Lesson 2 - Transport equation : characteristics method and DiPerna-Lions renormalization theory

Mathematical basis for evolution PDEs La Habana CIMPA school - 2013

Lesson 2 - Transport equation : characteristics method and DiPerna-Lions renormalization theory

7 octobre 2013

1 Introduction (June 25th, 26th and 27th, 2013)

We consider the evolution PDE (transport equation)

(1.1) ∂tf = Λf =−a(x)· ∇f(x) in (0,∞)×R^d, that we complement by an initial condition

f(0, x) =f₀(x) in R^d. We assume that the drift force field satisfies

a∈C¹(R^d)∩W^1,∞(R^d)

(aglobally Lipschitz would be suitable) and that the initial datum satisfies

(1.2) f0∈L^p(R^d), 1≤p≤ ∞.

We prove that there exists a unique solution in the renormalization sense to the transport equation (1.1) associated to the initial datumf0.

2 Characteristics method and existence of solutions

2.1 Smooth initial datum.

As a first step we considerf0∈C_c¹(R^d;R).

Thanks to the Cauchy-Lipschitz theorem on ODE, we know that for anyx∈R^d the equation

˙

x(t) =a(x(t)), x(0) =x,

admits a unique solutiont7→x(t) = Φt(x)∈C¹(R;R^d). Moreover, for anyt≥0, the vectors valued function Φ_t:R^d→R^d is aC¹-diffeomorphism and the application R+×R^d →R^d, (t, x)7→Φ_t(x) is globally Lipschitz.

The characteristics method makes possible to build a solution to the transport equation (1.1) thanks to the solutions (characteristics) of the above ODE problem.

We start with a simple case. Assumingf0∈C¹(R^d;R), we define the functionf ∈C¹(R×R^d;R) (2.1) ∀t≥0, ∀x∈R^d f(t, x) :=f₀(Φ⁻¹_t (x)).

From the associated implicit equationf(t,Φ_t(x)) =f₀(x), we deduce

0 = d

dt[f(t,Φt(x)] = (∂tf)(t,Φt(x)) + ˙Φt(x)·(∇xf)(t,Φt(x))

= (∂tf+a(x)· ∇xf)(t,Φt(x)).

The above equation holding true for anyt >0 andx∈R^d and the function Φ_tmapping R^d onto R^d, we deduce thatf ∈C¹((0, T)×R^d) satisfies the transport equation (1.1) in the sense of the classical differential calculus.

If furtheremore f0 ∈ C_c¹(R^d), by using that |Φt(x)−x| ≤L t for any x∈ R^d, t ≥ 0 we deduce that f(t)∈C_c¹(R^d) for anyt≥0, with suppf(t)⊂suppf0+B(0, L t). In other words, transport occurs with finite speed : that makes a great difference with the instantaneous positivity of solution (related of a “infinite speed” of propagation of particles) known for the heat equation and more generally for parabolic equations.

Exercise 2.1 Make explicit the construction and formulas in the three following cases : (1)a(x) =a∈R^d is a constant vector.

(2)a(x) =x

(3)a(x) =v,f0=f0(x, v)∈C¹(R^d×R^d)and we look for a solutionf =f(t, x, v)∈C¹((0,∞)× R^d×R^d).

(4) Prove that(S_t)is a group on C(R^d), where

(2.2) ∀f₀∈C(R^d), ∀t∈R, ∀x∈R^d (S_tf₀)(x) =f(t, x) :=f₀(Φ⁻¹_t (x)).

(5) Repeat a similar construction in the case of a time depending drift force field a = a(t, x) ∈ C¹([0, T]×R^d)and for the transport equation with a gain source term added at the RHS of (1.1).

2.2 L

initial datum.

As a second step we want to generalize the construction of solutions to more general initial data as announced in (1.2). We observe that at least formally the following computation holds for a given positive solutionf of the transport equation (1.1) :

d dt

Z

R^d

f^pdx = Z

R^d

∂_tf^pdx= Z

R^d

pf^p−1∂_tf dx

= Z

R^d

pf^p−1a· ∇_xdx= Z

R^d

a(x)· ∇_xf^pdx

= Z

R^d

(−div_xa)f^pdx≤ kdiv_xak_L^∞ Z

R^d

f^pdx.

With the help of the Gronwall lemma, we learn from that differential inequality that the following (still formal) estimate holds

(2.3) kf(t)kL^p≤e^bt/pkf0kL^p ∀t≥0,

withb:=kdivakL^∞. As a consequence, we may propose the following natural definition of solution.

Definition 2.2 We say thatf =f(t, x)is a weak solution to the transport equation(1.1)associated to the initial datumf₀∈L^p(R^d)if it satisfies the bound

f ∈L^∞(0, T;L^p(R^d)) and it satisfies the equation in the following weak sense :

Z T

0

Z

R^d

fn

∂tϕ+divx(aϕ)o dxdt+

Z

R^d

f0ϕ(0, .)dx= 0 for anyϕ∈C_c¹([0, T)×R^d).

Exercise 2.3 1. Prove that a classical solution is a weak solution.

2. Prove that a weak solution f is weakly continuous (after modification of f(t) on a time set of measure zero) in the following sense :

(i)f ∈C([0, T];D⁰(R^d)in general (and evenf ∈Lip([0, T];w∗ −(C_c¹(R^d))⁰)) ; (ii) f ∈C([0, T];w∗ −(Cc(R^d))⁰)when p= 1 (for the weak topology∗σ(M¹, Cc)) ; (ii) f ∈C([0, T];w−L^p(R^d))when p∈(1,∞)(for the weak topologyσ(L^p, L^p0)) ; (ii) f ∈C([0, T];w−L^p_loc(R^d))for any p∈[1,∞)whenp=∞.

Theorem 2.4 (Existence) For any f₀ ∈L^p(R^d), 1≤p≤ ∞, there exists a global (defined for anyT >0) weak solution to the transport equation (1.1)which furthermore satisfies

f ∈C([0,∞);L^p(R^d))whenp∈[1,∞); f ∈C([0,∞);L¹_loc(R^d))whenp=∞.

If moreoverf0≥0 thenf(t, .)≥0 for anyt≥0.

Step 1. Rigorous a priori bounds. Take f0 ∈ C_c¹(R^d). For any smooth (renormalizing) functionβ:R→R+,β(0) = 0, which areC¹and globally Lipschitz we clearly have thatβ(f(t, x)) is a solution to the same equation associated to the initial datum β(f0) andβ(f(t, .))∈ C_c¹(R^d) for anyt≥0. The function

(0, T)→R+, t7→

Z

R^d

β(f(t, x))dx

is clearlyC¹ (that is an exercise using the Lebesgue’s dominated convergence Theorem) and d

dt Z

R^d

β(f(t, x))dx = Z

R^d

∂_tβ(f(t, x))dx= Z

R^d

β⁰(f(t, x))∂_tf(t, x)dx

= Z

R^d

β⁰(f(t, x))a(x)· ∇xf(t, x)dx= Z

R^d

a(x)· ∇xβ(f(t, x))dx

= Z

R^d

(−divxa)(x)β(f(t, x))dx.

We deduce from that identity the differential inequality d

dt Z

R^d

β(f(t, x))dx≤b Z

R^d

β(f(t, x))dx, withb:=kdivxakL^∞, and then thanks to the Gronwall lemma

Z

R^d

β(f(t, x))dx≤e^bt Z

R^d

β(f0(x))dx.

Since f0 ∈ L¹(R^d)∩L^∞(R^d) by assumption, for any 1 ≤ p < ∞, we can define a sequence of renormalized functions (βn) such that 0≤βn(s)% |s|^p for anys∈Rand we can pass to the limit in the preceding inequality using the monotonous Lebesgue Theorem at the RHS and the Fatou Lemma at the LHS in order to get

Z

R^d

|f(t, x)|^pdx≤e^bt Z

R^d

|f₀(x)|^pdx, or in other words

kf(t, .)kL^p≤e^bt/pkf0kL^p ∀t≥0.

Passing to the limitp→ ∞in the above equation we obtain (maximum principle) kf(t, .)kL^∞≤ kf0kL^∞ ∀t≥0.

Moreover,f ∈C([0, T];L^p(R^d) for anyp∈[1,∞).

Step 2. Existence in the case p ∈ [1,∞). For any function f₀ ∈L^p(R^d), 1 ≤p < ∞, we may define a sequence of functionsf_0,n∈C_c¹(R^d) (take for instancef_0,n:= (χ_nf₀)∗ρ_n such that f_0,n→f₀ in L^p(R^d) : here comes the restriction p <∞). Because of the first step we may define fn(t) a solution to the transport equation. Moreover, thanks to the first step and because the equation is linear we have

sup

t∈[0,T]

kf_n(t, .)−f_m(t, .)k_Lp≤e^bt/pkf_0,n−f_0,mk_Lp→0 ∀T ≥0.

The sequence (fn) being a Cauchy sequence, there exists f ∈C([0, T];L^p(R^d)) such that fn →f inC([0, T];L^p(R^d)) asn→ ∞. Now, writing

0 = −

Z T

0

Z

R^d

ϕn

∂_tf_n+a· ∇fn

odxdt

= Z T

0

Z

R^d

fn

n

∂tϕ+ divx(aϕ)o dxdt+

Z

R^d

f0,nϕ(0, .)dx,

we may pass to the limit in the above equation and we get thatf is a solution in the convenient sense.

If moreoverf₀≥0 then the same holds forf_0,n, then forf_n and finally forf. tu Exercise 2.5 (1) Show that for any solution f to the transport equation associated to an initial datum f0 ∈C_c¹(R^d) built by the characterictics method, for any times T >0 and radius R there exists some constants CT, RT ∈(0,∞)such that

sup

t∈[0,T]

Z

B_R

|f(t, x)|dx≤C_T Z

B_RT

|f0(x)|dx.

(Hint. Use the proprty of finite speed propagation of the transport equation).

(2) Adapt the proof of existence to the case f0∈L^∞.

3 Weak solutions are renormalized solutions

Definition 3.1 LetΩ⊂R^dand assume herea∈W^1,∞((0, T)×Ω). We say thatg∈L¹_loc([0, T]×Ω) is a weak solution to the PDE evolution equation

(3.1) Lg:=∂tg−Λg:=∂tg+a· ∇xg−ν∆xg=G, g(0, .) =g0,

withG∈L¹_loc([0, T]×Ω),ν ∈R+,g₀∈L¹_loc(Ω), if for anyϕ∈C_c²([0, T)×Ω)there holds Z T

0

Z

Ω

g L^∗ϕ= Z

Ω

g0ϕ(0, .) + Z T

0

Z

Ω

ϕ G

whereL^∗ is the dual operator

L^∗ϕ:=−∂tϕ−divx(a ϕ)−ν∆xϕ.

In order to simplify the presentation, from now on, we may assume Ω =R^d, a∈W^1,∞(Ω).

Remarks 3.2 (1) In the above definition of weak solution, we do note need that a ∈W^1,∞, but just that a,divxa∈L^∞_loc. However, that additional information will be necessary in order to prove the renomalization result.

(2) One can take equivalentlyϕ∈ D([0, T)×R^d)in the definition : we usually call such a solution a “solution in the distributional sense”. We do not make the difference here between the different notions of weak solutions which are all equivalent in our framework.

(3) Any weak solution satisfies f ∈C([0, T];D⁰(R^d)) in the sense that for any given test function ϕ∈C_c²(R^d)the following quantity

t7→

Z

R^d

f(t, x)ϕ(x)dx is continuous.

For anyg∈C²classical solution of (3.1) andβ ∈C²(R;R), there holds

∂_tβ(g) +a· ∇_x(β(g))−∆β(g) =

=β⁰(g)∂tg+β⁰(g)a· ∇xg−div(β⁰(g)∇xg)

=β⁰(g)G−β⁰⁰(g)|∇g|².

Definition 3.3 We say thatg∈L¹_loc([0, T]×Ω)is a renormalized solution to the PDEs evolution equation (3.1)withG∈L¹_loc([0, T]×Ω),ν∈R+,g0∈L¹_loc(Ω) ifg satisfies the additionnal bound ν∇xg∈L²_loc([0, T]×Ω)as well as the equations

(3.2)

Z T

0

Z

Ω

β(g)L^∗ϕ= Z

Ω

β(g₀)ϕ(0, .) + Z T

0

Z

Ω

ϕ{β⁰(g)G−β⁰⁰(g)|∇g|²}

for any test function ϕ ∈ C_c²([0, T)×Ω) and any renormalizing function β ∈ C²(R) such that β⁰⁰∈C_c(R).

Theorem 3.4 With the above notations and assumptions, any weak solutiong∈C([0, T];L¹_loc(Ω)) to the equation (3.1)such that ν∇g∈L²_loc is a renormalized solution.

For the sake of simplicity, we only deal with the case Ω =R^d. We start with two elementary but fundamental lemmas.

Lemma 3.5 Given G∈L¹_loc([0, T]×R^d), letg∈L¹_loc([0, T]×R^d)be a weak solution to the PDE Λg=G on(0, T)×R^d.

For a mollifer sequence

ρ_ε(t, x) := 1 ε^d+1ρ(t

ε,x

ε), 0≤ρ∈ D(R^d+1), suppρ⊂(−1,0)×B(0,1), Z

R^d+1

ρ= 1, and forτ∈(0, T),ε∈(0, τ), we define the function

gε:= (ρε∗t,xg)(t, x) :=

Z T

0

Z

R^d

g(s, y)ρ(t−s, x−y)dsdy.

Theng_ε∈C^∞([0, T −τ)×R^d)and it satisfies the equation Lg_ε=G_ε+r_ε

in the classical differential calculus sense on[0, T −τ)×R^d, with Gε:=ρε∗t,xG, rε:=a· ∇xgε−(a· ∇g)∗ρε.

It is worth emphasizing that in the above formula the “commutator” rεis defined in a weak sense, namely

r_ε(t, x) :=

Z

R^d+1

g(s, y)n

a(x)· ∇xρ_ε(t−s, x−y) +div_y

a(y)ρ_ε(t−s, x−y)o dyds.

Proof of Lemma 3.5.DefineO:= [0, T −τ)×R^d. For any (t, x)∈ Ofixed and any ε∈(0, τ), we define

(s, y)7→ϕ(s, y) =ϕ^t,x_ε (s, y) :=ρε(t−s, x−y)∈ D((0, T)×R^d).

We then just write the weak formulation of equation (3.1) for that test function, and we get

0 =

Z T

0

Z

R^d

gΛ^∗ϕ− Z T

0

Z

R^d

G ϕ

= Z T

0

Z

R^d

g(s, y){ −∂sϕ^t,x(s, y)− ∇y(a(y)ϕ^t,x(s, y))−∆yϕ^t,x(s, y)}

− Z T

0

Z

R^d

G(s, y)ϕ^t,x(s, y)

= Z T

0

Z

R^d

g(s, y){∂tϕ^t,x(s, y) +a(x)· ∇xϕ^t,x(s, y)−∆_xϕ^t,x(s, y)}

+ Z T

0

Z

R^d

g(s, y){a(y)· ∇yϕ^t,x(s, y)−a(x)· ∇xϕ^t,x(s, y)} − Z T

0

Z

R^d

G(s, y)ϕ^t,x(s, y)

= ∂tgε(t, x) +a· ∇xgε(t, x)−∆xgε(t, x)−rε(t, x)−Gε(t, x),

which is the anounced equation. tu

Lemma 3.6 Under the assumptions B ∈W_loc^1,q(R^d)and g ∈L^p_loc(R^d) with 1/r = 1/p+ 1/q ≤1, then

R_ε:= (B· ∇g)∗ρ_ε−B· ∇(g∗ρ_ε)→0 L^r_loc, for any mollifer sequence(ρ_ε).

Remark 3.7 For a time dependent field vector a(t, x) satisfying the boundedness conditions of Theorem 3.4 the same result (with the same proof ) holds, so that the commutator rε defined in Lemma 3.5 satisfiesrε→0 in L¹_loc([0, T)×R^d).

Proof of Lemma 3.6.We only consider the casep= 1, q=∞andr= 1. We start writting R_ε(x) = −

Z g(y)n

div_y

B(y)ρ_ε(x−y)

+B(x)· ∇x ρ_ε(x−y)o dy

= Z

g(y)n

B(y)−B(x)

· ∇x(ρε(x−y))o

dy−((gdivB)∗ρε)(x)

=: R¹_ε(x) +R²_ε(x).

For the first term, we remark that

|R¹_ε(x)| ≤ Z

|g(y)|

B(y)−B(x) ε

|(∇ρ)ε(x−y)|dy

≤ k∇BkL^∞

Z

|x−y|≤1

|g(y)| |(∇ρ)ε(x−y)|dy, so that

(3.3)

Z

BR

|R¹_ε(x)|dx≤ k∇BkL^∞k∇ρk_L1kgk_L1(BR+1). On the other hand, ifgis a smooth (sayC¹) function

R_ε¹(x) = ∇x((gB)∗ρε)−B· ∇x(g∗ρε)

−→ ∇x(gB)−B· ∇xg= (divB)g.

Since every things make sense at the limit with the sole assumption divB ∈ L^∞ and g ∈ L¹, with the help of (3.3) we can use a density argument in order to get the same result without the additional smoothness hypothesis ong. More precisely, for a sequencegα inC¹ such thatgα→g inL¹_loc, we have

R¹_ε[gα]→(divB)g inL¹_loc, kR¹_ε[h]kL¹ ≤CkhkL¹ ∀h,

so that

R¹_ε[g]−(divB)g={R¹_ε[g]−R¹_ε[gα]}+{R¹_ε[gα]−(divB)gα)}+{(divB)gα−(divB)g)} →0 inL¹_loc as ε→0. For the second term, we clearly have

R²_ε= (gdivB)∗ρ_ε→gdivB

and we conclude by putting all the terms together. tu

Proof of Theorem 3.4. Step 1.We consider a weak solutiong∈L¹_loc to the PDE Lg=G in [0, T)×R^d.

By mollifying the functions with the sequence (ρε) defined in Lemma 3.5 and using Lemma 3.5, we get

Lgε=Gε+rε in [0, T)×R^d, rε→0 in L¹_loc.

Becauseg_εis a smooth function, we may perform the following computation (in the sense of the classical differential calculus)

Lβ(gε) =β⁰(gε)Gε−ν β⁰⁰(gε)|∇xgε|²+β⁰(gε)rε, so that

(3.4) Z

β(g_ε)L^∗ϕ= Z

Ω

β(g_ε(0, .))ϕ(0, .) + Z

β⁰(g_ε)G_εϕ−ν Z

β⁰⁰(g_ε)|∇_xg_ε|²ϕ+ Z

β⁰(g_ε)r_ε for anyϕ∈C_c²([0, T)×R^d. Using that

gε→g, Gε→G, rε→0, ν|∇gε|²→ν|∇g|²

inL¹_loc asε→0, which in turns implyβ⁰⁰(gε)→β⁰⁰(g) a.e., and that (β⁰⁰(gε)) is bounded inL^∞, we may pass to the limit ε → 0 in the last identity and we obtain (3.2) for any test function ϕ∈C_c²((0, T)×Ω).

In the case we consider a solutiongbuilt thanks to Theorem 2.4 above or Theorem 3.2 in chapter 1, we know that additionallyg∈C([0, T);L¹_loc(Ω)) and therefore

gε→g in C([0, T);L¹_loc(Ω)).

In particular gε(0, .)→g(0, .) in L¹_loc(Ω) and we may pass to the limit in equation (3.4) for any test functionϕ∈C_c²([0, T)×Ω), which ends the proof of (3.2).

4 Consequence of the renormalization result

In this section we present several immediate consequences of the renormalization formula establi- shed in Theorem 3.4 for the transport equation (1.1).

4.1 Uniqueness and C

-semigroup in L

( R

), 1 ≤ p < ∞

Corollary 4.1 Assume p ∈ [1,∞). For any initial datum g0 ∈ L^p(R^d), the transport equation admits a unique weak solutiong∈C([0, T];L^p(R^d)).

Proof of Corollary 4.1. Consider two weak solutions g₁ and g₂ to the transport equation (1.1) associated to the same initial datum g₀. The functiong :=g₂−g₁ ∈ C([0, T];L^p(R^d)) is then a weak solution to the transport equation (1.1) associated to the initial datumg(0) = 0. Thanks to Theorem 3.4 it is also a renormalized solution, which means

Z

R^d

β(g(t, .))ϕ dx= Z t

0

Z

R^d

β(g) divx(a ϕ)dxds

for any renormalizing functionβ∈W^1,∞(R),β(0) = 0, and any test functionϕ=ϕ(x)∈C_c¹(R^d).

We fixβ such that furthermore 0 < β(s)≤ |s|^p for anys 6= 0,χ ∈ D(R^d) such that 0≤χ ≤1, χ≡1 onB(0,1) and we takeψ(x) =χ_R(x) =χ(x/R), so that

Z

R^d

β(g(t, .))χRdx= Z t

0

Z

R^d

β(g) (divxa)χRdxds+ 1 R

Z t

0

Z

R^d

β(g)a(x)· ∇χ(x/R)dxds Observing thatβ(g)∈C([0, T];L¹(R^d)) and χ_R →1, we easilly pass to the limit R→ ∞ in the above expression, and we get

(4.1)

Z

R^d

β(g(t, .))dx= Z t

0

Z

R^d

β(g) (divxa)dxds

By the Gronwall lemma we conclude thatβ(g(t, .)) = 0 and theng(t, .) = 0 for anyt∈[0, T]. tu In the same way as in chapter 1, we can deduce from the above existence and uniqueness result on the linear transport equation (1.1) that the formula

(S_tg₀)(x) :=g(t, x)

defines aC₀-semigroup onL^p(R^d), 1≤p <∞, whereg is the solution to the transport equation (1.1) associated to the initial datumg₀.

4.2 Positivity

We can recover in a quite elegant way the positivity as an aposteriori property that we deduce from the renormalization formula.

Corollary 4.2 Consider a solution g ∈ C([0, T];L^p(R^d)), 1 ≤p <∞, to the transport equation (1.1). Ifg0≥0 theng(t, .)≥0 for any t≥0.

Proof of Corollary 4.3.We argue similarily as in the proof of Corollary 4.1 but fixing a renormalizing function β ∈ W^1,∞(R) such that β(s) = 0 for any s ≥ 0, β(s) > 0 for any s < 0. Since then β(g0) = 0, we deduce that (4.1) holds again with that choice of functionβ and then, thanks to Gronwall lemma,β(g(t, .)) = 0 for anyt≥0. That meansg(t, .)≥0 for anyt≥0. tu

4.3 A posteriori estimate

Corollary 4.3 Consider a solution g ∈ C([0, T];L^p(R^d)), 1 ≤p <∞, to the transport equation (1.1). Ifg0∈L^q(R^d),1≤q≤ ∞, theng∈L^∞(0, T;L^q(R^d))for any T >0.

Proof of Corollary 4.3.We argue similarily as in the proof of Corollary 4.1 but fixing an arbitray renormalizing function β ∈ W^1,∞(R), β(s) = 0 on a small neighbourhood of s = 0, so that β(g)∈C([0, T];L¹(R^d)). For such a choice, we obtain the time integrale inequality

Z

R^d

β(g(t, .))dx= Z

R^d

β(g0)dx+ Z t

0

Z

R^d

β(g) (divxa)dxds ∀t≥0.

From the Gronwall lemma, we obtain withb=kdivakL^∞, the estimate (4.2)

Z

R^d

β(g(t, .))dx≤e^{b t} Z

R^d

β(g0)dx ∀t≥0.

Because estimate (4.2) is uniform with respect to β, we may choose a sequence of renormalizing functions (β_n) such thatβ_n(s)% |s|^q in the case 1≤q <∞and we get

kg(t, .)kL^q ≤e^bt/qkg(t, .)kL^q ∀t≥0.

In the case q = ∞, we obtain the same conclusion by fixingβ ∈ W^1,∞ such that β(s) = 0 for any |s| ≤ kg₀k_L^∞, β(s)>0 for any |s|>kg₀k_L^∞ or by passing to the limitq → ∞in the above

inequality. tu

4.4 Continuity

We can recover the strong L^p continuity property from the renormalization formula for a given solution. We do not present that rather technical issue here.

5 Complementary results

In this section we state and give a sketch of the proof of several complementary results of existence and uniqueness.

Regarding to the existence issue, we may extend our analysis to more general - equations (modifying the assumptions on the coefficients) ;

- initial data ;

- equations again by adding a given source term / a nonlinear RHS term ;

- domains by considering the equation set on Ω ⊂ R^d (and we possibly have to add boundary conditions).

Regarding the uniqueness issue, we will explain how to obtain it in aL^∞framework.

5.1 Parabolic equation in L

We consider the parabolic equation

(5.1) ∂tg= Λg+G in (0,∞)×R^d,

with

Λ := ∆ +a· ∇

As a first step, we look for a priori estimates. Given any solution g and any convexfunctionβ, we (formally) have

d dt

Z

β(g) = Z

∆β(g)− Z

β⁰⁰(g)|∇g|²− Z

(diva)β(g) + Z

β⁰(g)G

≤ Z

|diva|β(g) + Z

β⁰(g)|G|

Particularizingβ(s) =|s|^p, 1≤p <∞, and using the Young inequalitya^p−1b≤a^p+b^p/p, we get d

dt Z

|g|^p≤p b Z

|g|^p+ Z

|G|^p, b:= 1 +kdivak_L^∞. The Gronwall lemma yields

kg(t)k^p_Lp = e^pbtkg0k^p_Lp+ Z t

0

e^pb(t−s)kG(s)k^p_Lpds

≤ e^pbt

kg0kL^p+Z t 0

kG(s)k^p_Lpds1/pp

,

or equivalently

(5.2) sup

[0,T]

kgkL^p ≤e^bT

kg0kL^p+kGk_L1(0,T;L^p)

.

Next, definingβ=βM,M ∈N, aC² even and convex function such that - ifM ≤2,β(s) :=s²/2 for s≤2,β⁰⁰(s) = 0 fors≥3 ;

- ifM ≥3,β⁰⁰(s) = 0 for 0≤s≤M −1/2 ands≥M+ 3/2,β⁰⁰(s) = 1 forM ≤s≤M+ 1, and fixingχ∈ D(R^d), 0≤χ≤1,|∇χ|²≤χ, we have

1 2

Z T

0

Z

β⁰⁰(g)

|∇g|²χ−|∇χ|² χ

≤ Z T

0

Z

β⁰⁰(g)(|∇g|²χ− ∇g· ∇χ)

=hZ

β(g)χi0 T

+ Z T

0

Z

β⁰(g)g∆χ− Z

β(g) div (a χ) + Z

β⁰(g)G χ

from which we deduce that for anyT, R >0 and anyM ∈Nthat there exists a constantCT ,R

(5.3)

Z T

0

Z

BR

|∇g|²1_{M≤|g|≤M+1}≤C_{T ,R} sup

[0,T]

kgk_L1(B_R+2)+kGk_L1((0,T)×B_R+2)

.

The corresponding existence and uniqueness result reads as follows.

Theorem 5.1 For any exponent1≤p≤ ∞, any initial datumg₀∈L^p(R^d) and any source term G∈L¹(0, T;L^p(R^d)), there exists a unique renormalized solutiongin the sense thatgsatisfies the estimates (5.2)and (5.3)as well as

g∈C([0, T);L^p(R^d))ifp∈[1,∞), g∈C([0, T);L¹_loc(R^d))ifp=∞, and satisfies equation (5.5)in the following renormalized sense

(5.4) Z

β(gt)ϕ(t, .) = Z t

0

Z β(g)

∂tϕ−div(aϕ) + ∆ϕ + Z t

0

Z ϕ

β⁰(g)G−β⁰⁰(g)|∇g|² for any test function ϕ ∈ C_c²([0, T)×Ω) and any renormalizing function β ∈ C²(R) such that β⁰⁰∈Cc(R).

Elements of proof of Theorem 5.1. Step 1. The general case. Consider some data g0 ∈ L^p and G∈L¹(0, T;L^p). We introduce two sequences (g0,n) and (Gn) such thatg0,n∈L²∩L^p,g0,n→g0

inL^p,Gn∈L¹(0, T;L²∩L^p) andGn→GinL¹(0, T;L^p) and the corresponding sequence (gn) of variational solutions inXT =C(L²)∩L²(H¹)∩H¹(H⁻¹) to equation (5.5) which existence has been proved in Chapter 1. Notice in particular thatgn ∈C([0, T];L¹_loc) and thatgnis a renormized solution so that the additionnal a posteriori estimates (5.2) and (5.3) holds uniformly inn.

Step 2. The casep∈(1,∞).We claim thatgn∈C([0, T];L^p(R^d)). Indeed, from the renormalizing formula, we easily show that|gn|^p−2|∇gn|²∈L¹((0, T)×R^d), next

d dt

Z |gn|^p

p =−(p−1) Z

|gn|^p−2|∇gn|²− Z

|gn|^pdiva+ Z

Gngn|gn|^p−2∈L¹(0, T) and thenkgkL^p ∈C([0, T]). Together with the fact thatgn∈C([0, T];L¹_loc), we proved the above claim.

Now, using (5.2) we classically get that (gn) is a Cauchy sequence in C([0, T];L^p(R^d)), it thus converges to a limitg inC([0, T];L^p(R^d)), from what we immediately pass to the limitn→ ∞in the renormalized formulation (5.4) and conclude to the existence. The uniqueness follows from the renormalization property.

Step 3. The case p= 1. Here again we can aslo prove that gn ∈ C([0, T];L¹(R^d)) and conclude exactly as in Step 2. The claimed continuity property comes from the fact that one can prove

sup

[0,T]

Z

R^d

|gn|1_|gn|≥M →0, sup

[0,T]

Z

R^d

|gn|1_|x|≥M →0, whenM → ∞for anyn(and in fact uniformly in n).

Step 4. The case p = ∞. Here we argue in a different way. From (5.2), the equation (5.5) and summing (5.3) up to M = kginkL^∞_tx, we get that (g_n) is bounded in L^∞((0, T)×R^d), (∇gn) is bounded inL²_loc((0, T)×R^d) and

Z

R^d

gnϕ dx is strongly compact inC([0, T]) for anyϕ∈C_c²([0, T)×R^d).

We may then extract a subsequence of (g_n) (still denoted in same way) such that g_n converges to a limit g in σ(L^∞_tx, L¹_tx) and also in C([0, T];w−L²_loc). We then can pass to the limit in the weak formulation (5.4) with β(s) =s and for any test functionϕ∈C_c²([0, T)×R^d). Thanks to Theorem 3.4, we recover a posteriori the fact that g is a renormalized solution and the local continuity ofg. Uniqueness follows from a duality argument that we present below. tu

5.2 Duhamel formula and existence for transport equation with an ad- ditional term

We consider the transport equation with an additional term

(5.5) ∂tg= Λg+G in (0,∞)×R^d,

with

(Λg)(x) :=−a(x)· ∇g(x) +c(x)g(x) + Z

R^d

b(x, y)g(y)dy We interpret that equation as a perturbation equation

∂_tg=Bg+ ˜G, G˜ =Ag+G, and we claim that the fiunction

(5.6) g(t) =S_B(t)g₀+

Z t

0

S_B(t−s) ˜G(s)ds

is a solution to equation (5.5). Indeed, the semigroup S_B satisfies by definition (and at least formally)

d

dtS_B(t)h=BS_B(t)h, so that (again formally)

d

dtg(t) = d

dtS_B(t)g₀+ Z t

0

d

dtS_B(t−s) ˜G(s)ds+S_B(0) ˜G(t)

= Bn

S_B(t)g₀+ Z t

0

S_B(t−s) ˜G(s)dso + ˜G(t)

= Bg(t) + ˜G(t).

All that computations can be justified when written in a weak sense. The method used here is nothing but the wellknown variation of the constant method in ODE, the expression (5.6) is called the“Duhamel formula”and a functiong(t) which satisfies (5.6) (in an appropriate and meaningful functional sense) is called a“mild solution”to the equation (5.5).

Theorem 5.2 Assume a∈W^1,∞,c∈L^∞,b∈L^∞_x (L^p_y⁰)∩L^∞_y (L^p_x⁰),1≤p <∞. For anyg0∈L^p andG∈L¹(0, T;L^p)there exists a unique mild (weak, renormalized) solution to equation (5.5).

Elements of proof of Theorem 5.2.For anyh∈C([0, T];L^p), we define the mapping (Uh)(t) :=SB(t)g0+

Z t

0

SB(t−s)n

Ah(s) +G(s)o ds and we observe that

U :C([0, T];L^p)→C([0, T];L^p)

with Lipschitz constant bounded byb T. We just point out that thanks to Young inequality (when 1< p <∞)

Z Z

b(x, y)h(y)g(x)^p−1dxdy ≤ Z Z

b(x, y)[h(y)^p+g(x)^pdxdy

≤ kbk_L∞

x(L^p_y⁰)khk^p_Lp+kbk_L∞

y(L^p_x⁰)kgk^p_Lp.

Choosing T small enough, we can apply the Banach-Picard contraction theorem and we get the existence of a fixed pointg∈C([0, T];L^p),g=Ug. Proceding by induction, we obtain in that way

a global mild solution to equation (5.5). tu

5.3 Duality and uniqueness in the case p = ∞

Theorem 5.3 Assume a ∈ W^1,∞. For any g0 ∈ L^∞ there exists at most one weak solution g∈L^∞((0, T)×R^d)to the transport equation (1.1).

Elements of proof of Theorem 5.3. Since the equation is linear we only have to prove that the unique weak solutiong∈L^∞((0, T)×R^d) associated to the initial datum g0= 0 isg= 0.

By definition, for anyψ∈C_c¹([0, T]×R^d), there holds Z T

0

Z

R^d

g L^∗ψ dxdt=− Z

R^d

g(T)ψ(T)dx

withL^∗ψ:=−∂_tψ−div(a ψ). We claim that for any Ψ∈C_c¹((0, T)×R^d) there exists a function ψ∈C_c¹([0, T]×R^d) such that

(5.7) Λ^∗ψ= Ψ, ψ(T) = 0.

If we accept that fact, we obtain Z T

0

Z

R^d

gΨdxdt= 0 ∀Ψ∈C_c¹((0, T)×R^d), which in turns impliesg= 0 and that ends the proof.

Here we can solve easily the backward equation (5.7) thanks to the characterics method which leads to an explicit representation formula. In order to make the discussion simpler we exhibit that formula for the associated forward problem (we do not want to bother with backward time, but one can pass from a formula to another just by changing time t→T −t). We then consider the equation

∂tψ+a· ∇ψ+c ψ= Ψ, ψ(0) = 0,

with c := diva. Introduction the flow Φt(x) associated to the ODE ˙x=a(x), if such a solution exists, we have

d dt

hψ(t,Φ_t(x))e^{R0tc(Φs(x))ds}i

= Ψ(t,Φ_t(x))e^{R0tc(Φs(x))ds}, from which we deduce

ψ(t,Φt(x)) =e^{−R0tc(Φτ(x))dτ} Z t

0

Ψ(s,Φs(x))e^{R0sc(Φτ(x))dτ},

or equivalently, observing that Φ⁻¹_t = Φ_−t because the ODE is time autonomous, we have ψ(t, x) :=

Z t

0

Ψ(s,Φ_s−t(x))e^{−Rstc(Φτ−t(x))dτ}ds.

It is clear thatψ defined by the above formula is the solution to our dual problem from which we get (reversing time) the solution to (5.7) we were trying to find. tu