Let Ω be a domain of with Lipschitzcontinuous boundary ∂Ω and consider and solving the following firstorder 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 Lipschitzcontinuous 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 wellposedness 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 wellposedness 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 socalled 1form in a threedimensional domain; see Bossavit [[3]].
In this paper, we analyze the wellposedness 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 wellposedness 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 wellposedness in Banach graph spaces. Following the analysis of Friedrichs system in Hilbert space proposed in the aforementioned works, we establish the wellposedness 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 socalled 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 wellposedness of the vector problem ((2a), (2b)) under similar assumptions on the Friedrichs tensor ((4)).
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]].
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 wellposedness of the continuous problem ((1a), (1b)) in Banach graph spaces and generalizes the sign condition on the Friedrichs tensor defined by ((3)).
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→Rv is Lebesgue measurable on ∂Ω and ∫∂Ωβ⋅nvp<∞}, 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 wellseparation 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.
The map with for all , extends continuously to , i.e. there exists such that
γ(v)Lp(β⋅n;∂Ω)≤CγvVβ;p(Ω),∀v∈Vβ;p(Ω).
Owing to the separation of the boundary from assumption ((9)), there exist such that on ∂Ω, , and . Proceeding as in [[8]], we infer that
∫∂Ωβ⋅nφp=∫∂Ω(β⋅n)φp(ψ+−ψ−)=∫Ω∇⋅(βφp(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∫∂Ωβ⋅nφp=p∫Ω(ψ+−ψ−)(β⋅∇φ)φφp−2+∫Ωφp∇⋅(β(ψ+−ψ−)). Next, Hölder's and Young's inequalities along with the identity yield∫Ω(β⋅∇φ)φφp−2≤β⋅∇φLp(Ω)φLp(Ω)p/p′≤1pβ⋅∇φLp(Ω)p+1p′φLp(Ω)p. It follows that with the constant . Then, observing that , we obtainφLp(β⋅n;∂Ω)≤CγφVβ;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. 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)vvp−2z+1p∫Ω(∇⋅β)vpz+1p∫Ω(β⋅∇z)vp=1p∫∂Ω(β⋅n)vpz.
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⋅∇(φφp−2z)=φφp−2β⋅∇z+(p−1)φp−2zβ⋅∇φ, for all and for all .
To examine the wellposedness 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)).
 
 Wellposedness 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 socalled Friedrichs positivity assumption ( ):
( )  . We define the reference time . 
Assume that ( ) holds. Then
(15)aβ,μ;p(v,vvp−2)≥τ−1vLp(Ω)p,∀v∈Vβ;p0(Ω).
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,vvp−2)=∫Ω(μ−1p∇⋅β)vp+1p∫∂Ω(β⋅n)vp, whence, using the definition ((14)) of the Friedrichs tensor and the fact that , we obtainaβ,μ;p(v,vvp−2)=∫Ωσβ,μ;pvp+1p∫∂Ω+(β⋅n)vp. The desired bound then follows from ( ) and the definition of .
To prove the wellposedness of ((13)) under the assumption ( ), we need to introduce the two Lipschitz spaces
(16)Lip0(Ω):={v:Ω→Rv∈Lip(Ω) and v∂Ω−≡0}, and(17)Lipc(∂Ω+):={v:∂Ω→Rv∈Lip(∂Ω) and v is compactly supported on ∂Ω+}, which satisfy the following Proposition.
For all , there is such that .
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 Lipschitzcontinuous, we introduce, for all , the local biLipschitz 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.
Assume that ( ) holds. Then the problem ((13)) is wellposed.
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)wLp′(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 thatβ⋅∇vLp(Ω):=supw∈Lp′(Ω)\{0}aβ,0;p(v,w)wLp′(Ω)=supw∈Lp′(Ω)\{0}aβ,μ;p(v,w)−∫ΩμvwwLp′(Ω). Then, we obtainβ⋅∇vLp(Ω)≤Sp+μL∞(Ω)vLp(Ω)≤Sp(1+μL∞(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)wp′+1p′∫Ω(∇⋅β)wp′−1p′∫∂Ω(β⋅n)wp′=∫Ωσβ,μ;pwp′−1p′∫∂Ω(β⋅n)wp′≥τ−1wLp′(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)).
 
 Wellposedness for nonpositive Friedrichs tensor 
Summarizing the results obtained so far, we have proved under assumption ( ) the wellposed 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 nondimensional function such that (19)ess infΩeζ(σβ,μ;p−1pβ⋅∇ζ)>0. We define the reference time . 
Assume that ( ) holds. Then
(20)aβ,μ;p(v,eζvvp−2)≥τ−1vLp(Ω)p,∀v∈Vβ;p0(Ω).
Let . Observing that
aβ,μ;p(v,eζvvp−2)=aβ˜,μ˜;p(v,vvp−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).
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 .
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 .
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 wellposedness of ((13)) under assumption ( ). Assume that ( ) holds. Then the problem ((13)) is well posed.
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)wwp′−2eζ(1−p′)=1p′∫∂Ω(β⋅n)wp′eζ(1−p′)−1p′∫Ω(∇⋅β)wp′eζ(1−p′)−1−p′p′∫Ω(β⋅∇ζ)wp′eζ(1−p′). Hence, testing the identity with the function , we infer that0=∫Ω(μ−∇⋅β)wp′eζ(1−p′)−∫Ω(β⋅∇w)wwp′−2eζ(1−p′)=∫Ω(μ−∇⋅β)wp′eζ(1−p′)+1p′∫Ω(∇⋅β)wp′eζ(1−p′)−1p∫Ω(β⋅∇ζ)wp′eζ(1−p′)−1p′∫∂Ω(β⋅n)wp′eζ(1−p′). Then, collecting these terms and using the fact that , we obtain0=∫Ωeζ(1−p′)(σβ,μ;p−1pβ⋅∇ζ)wp′−1p′∫∂Ω(β⋅n)wp′eζ(1−p′)≥∫Ωeζ(1−p′)(σβ,μ;p−1pβ⋅∇ζ)wp′. 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 wellposedness of the vectorvalued 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.
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 .
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 .
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→R3v is Lebesgue measurable on ∂Ω and ∫∂Ωβ⋅nvp<∞}. Equipped with the norm for all , this space defines a Banach space. The map with for all extends continuously to , i.e. there exists such that
γ(v)Lp(β⋅n;∂Ω)≤CγvVβ;p(Ω),∀v∈Vβ;p(Ω).
Owing to the existence of a trace in , we now extend Lemma 2.2 to a vectorvalued function.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)∫Ωvp−2zv⋅(β⋅∇)v+1p∫Ω(∇⋅β)vpz+1p∫Ωβ⋅∇zvp=1p∫∂Ω(β⋅n)vp.
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 .
 
 Wellposedness for positive and nonpositive 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.
Assume that ( ) holds. Then,
aβ,μ;p(v,vvp−2)≥τ−1vLp(Ω)p,∀v∈Vβ;p0(Ω).
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,vvp−2)=∫Ωvp−2v⋅σβ,μ;p⋅v+1p∫∂Ω(β⋅n)vp, 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 nondimensional function such that ess infΩeζ(ℵp−1pβ⋅∇ζ)>0. We define . 
Assume that ( ) holds. Then,
aβ,μ;p(v,eζvvp−2)≥τ−1vLp(Ω)p,∀v∈Vβ;p0(Ω).
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,eζvvp−2)≥∫Ωvp−2v⋅σβ˜,μ˜;p⋅v−∫Ωeζvp−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 wellposedness 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. Assume that ( ) or ( ) holds. Then the problem ((29)) is well posed.
In this paper, we have extended the wellposedness 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 1form 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 nonhomogeneous Dirichlet boundary condition, requiring to establish the surjectivity of the trace maps in these Banach graph spaces.
The author warmly thanks Alexandre Ern and JeanLuc Guermond for their advice and the fruitful discussions.