m0=1 a.e and p satisfies the hypothesis ℋ. Then for fixed ε>0, there exists a weak solution mε for the penalized problem (5) such that(i)
For allT>0, mε∈L∞0,T,W01,pΩ, ∂tmε∈L2Q;(ii)
For allΦ∈Lp0,T,W01,pΩ, there holds(13)β∫Q∂tmε⋅Φdxdt−∫Qmε∧∂tmε⋅Φdxdt=−1+β2∫Q∇mεp−2∇mε⋅∇Φdxdt−1+β2ε∫Qmε2−1mε⋅Φdxdt.(iii)
The following energy estimate holds(14)β1+β2∫Q∂tmε2dxdt+1p∫Ω∇mεpdx+14ε∫Ωmε2−12dx≤1p∫Ω∇m0pdx,for a.e. t∈0,T.Proof.
The spaceW01,pΩ is separable for 1≤p<+∞, then there exists a sequence of finite dimensional subspaces Vn=spanω1,…,ωn of W01,pΩ such that ∪n≥1Vn is dense in W01,pΩ. For a fixed positive integer n, we look for approximate solution mnε for (5) in the form(15)mnεt,x=∑i=1naintωix,such that(16)11+β2∫Ωβ∂tmnε−mnε∧∂tmnε⋅ωidx=−∫Ω∇mnεp−2∇mnε⋅∇ωidx−1ε∫Ωmnε2−1mnε⋅ωidx,for i=1,…,n with(17)mnε0,.=m0n,where m0n is the approximations of the initial data m0 with(18)m0nt,x=∑i=1na0intωix.We assume that(19)m0n⟶m0strongly inW01,pΩ.
We setan=a1n,…,ann and a0n=a01n,…,a0nn. Then the differential system (16) with initial condition (17) satisfying (18) and (19) can be written as(20)Aandandt=Fan,an0=a0n.
In fact, we have(21)β∂tmnε−mnε∧∂tmnε=Amnε∂tmnε,where(22)Amnε=β−mn3εmn2εmn3εβ−mn1ε−mn2εmn1εβ,such that mniε for i=1,2,3 are the components of the vector mnε. In addition(23)detAmnε=ββ2+mnε2≠0.
Then the matrixAan is invertible for all an, then the system (20) becomes(24)dandt=Aan−1Fan,an0=a0n.
SinceF satisfies the Caratheodory conditions, the existence of a solution to the system (24) is ensured due to Theorem 2. The global existence then follows from the following a priori estimates.(i)
A priori estimates.We multiply (16) by dain/dt, add up the obtained equations for i=1,…,n, we obtain(25)β1+β2∫Ω∂tmnε2dx+ddt1p∫Ω∇mnεpdx+14ε∫Ωmnε2−12dx=0.Integrating (25) with respect to t leads to(26)β1+β2∫Q∂tmnε2dxdt+1p∫Ω∇mnεpdx+14ε∫Ωmnε2−12dx=1p∫Ω∇m0npdx+14ε∫Ωm0n2−12dx.(ii)
Convergences and passage to the limit whenn goes to +∞.Following hypothesisℋ, we infer thatW01,pΩL4Ω. Thanks to (19), we deduce that the right-hand side of (26) is uniformly bounded with respect to n. Then by Poincaré’s inequality, we get(27)∂tmnεnis bounded inL2Q,mnεnis bounded inL∞0,T,W01,pΩ,(28)mnεnis bounded inL∞0,T,L4Ω.
Hence the following convergences hold up to a subsequence(29)∂tmnε⇀∂tmεweakly inL2Q,mnε⇀mεweakly−⋆inL∞0,T,W01,pΩ.
Since the Sobolev embeddingW01,pΩ⇀LpΩis compact and LpΩ⇀L2Ω is continuous, we use Aubin–Lions lemma to obtain the strong convergence as(30)mnε⟶mεstrongly inC0,T,LpΩand a.e.Proposition 2.
Ifmnε and mε satisfies the above convergences, then we have(31)Δpmnε⇀Δpmεweakly inLp′0,T,W−1,p′Ω.Proof.
Since1/p+1/p′=1, we get from (15)(32)∫0t∫Ω∇mnε|p−2∇mnεp′dxds=∫0t∇mnεppds≤C,for allt∈0,T,where C is a positive constant independent of n. Hence the sequence ∇mnεp−2∇mnεn is bounded in Lp′Q, so there exists a subsequence such that(33)∇mnεp−2∇mnε⇀ξεweakly inLp′Q,for some ξε∈Lp′Q. For the rest of the proof we use the following notation:(34)X=Lp0,T,W01,pΩ,X′=Lp′0,T,W−1,p′Ω,where .,.X′,X denote the duality pairing between X′ and X. As n⟶+∞, from (33), we have(35)−Δpmnε,ϕX′,X=∫0T∫Ω∇mnεp−2∇mnε⋅∇ϕdxdt⟶∫0T∫Ωξε⋅∇ϕdxdt.
Then(36)−Δpmnε⇀ηεweakly−⋆inX′.
In order to show thatηε coincides with −Δpmε, we will use the fact that the operator −Δp:X⟶X′ is a maximal monotone. It follows from Lemma 1 that −Δpmε=ηε if we prove the inequality, we get(37)lim supn⟶+∞−Δpmnε,mnεX′,X≤ηε,mεX′,X.
In order to verify (37) fix n. Then, multiply (16) by ai, and sum over i=1,…,n, to arrive at(38)β1+β2∫Ω∂tmnε⋅mnεdx=−∫Ω∇mnεp−2∇mnε⋅∇mnεdx−1ε∫Ωmnε2−1mnε2dx.
Integrate identity (38) over s∈0,t to obtain(39)∫0t∫Ω∇mnεp−2∇mnε⋅∇mnεdxds=−β1+β2∫0t∫Ω∂tmnε⋅mnεdxds+1ε∫0t∫Ωmnε2−1mnε2dxds.
The weak convergence of∂tmnεn in L2Q and the strong convergence of mnεn in L2Q imply that(40)limn⟶+∞∫0t∫Ω∂tmnε⋅mnεdxds=∫0t∫Ω∂tmε⋅mεdxds.
Now from the estimate (26), we deduce that mnε2−1mnεn is bounded in Lp′Q. Then mnε2−1mnε⇀ρε weakly in Lp′Q and using the fact that mnε⟶mε a.e, one can show that ρε=mε2−1mε. Therefore, we get(41)∫0t∫Ωmnε2−1mnε2dxds⟶∫0t∫Ωmε2−1mε2dxds.
Taking the limit superior in (39) as n goes to infinity, we obtain in particular that(42)lim supn⟶+∞∫0t∫Ω∇mnεp−2∇mnε⋅∇mnεdxds=−β1+β2∫0t∫Ω∂tmε⋅mεdxds+1ε∫0t∫Ωmε2−1mε2dxds.
In the following, we show that∫0tηε,mεX′,Xds is equal to the right-hand side of (42) for a.e t∈0,T, i.e.,(43)∫0tηε,mεX′,Xds=−β1+β2∫0t∫Ω∂tmε⋅mεdxds+1ε∫0t∫Ωmε2−1mε2dxds.
We multiply (16) by an arbitrary function σ∈C0,T, and integrate it from 0 to t obtain(44)−11+β2β∫0t∫Ω∂tmnε⋅σsωidxds−∫0t∫Ωmnε∧∂tmnε⋅σsωidxds=∫0tΔpmnε,σsωiX′,Xds+1ε∫0t∫Ωmnε2−1mnε⋅σsωidxds.
Taking the limit asn⟶+∞ on both sides of (44), we get(45)−11+β2β∫0t∫Ω∂tmε⋅σsωidxds−∫0t∫Ωmε∧∂tmε⋅σsωidxds=∫0tηε,σsωiX′,Xds+1ε∫0t∫Ωmε2−1mε⋅σsωidxds.
Since (45) holds for each i=1,…,n, we may replace σsωi by mnε to get(46)11+β2β∫0t∫Ω∂tmε⋅mnεdxds−∫0t∫Ωmε∧∂tmε⋅mnεdxds=−∫0tηε,mnεX′,Xds−1ε∫0t∫Ωmε2−1mε⋅mnεdxds.
Using the previous convergences, we can pass to the limit in (46) to obtain (43). Noting that for the convergence of the term ∫0t∫Ωmε∧∂tmε⋅mnεdxds, we have(47)∫0t∫Ωmε∧∂tmε⋅mnεdxds=∫0t∫Ωmnε∧mε⋅∂tmεdxds.
We know thatmnεn is bounded in L4Q and mε∈L4Q then the sequence mnε∧mεn is bounded in L2Q. Thanks to the convergence a.e of mnεn to mε, we obtain that mnε∧mε⇀mε∧mε=0 weakly in L2Q. Therefore, we get(48)∫0t∫Ωmnε∧mε⋅∂tmεdxds⟶0,asn⟶+∞.
Now, it follows from (42) and (43) that(49)limsupn⟶+∞−Δpmnε,mnεX′,X=ηε,mεX′,X.
Then again by Lemma1, we conclude that(50)ηε=−ΔpmεinX′,which completes the proof of Proposition 2.
We need to show thatmε satisfies equality (14) in Proposition 1. Let Φ∈Lp0,T,W01,pΩ. Using the fact that ηε=−Δpmε in (45), we get(51)11+β2β∫0t∫Ω∂tmε⋅σsωidxds−∫0t∫Ωmε∧∂tmε⋅σsωidxds=∫0tΔpmε,σsωiX′,Xds−1ε∫0t∫Ωmε2−1mε⋅σsωidxds.Let Sn=∑i=1naitωisuch thataiare smooth,n∈ℕ. Since ∪n≥1Sn is dense in Lp0,T,W01,pΩ, then for any Φ∈Lp0,T,W01,pΩ there exists a sequence Φnn⊂Sn such that limn⟶+∞Φn=Φ in Lp0,T,W01,pΩ. Hence by replacing ωi by Φn and taking σs=1 in (51), we obtain(52)11+β2β∫0t∫Ω∂tmε⋅Φndxds−∫0t∫Ωmε∧∂tmε⋅Φndxds=∫0tΔpmε,ΦnX′,Xds−1ε∫0t∫Ωmε2−1mε⋅Φndxds.
Take the limit asn⟶+∞ on both sides of (52) to verify that (71) holds for an arbitrary Φ in Lp0,T,W01,pΩ. Inequality (14) is a consequence of (26) due to the lower semi-continuity, hence the proof of Proposition 1 is complete.


### 3.2. Existence Result for the Problem (2)

#### 3.2.1. Some Properties of the Sequencemεε

Our aim here is the passage to the limit whenε goes to 0. For this we need to prove the two following results. The first one isLemma 3.
Ifmε is a solution of the penalized problem (5), then mε≤1 a.e in Q.Proof.
We test the first equation in (5) by 1−min1,mε/mεmε to obtain(53)β1+β2∫mε≥11−1mε∂tmε⋅mεdx+∫mε≥1∇mεp−2∇mε∇1−1mεmεdx=0.
A direct computation leads to(54)β21+β2ddt∫|mε|≥1mε−12dx+∫|mε|≥1∇mεpmεmε−1dx=0.
We conclude that(55)∫mε≥1mε−12dx≤∫mε≥1mε0,.−12dx=0,then mε≤1 a.e in Q.
The second result lies with heat flow ofp-harmonic maps (see [26, 28] for p=2).Proposition 3.
Letε>0 and mε satisfying in the sense of distribution the following equation:(56)β∂tmε−1+β2Δpmε=θε.
Assume boundedness ofmεε in L∞0,T,W01,pΩ, ∂tmεε in L2Q, and θεε in L1Q. Then for a subsequence ∇mε⟶∇m strongly in LrQ for all 1≤r<p.Proof.
Taking the limit in (15) when ε⟶0 by using the lower semi-continuity of the norm under the weak topology, we get(57)β1+β2∫Q∂tmε2dxdt+1p∫Ω∇mεpdx+14ε∫Ωmε2−12dx≤1p∫Ω∇m0pdx.
Then up to a subsequencemε⇀m weakly in L∞0,T,W01,pΩ and mε⟶m strongly in LpQ and a.e in Q. For a fixed 0<δ≤1, we define the following set Eδε by(58)Eδε≔t,x∈0,T×Ω,mεt,x−mt,x≥δ.
Letk>0 be fixed such that mεX≤k and θεL1Q≤k for all ε>0. Sobolev embedding and Hölder’s inequality imply that(59)∫Q∇mε−∇mrdxdt=∫Q\Eδε∇mε−∇mrdxdt+∫Eδε∇mε−∇mrdxdt≤C∫Q\Eδε∇mε−∇mpdxdt+krmeasEδεp−r/p,for 1≤r<p.
Next, we shall estimate∫Q\Eδε∇mε−∇mpdxdt.
LetTδu be the usual vector truncation function defined as(60)Tδu=uifu≤δ,δuuifu>δ.
So, by Lemma2, we obtain(61)2p−2∫Q\Eδε∇mε−∇mpdxdt≤∫Q\Eδε∇mεp−2∇mε−∇mp−2∇m⋅∇mε−∇mdxdt=∫Q∇mεp−2∇mε⋅∇Tδmε−mdxdt−δ∫Eδε∇mεp−2∇mε⋅∇mε−mmε−mdxdt−∫Q\Eδε∇mp−2∇m⋅∇mε−∇mdxdt=A1+A2+A3,where(62)A1=∫Q∇mεp−2∇mε⋅∇Tδmε−mdxdt,A2=−δ∫Eδε∇mεp−2∇mε⋅∇mε−mmε−mdxdt,(63)A3=−∫Q\Eδε∇mp−2∇m⋅∇mε−∇mdxdt.(i)
Estimate ofA1.From (56), we have(64)A1≤β1+β2∫Q∂tmε⋅Tδmε−mdxdt+∫Qθε⋅Tδmε−mdxdt≤β1+β2δ∂tmεL1Q+δθεL1Q≤Cδk.(ii)
Estimate ofA2.We have(65)A2=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−∇mi−miε−mimjε−mj∇mjε−∇mjdxdt=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−miε−mimjε−mj∇mjεdxdt+δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇mi−miε−mimjε−mj∇mjdxdt=A21+A22.We remark that(66)A21=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−miε−mimjε−mj∇mjεdxdt≤0.Using the fact thatδ/mε−m≤1 in Eδε, we get(67)A22≤2∫Eδε∇m‖∇mεp−1dxdt≤2kp−1∫Eδε∇mpdxdt1/p.(iii)
Estimate ofA3.By Hölder’s inequality and the weak convergence∇mε⇀∇m in LpQ, we obtain(68)A3≤∫Q∇mp−2∇m⋅∇mε−∇mdxdt+∫Eδε∇mp−2∇m⋅∇mε−∇mdxdt≤A3ε+2k∫Eδε∇mpdxdtp−1/p,where(69)A3ε=∫Q∇mp−2∇m⋅∇mε−∇mdxdt⟶0,asε⟶0.
Finally, we get(70)2p−2∫Q\Eδε∇mε−∇mpdxdt≤A1+A22+A3.
Using the fact thatm is absolutely continuous, we have limε⟶0measEδε  = 0 for enough small δ and from (59) and (70), we get the strong convergence of ∇mεε to ∇m in LrQ.


### 3.3. Passage to the Limit Whenε Goes to 0

In order to pass to the limit in (12), let φ∈Lp0,T,W01,pΩ∩L∞Q and Φε=mε∧φ. Since Φε belong to Lp0,T,W01,pΩ, there holds(71)β∫0t∫Ω∂tmε⋅mε∧φdxds−∫0t∫Ωmε∧∂tmε⋅mε∧φdxds=−1+β2∫0t∫Ω∇mεp−2∇mε⋅mε∧∇φdxds.On the other hand, we get(72)∫0t∫Ωmε∧∂tmε⋅mε∧φdxds=∫0t∫Ωmε⋅∂tmεmε⋅φ−mε2∂tmε⋅φdxds=∫0t∫Ωmε⋅∂tmεmε⋅φ−∂tmε⋅φdxds−∫0t∫Ωmε2−1∂tmε⋅φdxds⟶−∫0t∫Ω∂tm⋅φdxds,asε⟶0.From the estimate (57), we deduce that(73)∇mεp−2∇mε⇀ζweakly inLp′Q.Thanks to the convergence almost everywhere of∇mεε to ∇m in Proposition 3, we obtain that ζ=∇mp−2∇m. Since mε⟶m strongly in LpQ then(74)limε⟶0∫0t∫Ω∇mεp−2∇mε⋅mε∧∇φdxds=∫0t∫Ω∇mp−2∇m⋅m∧∇φdxds.Hence we can pass to the limit in (71), to obtain (5). Inequality (6) is a consequence of (14) by using the lower semi-continuity of the norms ⋅L2Q and ⋅L∞0,T,W01,pΩ under the weak topology as ε⟶0. To get the Hölder regularity, we use (4) to get(75)mt,.−mτ,.2=∫tτ∂tmds2≤t−τ1/2∫Q∂tm2dxds1/2≤Ct−τ1/2,which complete the proof of our main Theorem 1.


## 3.1. Existence for the Penalized Problem

Our aim is to prove Theorem1. We begin by constructing weak solutions to a penalized problem associated with (2) where the constraint m=1 is relaxed, i.e., for ε>0 we introduce(12)11+β2β∂tmε−mε∧∂tmε=Δpmε−1εmε2−1mεinQ,mε=0on0,T×∂Ω,mε0,.=m0inΩ..We have the following result.Proposition 1.
Letm0∈W01,pΩ such that m0=1 a.e and p satisfies the hypothesis ℋ. Then for fixed ε>0, there exists a weak solution mε for the penalized problem (5) such that(i)
For allT>0, mε∈L∞0,T,W01,pΩ, ∂tmε∈L2Q;(ii)
For allΦ∈Lp0,T,W01,pΩ, there holds(13)β∫Q∂tmε⋅Φdxdt−∫Qmε∧∂tmε⋅Φdxdt=−1+β2∫Q∇mεp−2∇mε⋅∇Φdxdt−1+β2ε∫Qmε2−1mε⋅Φdxdt.(iii)
The following energy estimate holds(14)β1+β2∫Q∂tmε2dxdt+1p∫Ω∇mεpdx+14ε∫Ωmε2−12dx≤1p∫Ω∇m0pdx,for a.e. t∈0,T.Proof.
The spaceW01,pΩ is separable for 1≤p<+∞, then there exists a sequence of finite dimensional subspaces Vn=spanω1,…,ωn of W01,pΩ such that ∪n≥1Vn is dense in W01,pΩ. For a fixed positive integer n, we look for approximate solution mnε for (5) in the form(15)mnεt,x=∑i=1naintωix,such that(16)11+β2∫Ωβ∂tmnε−mnε∧∂tmnε⋅ωidx=−∫Ω∇mnεp−2∇mnε⋅∇ωidx−1ε∫Ωmnε2−1mnε⋅ωidx,for i=1,…,n with(17)mnε0,.=m0n,where m0n is the approximations of the initial data m0 with(18)m0nt,x=∑i=1na0intωix.We assume that(19)m0n⟶m0strongly inW01,pΩ.
We setan=a1n,…,ann and a0n=a01n,…,a0nn. Then the differential system (16) with initial condition (17) satisfying (18) and (19) can be written as(20)Aandandt=Fan,an0=a0n.
In fact, we have(21)β∂tmnε−mnε∧∂tmnε=Amnε∂tmnε,where(22)Amnε=β−mn3εmn2εmn3εβ−mn1ε−mn2εmn1εβ,such that mniε for i=1,2,3 are the components of the vector mnε. In addition(23)detAmnε=ββ2+mnε2≠0.
Then the matrixAan is invertible for all an, then the system (20) becomes(24)dandt=Aan−1Fan,an0=a0n.
SinceF satisfies the Caratheodory conditions, the existence of a solution to the system (24) is ensured due to Theorem 2. The global existence then follows from the following a priori estimates.(i)
A priori estimates.We multiply (16) by dain/dt, add up the obtained equations for i=1,…,n, we obtain(25)β1+β2∫Ω∂tmnε2dx+ddt1p∫Ω∇mnεpdx+14ε∫Ωmnε2−12dx=0.Integrating (25) with respect to t leads to(26)β1+β2∫Q∂tmnε2dxdt+1p∫Ω∇mnεpdx+14ε∫Ωmnε2−12dx=1p∫Ω∇m0npdx+14ε∫Ωm0n2−12dx.(ii)
Convergences and passage to the limit whenn goes to +∞.Following hypothesisℋ, we infer thatW01,pΩL4Ω. Thanks to (19), we deduce that the right-hand side of (26) is uniformly bounded with respect to n. Then by Poincaré’s inequality, we get(27)∂tmnεnis bounded inL2Q,mnεnis bounded inL∞0,T,W01,pΩ,(28)mnεnis bounded inL∞0,T,L4Ω.
Hence the following convergences hold up to a subsequence(29)∂tmnε⇀∂tmεweakly inL2Q,mnε⇀mεweakly−⋆inL∞0,T,W01,pΩ.
Since the Sobolev embeddingW01,pΩ⇀LpΩis compact and LpΩ⇀L2Ω is continuous, we use Aubin–Lions lemma to obtain the strong convergence as(30)mnε⟶mεstrongly inC0,T,LpΩand a.e.Proposition 2.
Ifmnε and mε satisfies the above convergences, then we have(31)Δpmnε⇀Δpmεweakly inLp′0,T,W−1,p′Ω.Proof.
Since1/p+1/p′=1, we get from (15)(32)∫0t∫Ω∇mnε|p−2∇mnεp′dxds=∫0t∇mnεppds≤C,for allt∈0,T,where C is a positive constant independent of n. Hence the sequence ∇mnεp−2∇mnεn is bounded in Lp′Q, so there exists a subsequence such that(33)∇mnεp−2∇mnε⇀ξεweakly inLp′Q,for some ξε∈Lp′Q. For the rest of the proof we use the following notation:(34)X=Lp0,T,W01,pΩ,X′=Lp′0,T,W−1,p′Ω,where .,.X′,X denote the duality pairing between X′ and X. As n⟶+∞, from (33), we have(35)−Δpmnε,ϕX′,X=∫0T∫Ω∇mnεp−2∇mnε⋅∇ϕdxdt⟶∫0T∫Ωξε⋅∇ϕdxdt.
Then(36)−Δpmnε⇀ηεweakly−⋆inX′.
In order to show thatηε coincides with −Δpmε, we will use the fact that the operator −Δp:X⟶X′ is a maximal monotone. It follows from Lemma 1 that −Δpmε=ηε if we prove the inequality, we get(37)lim supn⟶+∞−Δpmnε,mnεX′,X≤ηε,mεX′,X.
In order to verify (37) fix n. Then, multiply (16) by ai, and sum over i=1,…,n, to arrive at(38)β1+β2∫Ω∂tmnε⋅mnεdx=−∫Ω∇mnεp−2∇mnε⋅∇mnεdx−1ε∫Ωmnε2−1mnε2dx.
Integrate identity (38) over s∈0,t to obtain(39)∫0t∫Ω∇mnεp−2∇mnε⋅∇mnεdxds=−β1+β2∫0t∫Ω∂tmnε⋅mnεdxds+1ε∫0t∫Ωmnε2−1mnε2dxds.
The weak convergence of∂tmnεn in L2Q and the strong convergence of mnεn in L2Q imply that(40)limn⟶+∞∫0t∫Ω∂tmnε⋅mnεdxds=∫0t∫Ω∂tmε⋅mεdxds.
Now from the estimate (26), we deduce that mnε2−1mnεn is bounded in Lp′Q. Then mnε2−1mnε⇀ρε weakly in Lp′Q and using the fact that mnε⟶mε a.e, one can show that ρε=mε2−1mε. Therefore, we get(41)∫0t∫Ωmnε2−1mnε2dxds⟶∫0t∫Ωmε2−1mε2dxds.
Taking the limit superior in (39) as n goes to infinity, we obtain in particular that(42)lim supn⟶+∞∫0t∫Ω∇mnεp−2∇mnε⋅∇mnεdxds=−β1+β2∫0t∫Ω∂tmε⋅mεdxds+1ε∫0t∫Ωmε2−1mε2dxds.
In the following, we show that∫0tηε,mεX′,Xds is equal to the right-hand side of (42) for a.e t∈0,T, i.e.,(43)∫0tηε,mεX′,Xds=−β1+β2∫0t∫Ω∂tmε⋅mεdxds+1ε∫0t∫Ωmε2−1mε2dxds.
We multiply (16) by an arbitrary function σ∈C0,T, and integrate it from 0 to t obtain(44)−11+β2β∫0t∫Ω∂tmnε⋅σsωidxds−∫0t∫Ωmnε∧∂tmnε⋅σsωidxds=∫0tΔpmnε,σsωiX′,Xds+1ε∫0t∫Ωmnε2−1mnε⋅σsωidxds.
Taking the limit asn⟶+∞ on both sides of (44), we get(45)−11+β2β∫0t∫Ω∂tmε⋅σsωidxds−∫0t∫Ωmε∧∂tmε⋅σsωidxds=∫0tηε,σsωiX′,Xds+1ε∫0t∫Ωmε2−1mε⋅σsωidxds.
Since (45) holds for each i=1,…,n, we may replace σsωi by mnε to get(46)11+β2β∫0t∫Ω∂tmε⋅mnεdxds−∫0t∫Ωmε∧∂tmε⋅mnεdxds=−∫0tηε,mnεX′,Xds−1ε∫0t∫Ωmε2−1mε⋅mnεdxds.
Using the previous convergences, we can pass to the limit in (46) to obtain (43). Noting that for the convergence of the term ∫0t∫Ωmε∧∂tmε⋅mnεdxds, we have(47)∫0t∫Ωmε∧∂tmε⋅mnεdxds=∫0t∫Ωmnε∧mε⋅∂tmεdxds.
We know thatmnεn is bounded in L4Q and mε∈L4Q then the sequence mnε∧mεn is bounded in L2Q. Thanks to the convergence a.e of mnεn to mε, we obtain that mnε∧mε⇀mε∧mε=0 weakly in L2Q. Therefore, we get(48)∫0t∫Ωmnε∧mε⋅∂tmεdxds⟶0,asn⟶+∞.
Now, it follows from (42) and (43) that(49)limsupn⟶+∞−Δpmnε,mnεX′,X=ηε,mεX′,X.
Then again by Lemma1, we conclude that(50)ηε=−ΔpmεinX′,which completes the proof of Proposition 2.
We need to show thatmε satisfies equality (14) in Proposition 1. Let Φ∈Lp0,T,W01,pΩ. Using the fact that ηε=−Δpmε in (45), we get(51)11+β2β∫0t∫Ω∂tmε⋅σsωidxds−∫0t∫Ωmε∧∂tmε⋅σsωidxds=∫0tΔpmε,σsωiX′,Xds−1ε∫0t∫Ωmε2−1mε⋅σsωidxds.Let Sn=∑i=1naitωisuch thataiare smooth,n∈ℕ. Since ∪n≥1Sn is dense in Lp0,T,W01,pΩ, then for any Φ∈Lp0,T,W01,pΩ there exists a sequence Φnn⊂Sn such that limn⟶+∞Φn=Φ in Lp0,T,W01,pΩ. Hence by replacing ωi by Φn and taking σs=1 in (51), we obtain(52)11+β2β∫0t∫Ω∂tmε⋅Φndxds−∫0t∫Ωmε∧∂tmε⋅Φndxds=∫0tΔpmε,ΦnX′,Xds−1ε∫0t∫Ωmε2−1mε⋅Φndxds.
Take the limit asn⟶+∞ on both sides of (52) to verify that (71) holds for an arbitrary Φ in Lp0,T,W01,pΩ. Inequality (14) is a consequence of (26) due to the lower semi-continuity, hence the proof of Proposition 1 is complete.


## 3.2. Existence Result for the Problem (2)

### 3.2.1. Some Properties of the Sequencemεε

Our aim here is the passage to the limit whenε goes to 0. For this we need to prove the two following results. The first one isLemma 3.
Ifmε is a solution of the penalized problem (5), then mε≤1 a.e in Q.Proof.
We test the first equation in (5) by 1−min1,mε/mεmε to obtain(53)β1+β2∫mε≥11−1mε∂tmε⋅mεdx+∫mε≥1∇mεp−2∇mε∇1−1mεmεdx=0.
A direct computation leads to(54)β21+β2ddt∫|mε|≥1mε−12dx+∫|mε|≥1∇mεpmεmε−1dx=0.
We conclude that(55)∫mε≥1mε−12dx≤∫mε≥1mε0,.−12dx=0,then mε≤1 a.e in Q.
The second result lies with heat flow ofp-harmonic maps (see [26, 28] for p=2).Proposition 3.
Letε>0 and mε satisfying in the sense of distribution the following equation:(56)β∂tmε−1+β2Δpmε=θε.
Assume boundedness ofmεε in L∞0,T,W01,pΩ, ∂tmεε in L2Q, and θεε in L1Q. Then for a subsequence ∇mε⟶∇m strongly in LrQ for all 1≤r<p.Proof.
Taking the limit in (15) when ε⟶0 by using the lower semi-continuity of the norm under the weak topology, we get(57)β1+β2∫Q∂tmε2dxdt+1p∫Ω∇mεpdx+14ε∫Ωmε2−12dx≤1p∫Ω∇m0pdx.
Then up to a subsequencemε⇀m weakly in L∞0,T,W01,pΩ and mε⟶m strongly in LpQ and a.e in Q. For a fixed 0<δ≤1, we define the following set Eδε by(58)Eδε≔t,x∈0,T×Ω,mεt,x−mt,x≥δ.
Letk>0 be fixed such that mεX≤k and θεL1Q≤k for all ε>0. Sobolev embedding and Hölder’s inequality imply that(59)∫Q∇mε−∇mrdxdt=∫Q\Eδε∇mε−∇mrdxdt+∫Eδε∇mε−∇mrdxdt≤C∫Q\Eδε∇mε−∇mpdxdt+krmeasEδεp−r/p,for 1≤r<p.
Next, we shall estimate∫Q\Eδε∇mε−∇mpdxdt.
LetTδu be the usual vector truncation function defined as(60)Tδu=uifu≤δ,δuuifu>δ.
So, by Lemma2, we obtain(61)2p−2∫Q\Eδε∇mε−∇mpdxdt≤∫Q\Eδε∇mεp−2∇mε−∇mp−2∇m⋅∇mε−∇mdxdt=∫Q∇mεp−2∇mε⋅∇Tδmε−mdxdt−δ∫Eδε∇mεp−2∇mε⋅∇mε−mmε−mdxdt−∫Q\Eδε∇mp−2∇m⋅∇mε−∇mdxdt=A1+A2+A3,where(62)A1=∫Q∇mεp−2∇mε⋅∇Tδmε−mdxdt,A2=−δ∫Eδε∇mεp−2∇mε⋅∇mε−mmε−mdxdt,(63)A3=−∫Q\Eδε∇mp−2∇m⋅∇mε−∇mdxdt.(i)
Estimate ofA1.From (56), we have(64)A1≤β1+β2∫Q∂tmε⋅Tδmε−mdxdt+∫Qθε⋅Tδmε−mdxdt≤β1+β2δ∂tmεL1Q+δθεL1Q≤Cδk.(ii)
Estimate ofA2.We have(65)A2=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−∇mi−miε−mimjε−mj∇mjε−∇mjdxdt=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−miε−mimjε−mj∇mjεdxdt+δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇mi−miε−mimjε−mj∇mjdxdt=A21+A22.We remark that(66)A21=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−miε−mimjε−mj∇mjεdxdt≤0.Using the fact thatδ/mε−m≤1 in Eδε, we get(67)A22≤2∫Eδε∇m‖∇mεp−1dxdt≤2kp−1∫Eδε∇mpdxdt1/p.(iii)
Estimate ofA3.By Hölder’s inequality and the weak convergence∇mε⇀∇m in LpQ, we obtain(68)A3≤∫Q∇mp−2∇m⋅∇mε−∇mdxdt+∫Eδε∇mp−2∇m⋅∇mε−∇mdxdt≤A3ε+2k∫Eδε∇mpdxdtp−1/p,where(69)A3ε=∫Q∇mp−2∇m⋅∇mε−∇mdxdt⟶0,asε⟶0.
Finally, we get(70)2p−2∫Q\Eδε∇mε−∇mpdxdt≤A1+A22+A3.
Using the fact thatm is absolutely continuous, we have limε⟶0measEδε  = 0 for enough small δ and from (59) and (70), we get the strong convergence of ∇mεε to ∇m in LrQ.


## 3.2.1. Some Properties of the Sequencemεε

Our aim here is the passage to the limit whenε goes to 0. For this we need to prove the two following results. The first one isLemma 3.
Ifmε is a solution of the penalized problem (5), then mε≤1 a.e in Q.Proof.
We test the first equation in (5) by 1−min1,mε/mεmε to obtain(53)β1+β2∫mε≥11−1mε∂tmε⋅mεdx+∫mε≥1∇mεp−2∇mε∇1−1mεmεdx=0.
A direct computation leads to(54)β21+β2ddt∫|mε|≥1mε−12dx+∫|mε|≥1∇mεpmεmε−1dx=0.
We conclude that(55)∫mε≥1mε−12dx≤∫mε≥1mε0,.−12dx=0,then mε≤1 a.e in Q.
The second result lies with heat flow ofp-harmonic maps (see [26, 28] for p=2).Proposition 3.
Letε>0 and mε satisfying in the sense of distribution the following equation:(56)β∂tmε−1+β2Δpmε=θε.
Assume boundedness ofmεε in L∞0,T,W01,pΩ, ∂tmεε in L2Q, and θεε in L1Q. Then for a subsequence ∇mε⟶∇m strongly in LrQ for all 1≤r<p.Proof.
Taking the limit in (15) when ε⟶0 by using the lower semi-continuity of the norm under the weak topology, we get(57)β1+β2∫Q∂tmε2dxdt+1p∫Ω∇mεpdx+14ε∫Ωmε2−12dx≤1p∫Ω∇m0pdx.
Then up to a subsequencemε⇀m weakly in L∞0,T,W01,pΩ and mε⟶m strongly in LpQ and a.e in Q. For a fixed 0<δ≤1, we define the following set Eδε by(58)Eδε≔t,x∈0,T×Ω,mεt,x−mt,x≥δ.
Letk>0 be fixed such that mεX≤k and θεL1Q≤k for all ε>0. Sobolev embedding and Hölder’s inequality imply that(59)∫Q∇mε−∇mrdxdt=∫Q\Eδε∇mε−∇mrdxdt+∫Eδε∇mε−∇mrdxdt≤C∫Q\Eδε∇mε−∇mpdxdt+krmeasEδεp−r/p,for 1≤r<p.
Next, we shall estimate∫Q\Eδε∇mε−∇mpdxdt.
LetTδu be the usual vector truncation function defined as(60)Tδu=uifu≤δ,δuuifu>δ.
So, by Lemma2, we obtain(61)2p−2∫Q\Eδε∇mε−∇mpdxdt≤∫Q\Eδε∇mεp−2∇mε−∇mp−2∇m⋅∇mε−∇mdxdt=∫Q∇mεp−2∇mε⋅∇Tδmε−mdxdt−δ∫Eδε∇mεp−2∇mε⋅∇mε−mmε−mdxdt−∫Q\Eδε∇mp−2∇m⋅∇mε−∇mdxdt=A1+A2+A3,where(62)A1=∫Q∇mεp−2∇mε⋅∇Tδmε−mdxdt,A2=−δ∫Eδε∇mεp−2∇mε⋅∇mε−mmε−mdxdt,(63)A3=−∫Q\Eδε∇mp−2∇m⋅∇mε−∇mdxdt.(i)
Estimate ofA1.From (56), we have(64)A1≤β1+β2∫Q∂tmε⋅Tδmε−mdxdt+∫Qθε⋅Tδmε−mdxdt≤β1+β2δ∂tmεL1Q+δθεL1Q≤Cδk.(ii)
Estimate ofA2.We have(65)A2=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−∇mi−miε−mimjε−mj∇mjε−∇mjdxdt=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−miε−mimjε−mj∇mjεdxdt+δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇mi−miε−mimjε−mj∇mjdxdt=A21+A22.We remark that(66)A21=−δ∫Eδε∇mεp−2mε−m3∇miεmε−m2∇miε−miε−mimjε−mj∇mjεdxdt≤0.Using the fact thatδ/mε−m≤1 in Eδε, we get(67)A22≤2∫Eδε∇m‖∇mεp−1dxdt≤2kp−1∫Eδε∇mpdxdt1/p.(iii)
Estimate ofA3.By Hölder’s inequality and the weak convergence∇mε⇀∇m in LpQ, we obtain(68)A3≤∫Q∇mp−2∇m⋅∇mε−∇mdxdt+∫Eδε∇mp−2∇m⋅∇mε−∇mdxdt≤A3ε+2k∫Eδε∇mpdxdtp−1/p,where(69)A3ε=∫Q∇mp−2∇m⋅∇mε−∇mdxdt⟶0,asε⟶0.
Finally, we get(70)2p−2∫Q\Eδε∇mε−∇mpdxdt≤A1+A22+A3.
Using the fact thatm is absolutely continuous, we have limε⟶0measEδε  = 0 for enough small δ and from (59) and (70), we get the strong convergence of ∇mεε to ∇m in LrQ.


## 3.3. Passage to the Limit Whenε Goes to 0

In order to pass to the limit in (12), let φ∈Lp0,T,W01,pΩ∩L∞Q and Φε=mε∧φ. Since Φε belong to Lp0,T,W01,pΩ, there holds(71)β∫0t∫Ω∂tmε⋅mε∧φdxds−∫0t∫Ωmε∧∂tmε⋅mε∧φdxds=−1+β2∫0t∫Ω∇mεp−2∇mε⋅mε∧∇φdxds.On the other hand, we get(72)∫0t∫Ωmε∧∂tmε⋅mε∧φdxds=∫0t∫Ωmε⋅∂tmεmε⋅φ−mε2∂tmε⋅φdxds=∫0t∫Ωmε⋅∂tmεmε⋅φ−∂tmε⋅φdxds−∫0t∫Ωmε2−1∂tmε⋅φdxds⟶−∫0t∫Ω∂tm⋅φdxds,asε⟶0.From the estimate (57), we deduce that(73)∇mεp−2∇mε⇀ζweakly inLp′Q.Thanks to the convergence almost everywhere of∇mεε to ∇m in Proposition 3, we obtain that ζ=∇mp−2∇m. Since mε⟶m strongly in LpQ then(74)limε⟶0∫0t∫Ω∇mεp−2∇mε⋅mε∧∇φdxds=∫0t∫Ω∇mp−2∇m⋅m∧∇φdxds.Hence we can pass to the limit in (71), to obtain (5). Inequality (6) is a consequence of (14) by using the lower semi-continuity of the norms ⋅L2Q and ⋅L∞0,T,W01,pΩ under the weak topology as ε⟶0. To get the Hölder regularity, we use (4) to get(75)mt,.−mτ,.2=∫tτ∂tmds2≤t−τ1/2∫Q∂tm2dxds1/2≤Ct−τ1/2,which complete the proof of our main Theorem 1.


## 4. Concluding Remarks

We were concerned with a LL equation involving ap-Laplacian as the exchange field. Global existence of weak solutions is obtained and the key of the proof is penalization combined with some compactness arguments and monotonicity method. As a perspective to the present work, it will be interesting to investigate the limiting behaviors of the obtained solutions with respect to a small and a large damping parameter. Existence of weak solutions to the LL model in the case where the exchange field is expressed in term of a p-Laplacian operator with 1<p<2 is also a very challenging question.



---

*Source: 9968295-2022-09-17.xml*