We revisit the classical Velocity Averaging Lemmas in this section. Throughout the section, ψ ∈ L∞c (RN) is any fixed bounded function with compact support. We denote by C various positive constants which may depend on ψ and we use the notation
¯ u =
Z
RN
u(v)ψ(v)dv.
Before we introduce Velocity Averaging Lemmas, we first prove a basic fact regarding transport equation., which is fundamental in velocity averaging theory.
Proposition 3.1. Let 1 ≤ p ≤ ∞. For any f ∈ Lp(RNx × RNv ), the transport equation
u + v · ∂xu = f in RNx × RNv (5) has a unique solution uf in Lp(RNx × RNv ). The linear operator T : Lp(RNx × RNv ) → Lp(RNx) defined by
T (f ) = ¯uf = Z
RN
u(·, v)ψ(v)dv is continuous.
doi:10.6342/NTU201901751
Hence uf is a weak solution to (5). By H¨older’s inequality, kufkpLp(RNx×RNv)
We also have, by H¨older’s inequality again, k ¯ufkpLp(RNx) =
Pick a φ ∈ Cc∞(RN × RN). Then e−sφ(x + sv, v) ∈ Cc∞(RN × RN) for each s > 0. By H¨older’s inequality,
Z
Since φ is arbitrary, we conclude w(x, v) = 0 almost everywhere. In other words, u1 = u2.
Remark. The transport equation
u + v · ∂xu = 0 has nontrivial solutions in the space L1loc(RNx × RNv ).
Next, we consider L2 Stationary Velocity Averaging Lemma, the main theorem in [16]. We provide two proofs here. Both proofs rely on Fourier analysis.
Theorem 3.2. Let u = u(x, v) be such that u and v · ∂xu (= ∂s∂u(x + sv, v) Proof 1 of Theorem 3.2. The assumption on u may be formulated as F u and (v · ξ)F u belong to L2(RNξ × RNv ). Notice that
doi:10.6342/NTU201901751 By Cauchy-Schwarz inequality, we obtain
By Cauchy-Schwarz inequality again, we have
From this, (8), (9) and (10), we conclude k¯uk
Proof 2 of Theorem 3.2. Since u ∈ L2(RNx × RNv ) and v · ∂xu ∈ L2(RNx × RNv ), we have
u + v · ∂xu = f ∈ L2(RNx × RNv ).
By Proposition 3.1, we have
u(x, v) =
Considering the Fourier transform of ¯u, we have F ¯u(ξ) =Z by Cauchy-Schwarz inequality. Notice that
Z Change of the variable s = |ξ|vkξ gives
Z ∞ Substituting this into (11) gives
Z
doi:10.6342/NTU201901751 by Plancherel’s identity. Consequently,
k¯uk
H12(RN)=
Z
RN
(1 + |ξ|)|F ¯u(ξ)|2dξ
1/2
≤C k¯ukL2(RN)+
Z
RN
|ξ||F ¯u(ξ)|2dξ
1/2!
≤C(kukL2 + kf kL2)
=C(kukL2 + ku + v · ∂xukL2)
≤C(kukL2 + kv · ∂xukL2).
Notice that L2 Velocity Averaging Lemma can be formulated as follows.
Corollary 3.3. Under the hypotheses of Proposition 3.1, T : L2(RNx × RNv ) → H12(RN) is continuous.
Proof. If f ∈ L2(RNx × RNv ), then uf ∈ L2(RNx × RNv ) and v · ∂xuf = f − uf ∈ L2(RNx × RNv ).
Furthermore,
kv · ∂xufkL2 =kf − ufkL2
≤kf kL2 + kufkL2 ≤ Ckf kL2. According to Theorem 3.2, ¯uf ∈ H12(RN) and
k ¯ufk
H12 ≤ C(kufkL2 + kv · ∂xufkL2) ≤ Ckf kL2.
With the above corollary, we are able to generalize Theorem 3.2 to any Lp space with 1 < p < ∞ via the real method of interpolation.
Theorem 3.4. Let u = u(x, v) be such that u and v · ∂xu both belong to Lp(RNx × RNv ) with 1 < p < ∞. Then ¯u belongs to Ws,p(RN) with s = min(1p,p10) and
k¯ukWs,p(RN)≤ C(kukLp+ kv · ∂xukLp). (12) Proof. The proof is based on the following real interpolation theorem in [10].
Theorem 3.5. We have the following relations between interpolation spaces and Sobolev spaces.
L1(RN), H12(RN)
2
p0,p = Wp01,p(RN), for 1 < p < 2, (13)
and
L∞(RN), H12(RN)
2 p,p
⊂ W1p,p(RN), for 2 < p < ∞. (14) Moreover, the norm for the interpolation space on the left hand side of (13) is equivalent to the norm of the Sobolev space on the right side. Likewise, the norm of the Sobolev space on the right side of (14) is less than a fixed multiple of the norm of the interpolation space on the left side.
Consider the operator T in Corollary 3.3. We have T : L1(RNx × RNv ) → L1(RN), T : L2(RNx × RNv ) → H12(RN).
By Theorem 3.5 and the fact (L1, L2)2
p0,p= Lp for 1 < p < 2, T : Lp(RNx × RNv ) → Wp01,p(RN)
is bounded for 1 < p < 2. Hence if u + v · ∂xu =: f ∈ Lp(RNx × RNv ), then
¯
u = T (f ) ∈ Wp01,p(RN) and k¯uk
W
1 p0,p
(RN)≤ Ckf kLp ≤ C(kukLp+ kv · ∂xukLp).
The case 2 < p < ∞ follows similarly.
Unfortunately, the above Lp Velocity Averaging Lemma fails for the endpoint cases p = 1 and p = ∞. [16] provided an example to illustrate this phenomenon.
Theorem 3.6. Under the hypothesis of Proposition 3.1. Suppose also ψ = 1{|v|≤1}. There exists a bounded subset K ⊂ L1(RNx × RNv ) such that T (K) is not weakly precompact in L1(B1).
Proof. Let v0 be an arbitrary point in {0 < |v| < 1}. Consider a sequence of functions
fn(x, v) = n2Nψ1(n(0 − x)) ψ2(n(v0− v)) , where ψ1 and ψ2 are two mollifiers in Cc∞(B1). Therefore,
T (fn)(x) = Z
{|v|≤1}
Z ∞ 0
fn(x − tv, v)e−tdtdv.
If T (fn) converges weakly in L1(B1) to a function w. Given θ ∈ Cc∞(B1). On the one hand, as n → ∞,
Z
|x|≤1
T (fn)(x)θ(x)dx = Z
|x|≤1
Z
|v|≤1
Z ∞ 0
fn(x − tv, v)e−tθ(x)dtdvdx
→ Z ∞
0
e−tθ(tv0)dt.
On the other hand, as n → ∞, Z
|x|≤1
T (fn)(x)θ(x)dx → Z
|x|≤1
w(x)θ(x)dx.
Therefore,
Z ∞ 0
e−tθ(tv0)dt = Z
|x|≤1
w(x)θ(x)dx (15)
for any θ ∈ Cc∞(B1). Let θ be a mollifier in Cc∞(B1) and y a Lebesgue point of w not on the line {tv0 : t > 0}. Consider θm(x) = mNθ (N (y − x)). Hence, as m → ∞,
Z
|x|≤1
w(x)θm(x)dx → w(y).
doi:10.6342/NTU201901751 We also have, as m → ∞,
Z ∞ 0
e−tθm(tv0)dt → 0.
Therefore, (15) implies w(y) = 0 for any Lebesgue point y /∈ {tv0 : t > 0}, which means w = 0 almost everywhere, a contradiction to (15). Hence we let K = {fn}n∈N.
Remark. Theorem 3.4 implies
T : Lp(RNx × RNv ) → Ws,p(RNx) (16) is bounded with 1 < p < ∞ and s = min(1p,p10). However, Theorem 3.6 shows that (16) fails when p = 1 no matter how small s > 0 is. For if for some s > 0 such that (16) holds, then consider K in Theorem 3.6. Therefore, T (K) ⊂ Ws,1(RNx) is bounded. Let M = { T (f )|B
1 : f ∈ K}. Hence, M ⊂ Ws,1(B1) is bounded. We have Ws,1(B1) ⊂⊂ L1(B1) by compact embedding property, so M is precompact in L1(B1), which contradicts Theorem 3.6.
For the case p = 1, nonetheless, we can still make additional assumptions to yield a regularity result via compact embedding property of Sobolev space.
Theorem 3.7. Under the hypotheses of Proposition 3.1, if K ⊂ L1(RNx × RNv ) is bounded and uniformly integrable, then T (K) is precompact in L1(S) for any C1 bounded open set S ⊂ RN.
Proof. For any f ∈ K and α > 0, we define
χα,f = 1{(x,v):|f (x,v)|<α}, ωα,f = 1 − χα,f.
Note that f · ωα,f and f · χα,f are the large part and the small part of f respectively.
We can write
¯
uf = T (f ) = T (f · ωα,f) + T (f · χα,f)
=: φf + ψf.
The idea is that we deal with large parts by uniform integrability and small parts by L2 Velocity Averaging Lemma. Clearly, we have
Z
|φf(x + h) − φf(x)|dx ≤ 2 Z
|φf(x)|dx
= 2 Z
|T (f · ωα,f)|dx
≤ 2 Z Z
|f · ωα,f|dvdx.
We claim
sup
f ∈K
Z Z
|f · ωα,f|dvdx → 0
as α → ∞. For given ε > 0, choose δ as in Definition 2.6. Since K is bounded in L1(RNx × RNv ),
sup
f ∈K
|{|f (x, v)| ≥ α}| ≤ sup
f ∈K
1 α
Z Z
{|f |≥α}
|f |dxdv
≤1 α sup
f ∈K
Z Z
|f |dxdv
<δ,
if α is large enough. The uniform integrability implies
sup
f ∈K
Z Z
|f · ωα,f|dxdv = sup
f ∈K
Z Z
{|f (x,v)|≥α}
|f |dxdv
<ε.
Thus we can choose α > 0 such that Z
|φf(x + h) − φf(x)|dx ≤ 2 Z
|φf(x)|dx < 2ε (17)
for every f ∈ K and h ∈ RN. With the α chosen above, sup
f ∈K
Z Z
|f · χα,f|2dxdv ≤ sup
f ∈K
α Z Z
|f |dxdv
< ∞.
In other words, {f · χα,f : f ∈ K} is bounded in L2(RNx × RNv ). According to Corollary 3.3, the set {ψf : f ∈ K} is bounded in H12. Since H12(S) ⊂⊂ L1(S) for any C1 bounded open set S ⊂ RN. By Theorem 2.7,
sup
f ∈K
Z Z
S
|ψf(x + h) − ψf(x)|dx → 0 (18)
as h → 0. Hence T (K) satisfies (3) by (17) and (18). For (4), we note that K is uniformly integrable and
Z
S− ¯G
|¯u(x)|dx ≤ C Z
S− ¯G
Z
D
|f (x, v)|dvdx < ε,
where D is the support of ψ, if |S − ¯G| is small enough. Finally, we conclude that T (K) is precompact in L1(S) for any C1 bounded open set S.
The Velocity Averaging Lemmas for evolutionary case are parallel to those for stationary case. However, since the transport operators ∂t+ v · ∂x and v · ∂x have different (but similar) structures, the proofs for stationary case can not be applied to evolutionary case. Therefore, we choose to modify the second proof for Theorem 3.2. As in the stationary case, we first consider the velocity averaging operator with time t and then introduce L2 Evolutionary Velocity Averaging Lemma.
doi:10.6342/NTU201901751
is a solution to (19). The rest is similar to the proof in Proposition 3.1.
Theorem 3.9. Let u = u(t, x, v) be such that u and ∂tu + v · ∂xu (= Proof. In this proof, we denote byF f the Fourier transform of f in t and x variable only, and by τ and ξ the corresponding Fourier variables. Since u ∈ L2(Rt× RNx × RNv ) and ∂tu + v · ∂xu ∈ L2(Rt× RNx × RNv ), it follows that
u + ∂tu + v · ∂xu = f ∈ L2(Rt× RNx × RNv ).
By Proposition 3.8, we have u(t, x, v) =
Pick R > 0 sufficiently large so that
follows similarly. If |ξ| = 0, Z R
If |ξ| 6= 0, then we have the following lemma.
Lemma 3.10. There exists a constant C > 0 depending only on R > 0 such that Z R
−R
1
1 + (τ + |ξ|s)2ds ≤ C · 1
|τ |. Proof of Lemma 3.10. Clearly,
Z R as |b| → ∞, the maximum and the minimum of D(a, ·) must occur at critical points b = ±1/√
a2− 1. However, D
a, ±√ 1
a2−1
is a bounded function for |a| ≥ 2. Thus Lemma 3.10 follows.
doi:10.6342/NTU201901751 Go back to the proof of Theorem 3.9. Hence, according to Lemma 3.10, we have
|τ |
Z |ψ(v)|2
1 + (τ + (v · ξ))2dv ≤ C and
J ≤ C Z Z Z
|F f(τ, ξ, v)|2dvdξdτ. (22) By (21) and (22), we conclude that
k¯uk
H12(Rt×RNx)≤C(k¯ukL2(Rt×RN
x)+ kf kL2(Rt×RN
x×RNv))
≤C(kukL2 + k∂tu + v · ∂xukL2).
Other proofs for Theorem 3.9 can be found in, for example, [4, 9].
Also notice that Theorem 3.9 can be formulated as follows.
Corollary 3.11. Under the hypotheses of Proposition 3.8, T : L2 Rt× RNx × RNv → H12(Rt× RNx) is continuous.
By Corollary 3.11, we can prove Lp (1 < p < ∞) and L1 Evolutionary Veloc-ity Averaging Lemmas as in the stationary case. We omit proofs since they are essentially the same.
Theorem 3.12. Let u = u(t, x, v) be such that u and ∂tu + v · ∂xu both belong to Lp(Rt× RNx × RNv ). Then ¯u(t, x) belongs to Ws,p(Rt× RNx) with s = min(1p,p10) and k¯ukWs,p(Rt×RNx)≤ C(kukLp + k∂tu + v · ∂xukLp). (23) Theorem 3.13. Under the hypotheses of Proposition 3.8, if K ⊂ L1(Rt×RNx ×RNv ) is bounded and uniformly integrable, then T (K) is precompact in L1((0, T ) × S) for any T > 0 and C1 bounded open set S ⊂ RN.
So far, we only dealt with functions defined on whole space RNx . Now we consider a localized version of the above results. Let X be a C1 bounded convex domain in RNx. Denote by dΣ the surface measure on ∂X, and by n(q) the unit outward normal vector to X at q ∈ ∂X. We define
Γ := ∂X × RNv ,
Γ+ := {(q, v) ∈ Γ |n(q) · v > 0 } , Γ− := {(q, v) ∈ Γ |n(q) · v < 0 } , Γ0 := {(q, v) ∈ Γ |n(q) · v = 0 } ,
and also define the backward exit time τx,v > 0 and the f orward exit time γx,v > 0 by relations
x − τx,vv ∈∂X, x + γx,vv ∈∂X,
where (x, v) ∈ ¯X × RN. τx,v and γx,v are well-defined since X is convex. Denote by dσ the measure
dσ = |v · n(q)|dΣ(q)dv.
We have the following change of variables formulas.
Lemma 3.14. For any h ∈ L1(X × RNv ), the following hold.
Therefore, the Jacobian for the change of vatiables is
J (α, t, v) = det
Hence, it follows that
dxdv =|J (α, t, v)|dαdtdv
=|v · n(q)|dtdΣ(q)dv
=dtdσ.
Consequently, we have (24) and (25).
For 1 < p < ∞, we define
Wp(X) :=u(x, v) : u and v · ∂xu both belong to Lp(X × RN) , kukWp(X) := kukLp(X×RN)+ kv · ∂xukLp(X×RN).
Let us discuss the boundary value u|Γ− of u ∈ Wp(X) on Γ− with measure dσ. The idea follows from [5, 6] and [20]. From the identity (24), we have
Z
doi:10.6342/NTU201901751 function of t if we modify the values of u(q + tv, v) on a set of zero measure. By the
Fundamental Theorem of Calculus, for such (q, v) ∈ Γ−, u(q + s2v, v) − u(q + s1v, v) =
Z s2
s1
∂
∂tu(q + tv, v)dt
= Z s2
s1
v · ∂xu(q + tv, v)dt
holds for almost every s1, s2. This fact motivates the concept of u|Γ−, which is defined as follows.
Definition 3.15. For u ∈ Wp(X),
u|Γ−(q, v) = u|Γ−(q(x, v), v) := u(x, v) − Z τx,v
0
v · ∂xu(x − sv, v)ds, where q(x, v) := x − τx,vv.
Thanks to the previous discussion, u|Γ−(q, v) is well-defined for almost every (q, v) ∈ Γ− if we think of u(x − sv, v) as an absolutely continuous function of s.
To obtain Velocity Averaging Lemmas in bounded domains, we require Extension Lemma, which allows us to extend a function defined only in a bounded domain to a function in whole space without loss of regularity. The Extension Lemma was first proved by Cessenat and then presented in [16]. Here, we have an independent proof for the lemma. We further define the space W−p(X) as follows.
W−p(X) :=u ∈ Wp(X) : u|Γ− ∈ Lp(Γ−; dσ) , kukWp
−(X) := kukWp(X)+ ku|Γ−kLp(Γ−;dσ).
Lemma 3.16 (Extension Lemma). There exists a continuous extension operator Π : W−p(X) → Wp(RN),
i.e., (Πu)|X = u and kΠukWp(RN) ≤ CkukWp
−(X) for some constant C depending only on p.
Proof. Suppose u ∈ W−p(X). We construct Πu straightforwardly. Define f, g by f := u + v · ∂xu in X,
g := u|Γ− respectively. We verify that
u(x, v) = e−τx,vg(x − τx,vv, v) + Z τx,v
0
e−tf (x − tv, v)dt.
Define
u0(x, v) := e−τx,vg(x − τx,vv, v), u1(x, v) :=
Z τx,v
0
e−tf (x − tv, v)dt, w(x, v) := u0+ u1.
We prove that u = w. Let q(x, v) = x − τx,vv. First, boundary condition z|Γ− = 0. Observe that
∂
doi:10.6342/NTU201901751 Therefore, eτx,vz(x, v) is constant along the characteristic line {(q + tv) : 0 < t <
γq,v}, which implies z(x, v) = C(q(x, v), v)e−τx,v for some constant C(q, v) depending only on (q, v) ∈ Γ− for almost every (x, v) and (q, v). The boundary condition z|Γ− = 0 forces C(q, v) = 0. We conclude that u = w. We now define a function U = U (x, v) on RNx × RNv as follows. If (x, v) /∈ S := {(y + tv, v) | y ∈ X, t ∈ R}, then U (x, v) = 0. Also set U0(x, v) = U1(x, v) = 0 in this case. Otherwise, there exists a unique number τx,v (possibly negative) such that (x − τx,vv, v) ∈ Γ−. In this case, we set
U (x, v) = e−|τx,v|g(x − τx,vv, v) + Z ∞
0
e−tF (x − tv, v)dt
=: U0+ U1. Recall
F (x, v) = f (x, v) if x ∈ X, 0 otherwise.
Clearly, U1 satisfies the equation U1+ v · ∂xU1 = F , and we have kU1kLp(RN×RN)+ kv · ∂xU1kLp(RN×RN)≤ CkF kLp(RN×RN)
= Ckf kLp(X×RN). Now we claim
v · ∂xU0 =
0 if (x, v) /∈ S,
−e−τx,vg(x − τx,vv, v) if τx,v ≥ 0, eτx,vg(x − τx,vv, v) if τx,v < 0.
In other words, if φ ∈ Cc∞(RN × RN), we need to show
− Z
RN
Z
RN
U0(x, v)v · ∂xφ(x, v)dxdv = − Z
RN
Z
P (v)
e−τx,vg(x − τx,vv, v)φ(x, v)dxdv +
Z
RN
Z
N (v)
eτx,vg(x − τx,vv, v)φ(x, v)dxdv, where
P (v) =x ∈ RN : (x, v) ∈ S and τx,v ≥ 0 , N (v) =x ∈ RN : (x, v) ∈ S and τx,v < 0 . According to the definition of U0,
− Z
RN
Z
RN
U0(x, v)v · ∂xφ(x, v)dxdv = − Z
RN
Z
P (v)
e−τx,vg(x − τx,vv, v)v · ∂xφ(x, v)dxdv
− Z
RN
Z
N (v)
eτx,vg(x − τx,vv, v)v · ∂xφ(x, v)dxdv.
By an analogy of (24),
Adding (28) and (29) proves the claim. Finally, we prove kU0kLp(RN×RN) = CkgkLp(Γ−;dσ). so we have the inequality
kU kWp(RN) ≤kU0kLp(RN×RN)+ kv · ∂xU0kLp(RN×RN)
doi:10.6342/NTU201901751 With Lemma 3.16, we can derive Lp Velocity Averaging Lemma on bounded
domains from Theorem 3.4.
Theorem 3.17. Let u = u(x, v) be such that u and v · ∂xu both belong to Lp(X × RN) and u|Γ− belongs to Lp(Γ−; dσ) with 1 < p < ∞. Then ¯u belongs to Ws,p(RN) with s = min(1p,p10). Moreover, we have the inequality
k¯ukWs,p(X) ≤ C(kukLp(X×RN)+ kv · ∂xukLp(X×RN)+ ku|Γ−kLp(Γ−;dσ)). (30) Proof. By Lemma 3.16, we extend u to Πu ∈ Wp(RN). By Theorem 3.4, Πu ∈ Ws,p(RN) and
kΠukWs,p(RN) ≤ CkΠukWp(RN).
Clearly, we have
k¯ukWs,p(X) ≤ kΠukWs,p(RN) and
kΠukWp(RN) ≤CkukWp
−(X)
=C(kukLp(X×RN)+ kv · ∂xukLp(X×RN)+ ku|Γ−kLp(Γ−;dσ)).
Therefore (30) holds.
Applying compact embedding property, we can deduce the following conclusion.
Corollary 3.18. For 1 < p < ∞, the operator A defined by Au =
Z
RN
u(·, v)dv is compact from W−p(X) into Lp(X).
Proof. This follows immediately from Theorem 3.17 and the fact Ws,p(X) ⊂⊂ Lp(X).