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Comptes Rendus Mathématique
Volume 355, n° 8
pages 892-902 (août 2017)
Doi : 10.1016/j.crma.2017.07.009
Received : 1 December 2016 ;  accepted : 20 July 2017
Well-posedness of the scalar and the vector advection–reaction problems in Banach graph spaces
Analyse des problèmes d'advection–réaction scalaire et vectoriel dans les espaces de Banach du graphe
 

Pierre Cantin
 Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée cedex 2, France 

Abstract

An extension of the well-posedness analysis of the scalar and the vector advection–reaction problem in Banach graph spaces of power   is proposed. This analysis is based on the sign of the associated Friedrichs tensor, taking positive, null or reasonably negative values.

The full text of this article is available in PDF format.
Résumé

Cette Note propose une extension de l'analyse de la bonne position des problèmes d'advection–réaction scalaire et vectorielle dans les espaces du graphe de Banach de puissance  . Cette analyse étend l'hypothèse sur le signe du tenseur de Friedrichs associé à ces problèmes, permettant ainsi de considérer le cas où ce tenseur prend des valeurs positives, nulles ou raisonnablement négatives.

The full text of this article is available in PDF format.
Introduction

Let Ω be a domain of   with Lipschitz-continuous boundary ∂Ω and consider   and   solving the following first-order homogeneous boundary valued problems
(1a)β⋅∇u+μu=s a.e. in Ω,(1b)u=0  a.e. on ∂Ω−, and(2a)∇(β⋅u)+(∇×u)×β+μu=s a.e. in Ω,(2b)u=0  a.e. on ∂Ω−. In this paper, β denotes a Lipschitz-continuous  -valued vector field on Ω, and μ and μ denote two bounded reaction coefficients taking   and   values, respectively. The inflow boundary   is defined as  , with n the exterior unit normal of ∂Ω.

The first problem ((1a), (1b)) has been studied several times in the literature. We mention in particular the pioneering work of Bardos [[1]] and Beirão da Veiga [[2]] for the well-posedness analysis in smooth domains with regular model parameters and also the work of DiPerna & Lions [[6]] when the problem is expressed in unbounded domains with irregular model parameters. More recently, Girault & Tartar [[10]] proved (using a viscous and a Yosida regularization) the well-posedness of these problem in   for all   under the assumption  , and also the  -regularity of solution to ((1a), (1b)) if   and if   is sufficiently small. Regarding the problem ((2a), (2b)), there is very little work in the literature. However, this problem models physical situations, such as the static advection of a magnetic field in a conductor of conductivity μ and of velocity β . In the context of the differential geometry, this problem is also very important, since it represents the Lie derivative of a so-called 1-form in a three-dimensional domain; see Bossavit [[3]].

In this paper, we analyze the well-posedness of problems ((1a), (1b)) and ((2a), (2b)) in Banach graph spaces of power  . Observing that these problems define two Friedrichs systems (see Ern & al. [[8], [9]]), the well-posedness is a consequence of the positivity of the  -valued Friedrichs tensor
(3)σβ,μ;p:=μ−1p∇⋅β, for the first problem ((1a), (1b)) and of the positivity of the lowest eigenvalue of the  -valued Friedrichs tensor(4) for the second problem ((2a), (2b)). The first contribution of this work concerns the well-posedness in Banach graph spaces. Following the analysis of Friedrichs system in Hilbert space proposed in the aforementioned works, we establish the well-posedness of these two problems for positive (in the sense above) Friedrichs tensors ((3)) and ((4)). The second part of this work is devoted to the analysis when these assumptions are not satisfied. Introducing a so-called potential (whose existence follows from the regularity and the trajectory of the vector field in Ω, see Devinatz & al. [[5]]), we prove that one may extend the Friedrichs positivity assumptions so as to consider null or reasonably negative tensors.

This paper is organized as follows. First, some notations are introduced and we recall the classical statement of the Banach–Nečas–Babuška (BNB) theorem. Section 2 is concerned with the scalar problem ((1a), (1b)); we prove that this problem is well posed in the Banach graph space of power   if the infimum of the Friedrichs tensor ((3)) takes positive values, and we extend this result to consider null or reasonably negative values. In Section 3, we extend these results to prove the well-posedness of the vector problem ((2a), (2b)) under similar assumptions on the Friedrichs tensor ((4)).

Notations

In this paper, p denotes any real number in   with   its conjuguate number such that  . The inner, the cross and the tensor products in   are denoted by ⋅, × and ⊗ respectively. To alleviate the notation,   denotes either the Lebesgue measure of a set, the absolute value of a real number, the Euclidean norm of a vector or the Frobenius norm of a tensor. As usual, the Banach space   collects all measurable functions  , whose absolute value raised to the power p is Lebesgue integrable, i.e.  . Similarly, the Banach space   collects all measurable functions   whose Euclidean norm raised to the power p is Lebesgue integrable, i.e.  . We denote by   (resp.  ) the space of infinitely differentiable  -valued functions (resp.  -valued) on Ω and   (resp.  ) the subspace of those that are compactly supported in Ω.

Banach–Nečas–Babuška (BNB) theorem

Consider the following abstract variational problem
(5)Find u∈U  s.t.  a(u,v)=〈f,v〉V′,V,∀v∈V, where U and V are two Banach spaces equipped with   and  , respectively, V is reflexive,  ,   and   is the duality pairing between   and V . A necessary and sufficient condition for ((5)) to be well posed is given by the (BNB) theorem, see, e.g., Ern & Guermond [[7]].

Theorem 1.1

The abstract problem ((5)) is well posed if and only if:

(BNB1) there exists such that

(BNB2) for all  ,  .

Scalar advection–reaction problem

This section analyzes the well-posedness of the continuous problem ((1a), (1b)) in Banach graph spaces and generalizes the sign condition on the Friedrichs tensor   defined by ((3)).

The graph space

The Banach graph space of power p associated with ((1a), (1b)) is defined by
(6)Vβ;p(Ω):={v∈Lp(Ω)|β⋅∇v∈Lp(Ω)}, and is equipped with the norm   for all  . This space defines a reflexive Banach space owing to the first and the second Clarkson inequalities (see Brezis [[4]]) where for all  ,   means that the linear form(7)Cc∞(Ω)∋φ↦−∫Ωv∇⋅(βφ), is bounded in  , so that   is the Riesz representative of ((7)) in  . To specify the meaning of the trace of a function in  , we introduce the space   given by(8)Lp(|β⋅n|;∂Ω):={v:∂Ω→R|v is Lebesgue measurable on ∂Ω and ∫∂Ω|β⋅n||v|p<∞}, which is a Banach space when equipped with the norm   for all  . As observed by Ern & Guermond [[8]], the existence of traces in   for a function in   is not always guaranteed. A necessary and sufficient condition is the well-separation of the boundary ∂Ω with respect to the vector field β , i.e.(9)dist(∂Ω−,∂Ω+)>0 with ∂Ω±={x∈∂Ω|±β(x)⋅n(x)>0}. In this following, we always assume that this condition is satisfied. Let us adapt the proof of [[8]] to the general case   to prove the existence of such traces.

Lemma 2.1

The map   with   for all  , extends continuously to  , i.e. there exists   such that
||γ(v)||Lp(|β⋅n|;∂Ω)≤Cγ||v||Vβ;p(Ω),∀v∈Vβ;p(Ω).

Proof

Owing to the separation of the boundary from assumption ((9)), there exist   such that   on ∂Ω,  ,   and  . Proceeding as in [[8]], we infer that
∫∂Ω|β⋅n||φ|p=∫∂Ω(β⋅n)|φ|p(ψ+−ψ−)=∫Ω∇⋅(β|φ|p(ψ+−ψ−)),∀φ∈C∞(Ω‾), where we have used the partition of the unity on the boundary and the Stokes formula. Applying now the Leibniz product rule and recalling that  , we obtain∫∂Ω|β⋅n||φ|p=p∫Ω(ψ+−ψ−)(β⋅∇φ)φ|φ|p−2+∫Ω|φ|p∇⋅(β(ψ+−ψ−)). Next, Hölder's and Young's inequalities along with the identity   yield∫Ω|(β⋅∇φ)φ|φ|p−2|≤||β⋅∇φ||Lp(Ω)||φ||Lp(Ω)p/p′≤1p||β⋅∇φ||Lp(Ω)p+1p′||φ||Lp(Ω)p. It follows that   with the constant  . Then, observing that  , we obtain||φ||Lp(|β⋅n|;∂Ω)≤Cγ||φ||Vβ;p(Ω),∀φ∈C∞(Ω‾), with  . Finally, recalling that   is dense in   for all   (see Jensen [[12]]), this inequality holds as well for any function in  .

Owing to the existence of traces in  , the following integration by parts formulae hold.
Lemma 2.2

For all   and for all  ,
(10a)∫Ω(β⋅∇v)w+∫Ω(β⋅∇w)v+∫Ω(∇⋅β)vw=∫∂Ω(β⋅n)vw. In addition, for all   and for all  , (10b)∫Ω(β⋅∇v)v|v|p−2z+1p∫Ω(∇⋅β)|v|pz+1p∫Ω(β⋅∇z)|v|p=1p∫∂Ω(β⋅n)|v|pz.

Proof

These formulae follow from the density of   in   for all  . The first one results from the Leibniz product rule, while the second one is a consequence of the identity
(11)β⋅∇(φ|φ|p−2z)=φ|φ|p−2β⋅∇z+(p−1)|φ|p−2zβ⋅∇φ, for all   and for all  .

Weak formulation

To examine the well-posedness of ((1a), (1b)), we introduce the bilinear form  , where  , and such that for all   and all  ,
(12)aβ,μ;p(v,w):=∫Ω(β⋅∇v)w+∫Ωμvw. Observe that, for all  ,   is a closed subspace of   owing to Lemma 2.1. Assuming that  , the weak formulation of ((1a), (1b)) in the graph space   is:(13)Find u∈Vβ;p0(Ω)  s.t.  aβ,μ;p(u,v)=∫Ωsv,∀v∈Lp′(Ω). It is readily seen that if   solves ((13)), the equation ((1a)) holds in   and the boundary condition ((1b)) holds in  . Note that the boundary conditions are strongly enforced in ((13)).

Well-posedness for positive Friedrichs tensor

To examine the uniqueness of the weak solution u to ((13)) in the graph space  , we recall the  -valued Friedrichs tensor
(14)σβ,μ;p:=μ−1p∇⋅β. Hereafter, we assume that this tensor satisfies the so-called Friedrichs positivity assumption ( ):

( )
 . We define the reference time  .

Proposition 2.3

Assume that (  ) holds. Then
(15)aβ,μ;p(v,v|v|p−2)≥τ−1||v||Lp(Ω)p,∀v∈Vβ;p0(Ω).

Proof

Let  . Observing that  , the quantity   is well defined. Owing to the integration by parts formula ((10b)) with   on Ω (so that  ), we infer that
aβ,μ;p(v,v|v|p−2)=∫Ω(μ−1p∇⋅β)|v|p+1p∫∂Ω(β⋅n)|v|p, whence, using the definition ((14)) of the Friedrichs tensor   and the fact that  , we obtainaβ,μ;p(v,v|v|p−2)=∫Ωσβ,μ;p|v|p+1p∫∂Ω+(β⋅n)|v|p. The desired bound then follows from ( ) and the definition of  .

To prove the well-posedness of ((13)) under the assumption ( ), we need to introduce the two Lipschitz spaces
(16)Lip0(Ω):={v:Ω→R|v∈Lip(Ω) and v|∂Ω−≡0}, and(17)Lipc(∂Ω+):={v:∂Ω→R|v∈Lip(∂Ω) and v is compactly supported on ∂Ω+}, which satisfy the following Proposition.

Proposition 2.4

For all  , there is   such that  .

Proof

In order to stay general, we denote   the dimension of Ω. Let   and denote K its compact support on  . Owing to the Borel–Lebesgue property, we define   the finite family of open sets in   covering K , i.e.
(18)K⊂⋃1≤i≤N(Bi∩∂Ω+)⊊∂Ω+, and we denote   the partition of the unity subordinate to this covering, i.e.  for all  ,  ,   and  .

Recalling that the boundary ∂Ω is assumed to be Lipschitz-continuous, we introduce, for all  , the local bi-Lipschitz charts   where  , such that   with   and   with  . Denoting  , we introduce the function   defined as the extrusion of   in  , i.e.   for all  ,  . Next, mapping back to Ω, we consider   such that   for all   and   for all  . Finally, collecting these functions  , we observe that the function   satisfies the desired conditions.

Theorem 2.5

Assume that (  ) holds. Then the problem ((13)) is well-posed.

Proof

We apply Theorem 1.1. Adapting the proof of Ern & Guermond [[7]], we first consider   and we denote
Sp:=supw∈Lp′(Ω)\{0}⁡aβ,μ;p(v,w)||w||Lp′(Ω). Owing to Proposition 2.3, we infer that  , recalling that  . Hence, we obtain  . To bound the second part of the graph norm, i.e. the advective derivative, we observe that||β⋅∇v||Lp(Ω):=supw∈Lp′(Ω)\{0}⁡aβ,0;p(v,w)||w||Lp′(Ω)=supw∈Lp′(Ω)\{0}⁡aβ,μ;p(v,w)−∫Ωμvw||w||Lp′(Ω). Then, we obtain||β⋅∇v||Lp(Ω)≤Sp+||μ||L∞(Ω)||v||Lp(Ω)≤Sp(1+||μ||L∞(Ω)τ), yielding the first condition (BNB1) with the constant .

Let us prove now the second condition (BNB2). Consider   such that   for all  . Owing to the inclusion  , it follows that   a.e. in Ω, so that the dense inclusion   implies that  , whence  . Applying now the integration by parts formula ((10a)), we observe that
∫∂Ω(β⋅n)vw=aβ,μ;p(v,w)−∫Ω(μ−∇⋅β)vw+∫Ω(β⋅∇w)v=0, for all  . In particular, observing that   for all  , and owing to Proposition 2.4, we have∫∂Ω+(β⋅n)vw=0,∀v∈Lipc(∂Ω+). Owing to the vanishing integral property, this identity yields  . Now, testing the identity   by an arbitrary   and using the chain rule  , we infer that0=∫Ω(μw−∇⋅(βw))y=∫Ω(μ−∇⋅β)wy−∫Ωβ⋅∇wy. Hence, the particular choice   along with the identity ((10b)) with p replaced by   and with   yields0=∫Ω(μ−∇⋅β)|w|p′+1p′∫Ω(∇⋅β)|w|p′−1p′∫∂Ω(β⋅n)|w|p′=∫Ωσβ,μ;p|w|p′−1p′∫∂Ω(β⋅n)|w|p′≥τ−1||w||Lp′(Ω)p′, where we have used that   and the assumption ( ). As a result,   a.e. in Ω, so that the condition (BNB2) is satisfied. Owing to Theorem 1.1, there exists a unique solution solving the problem ((13)).

Well-posedness for non-positive Friedrichs tensor

Summarizing the results obtained so far, we have proved under assumption ( ) the well-posed of ((1a), (1b)) in the graph space  . This section aims to extend this result under the new assumption ( ) so as to include the situation where the infimum of the Friedrichs tensor   takes null or slightly negative values.

( )
  and there exists a non-dimensional function   such that
(19)ess infΩeζ(σβ,μ;p−1pβ⋅∇ζ)>0. We define the reference time  .
Proposition 2.6

Assume that (  ) holds. Then
(20)aβ,μ;p(v,eζv|v|p−2)≥τ−1||v||Lp(Ω)p,∀v∈Vβ;p0(Ω).

Proof

Let  . Observing that
aβ,μ;p(v,eζv|v|p−2)=aβ˜,μ˜;p(v,v|v|p−2), where we have denoted   and  , the inequality ((20)) follows from Proposition 2.3 if  , i.e. if ( ) holds, since we haveσβ˜,μ˜;p=eζ(σβ,μ;p−1pβ⋅∇ζ).

Remark 1

Instead of using Proposition 2.3 in the proof of Proposition 2.6, it is also possible to obtain this result by applying the general integration by parts formula ((10b)) with  .

Example 1

Assumption ( ) indeed generalizes the assumption ( ) since it is now possible to consider situations that cannot be handled under ( ). For example, considering the rotating field   expressed in the Cartesian coordinates of   and a reaction coefficient  , we have  . If  , this Friedrichs tensor does not satisfy ( ), whereas ( ) does, for example with the potential   for all   such that  .

Remark 2

Following Devinatz & al. [[5]] and considering a continuously differentiable field  , the existence of the potential ζ relies on the assumption that every solution to the Cauchy problem  ,   remains in the domain Ω for a finite time only. Observing that the proof in this reference is based on the flow box theorem, the extension to a less regular field (e.g.  ) is a priori not obvious.

We are now in a position to state the well-posedness of ((13)) under assumption ( ).
Theorem 2.7

Assume that (  ) holds. Then the problem ((13)) is well posed.

Proof

We follow the same ideas as in the proof of Theorem 2.5. The condition (BNB1) is inferred from Proposition 2.6 with . Turning to the second condition (BNB2), we consider   such that   for all  . Proceeding as in the proof of Theorem 2.5, this implies that w belongs to the graph space   and that it satisfies   a.e. in Ω and  . Let us prove that   a.e. in Ω. First, we observe that replacing p by   and choosing   in ((10b)) yields
∫Ω(β⋅∇w)w|w|p′−2eζ(1−p′)=1p′∫∂Ω(β⋅n)|w|p′eζ(1−p′)−1p′∫Ω(∇⋅β)|w|p′eζ(1−p′)−1−p′p′∫Ω(β⋅∇ζ)|w|p′eζ(1−p′). Hence, testing the identity   with the function  , we infer that0=∫Ω(μ−∇⋅β)|w|p′eζ(1−p′)−∫Ω(β⋅∇w)w|w|p′−2eζ(1−p′)=∫Ω(μ−∇⋅β)|w|p′eζ(1−p′)+1p′∫Ω(∇⋅β)|w|p′eζ(1−p′)−1p∫Ω(β⋅∇ζ)|w|p′eζ(1−p′)−1p′∫∂Ω(β⋅n)|w|p′eζ(1−p′). Then, collecting these terms and using the fact that  , we obtain0=∫Ωeζ(1−p′)(σβ,μ;p−1pβ⋅∇ζ)|w|p′−1p′∫∂Ω(β⋅n)|w|p′eζ(1−p′)≥∫Ωeζ(1−p′)(σβ,μ;p−1pβ⋅∇ζ)|w|p′. As a result, owing to   and the fact that  , it follows that   a.e. in Ω. Owing to Theorem 1.1, we conclude that ((13)) is well posed under assumption  .

Vector advection–reaction problem

In this section, we apply similar ideas to analyze the well-posedness of the vector-valued problem ((2a), (2b)) in Banach graph spaces where we generalize the assumption of the sign of the Friedrichs tensor   defined by ((4)). For the sake of brevity, the proofs are omitted if they are straightforwardly adapted from those of the scalar case in Section 2.

The graph space

Let us introduce the graph space
(21)Vβ;p(Ω):={v∈Lp(Ω)|(β⋅∇)v∈Lp(Ω)},where the i -th component in the Cartesian basis of   is given by   (where repeated indices are summed) and where   means that the linear form(22)Cc∞(Ω)∋φ↦−∫Ω∇⋅(β⊗φ)⋅v, is bounded in  , so that   is the Riesz representative of ((22)) in  . Equipped with the norm   for all  , this space defines a reflexive Banach space. The following proposition states that the problem ((2a), (2b)) is well defined in the graph space  .

Proposition 3.1

The following holds:
Vβ;p(Ω)={v∈Lp(Ω)|∇(β⋅v)+(∇×v)×β∈Lp(Ω)}, where   means that the linear form (23)Cc∞(Ω)∋φ↦−∫Ω(β∇⋅φ+∇×(φ×β))⋅v, is bounded in  .

Proof

Let  . By definition, we have
∫Ω(β⋅∇)v⋅φ=−∫Ω∇⋅(β⊗φ)⋅v,∀φ∈Cc∞(Ω). Now, recalling the identity   for all   and for all  , it follows that∫Ω(β⋅∇)v⋅φ=−∫Ω(β∇⋅φ+∇×(φ×β))⋅v−∫Ω((φ⋅∇)β)⋅v. Hence, observing that  , we obtain Hence, the linear form ((23)) is bounded, yielding  , so that the inclusion holds. Note that we have the identity a.e. in Ω. Since the proof of the converse inclusion is similar, the proof is completed.

Recalling now that the boundary ∂Ω is well separated in the sense of ((9)), functions in the graph space   have a trace in the space(24)Lp(|β⋅n|;∂Ω):={v:∂Ω→R3|v is Lebesgue measurable on ∂Ω and ∫∂Ω|β⋅n||v|p<∞}. Equipped with the norm   for all  , this space defines a Banach space.
Lemma 3.2

The map   with   for all   extends continuously to  , i.e. there exists   such that
||γ(v)||Lp(|β⋅n|;∂Ω)≤Cγ||v||Vβ;p(Ω),∀v∈Vβ;p(Ω).

Owing to the existence of a trace in  , we now extend Lemma 2.2 to a vector-valued function.
Lemma 3.3

For all   and for all  ,
(25a)∫Ωw⋅(β⋅∇)v+∫Ωv⋅(β⋅∇)w+∫Ω(∇⋅β)v⋅w=∫∂Ω(β⋅n)v⋅w. In addition, for all   and for all  , (25b)∫Ω|v|p−2zv⋅(β⋅∇)v+1p∫Ω(∇⋅β)|v|pz+1p∫Ωβ⋅∇z|v|p=1p∫∂Ω(β⋅n)|v|p.

Weak formulation

Following the route of Section 2, we introduce the bilinear form   with   such that for all   and for all  ,
(26)aβ,μ;p(v,w):=∫Ω(∇(β⋅v)+(∇×v)×β)⋅w+∫Ωμv⋅w, with   the reaction tensor. Following the proof of Proposition 3.1, we observe that the bilinear form ((26)) can be reformulated as(27) and the writing yields   and(28)aβ,μ;p(v,w)=∫Ω(β⋅∇)v⋅w+∫Ωμ′v⋅w. Assuming now that  , the weak formulation of ((2a), (2b)) in the graph space   is:(29)Find u∈Vβ;p0(Ω)   s.t.  aβ,μ;p(u,v)=∫Ωs⋅v,∀v∈Lp′(Ω). We readily see that if   solves ((29)), the problem ((2a)) holds in  , and the boundary condition ((2b)) holds in  .

Well-posedness for positive and non-positive Friedrichs tensors

The uniqueness of the solution to problem ((29)) relies on the sign of the lowest eigenvalue of the  -valued Friedrichs tensor
(30) For all  , this lowest eigenvalue is denoted by   and is defined asℵp(x)=min⁡{(σβ,μ;p(x)y,y)|y∈R3 s.t. |y|=1}, where   denotes the classical Euclidean inner product in  . Hereafter, we assume that this eigenvalue satisfies the following assumption.

( )
 . We define  .
Proposition 3.4

Assume that (  ) holds. Then,
aβ,μ;p(v,v|v|p−2)≥τ−1||v||Lp(Ω)p,∀v∈Vβ;p0(Ω).

Proof

Let   and consider   (since  ). Owing to the identity ((27)), we infer that
Using now the integration by parts formula ((25b)) with  , we obtain(31)aβ,μ;p(v,v|v|p−2)=∫Ω|v|p−2v⋅σβ,μ;p⋅v+1p∫∂Ω(β⋅n)|v|p, whence the result follows using ( ) and recalling that  .

To take into account the situation where the smallest eigenvalue   takes null or slightly negative values in Ω, we consider the new assumption ( ).
( )
  and there exists a non-dimensional function   such that
ess infΩeζ(ℵp−1pβ⋅∇ζ)>0. We define  .
Proposition 3.5

Assume that (  ) holds. Then,
aβ,μ;p(v,eζv|v|p−2)≥τ−1||v||Lp(Ω)p,∀v∈Vβ;p0(Ω).

Proof

Let  . Denoting  ,   and observing that
∇(β˜⋅v)=eζ(∇(β⋅v)+(∇ζ⊗β)v) a.e. in Ω, we infer that  . Owing to ((31)), this identity yieldsaβ,μ;p(v,eζv|v|p−2)≥∫Ω|v|p−2v⋅σβ˜,μ˜;p⋅v−∫Ωeζ|v|p−2v⋅(∇ζ⊗β+β⊗∇ζ2)⋅v. In addition, observing thatσβ˜,μ˜;p=eζ(σβ,μ;p−1p(β⋅∇ζ)Id)+eζ(∇ζ⊗β+β⊗∇ζ2), the expected result follows from assumption ( ).

Finally, the well-posedness of ((2a), (2b)) holds under assumption ( ) or ( ). The proof follows the same ideas used to prove Theorem 2.5, Theorem 2.7, this time using Proposition 3.4, Proposition 3.5, respectively.
Theorem 3.6

Assume that (  ) or (  ) holds. Then the problem ((29)) is well posed.

Conclusion

In this paper, we have extended the well-posedness of problems ((1a), (1b)) and ((2a), (2b)), not only in the Banach graph space of exponent  , but also under new assumptions regarding the classical Friedrichs tensor, so as to consider the situation when it takes positive, null, or slightly negative values. Observing that equations ((1a)) and ((2a)) are the proxy of the Lie derivative in   of a 0- and a 1-form respectively, the present analysis could be extended in a future work within the more general framework proposed by Heumann [[11]] to treat the advection of a differential k -form within a manifold of  . However, the question of the existence of the potential ζ in such a more general context is still open. Another extension of this work concerns the case of the non-homogeneous Dirichlet boundary condition, requiring to establish the surjectivity of the trace maps in these Banach graph spaces.


Acknowledgements

The author warmly thanks Alexandre Ern and Jean-Luc Guermond for their advice and the fruitful discussions.

References

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