Quantitative estimates for advective equations with

Introduction Application to compressible fluids Application: Anelastic constraints A new method for compactness

Quantitative estimates for advective equationswith compressible flows

P-E Jabin, Univ. of Maryland

In collaboration with D. Bresch, CNRS — Univ. of SavoieMont-Blanc France

International Congress of Mathematicians 2018, Rio de Janeiro


Our goal

Consider the advection equation on the density ρ(t, x):

∂tρ+ div (ρ u) = 0,

for a velocity field u(t, x) in some Sobolev space L1tW

1,px .

The field u is either given or coupled with ρ.

−→ Many examples of such equations, linear or non-linear inphysics (Fluid mechanics...), the Bio-Sciences and other fields(transport models in biophysics, animal migration, pedestriandynamics...).

But we are interested in the case where div u 6∈ L∞ so that theflow may be compressible. Instead we only know a priori thatρ ∈ Lqt,x for some q > 1.

Question: Can one propagate some explicit regularity on ρ?


Renormalized solutions

• Well posedness for linear advection eqs (u given) was obtainedby DiPerna and Lions for u ∈ L1

tW1,px with p ≥ 1, extended to

BV by Ambrosio, through renormalized solutions.

• Renormalization: Prove that for any weak solution ρ then

∂tχ(ρ) + div (χ(ρ) u) + div u (χ′(ρ) ρ− χ(ρ)) = 0.

• Main idea: For some convolution kernel Kh, write an equationon Kh ? ρ and show a commutator estimate∫

Kh(x − y) (u(x)− u(y)) ρ(y) dy −→ 0, in L1.

• There is a need for more quantitative estimates but those areessentially only available for nearly incompressible flows(Crippa-DeLellis, J., Leger, Seeger-Smart-Street, Seis...).


A first application: Compressible Fluid dynamics

The compressible Navier-Stokes system reads

∂tρ+ div (u ρ) = 0,

∂t(ρ u) + div (ρ u ⊗ u)− µ∆u − (λ+ µ)∇div u = −∇p(ρ),

written here in the barotropic case, in a bounded domain Ω ⊂ Rd

with for instance Dirichlet boundary conditions

u|∂Ω = 0.




∂tρ+ div (u ρ) = 0,

∂t(ρ u) + div (ρ u ⊗ u)− div (µ(θ)D u) +∇(λ(θ) div u) = −∇p(ρ, θ),

∂t(ρE (ρ, θ)) + div (ρ u E ) + div (p u) = div (S u) + div (κ(θ)∇θ).

where D u = (∇u +∇uT )/2.Variants of course exist:• With temperature for the Navier-Stokes-Fourier system




∂tρ+ div (u ρ) = 0,

∂t(ρ u) + div (ρ u ⊗ u)− div (µ(ρ)D u) +∇(λ(ρ) div u) = −∇p(ρ),

where D u = (∇u +∇uT )/2.Variants of course exist:• With temperature for the Navier-Stokes-Fourier system• With density dependent viscosity(see nevertheless Bresch-Desjardins, Mellet-Vasseur, Vasseur-Yu...)


The difficulty

One has the following a priori estimatesEnergy estimate: For P(ρ) s.t. P ′ ρ− P = p(ρ),∫ (

P(ρ(t, x)) +1

2ρ u2

)dx +

∫ t

0

∫|∇u|2 = const.

Note that if C−1ργ ≤ p ≤ C ργ then P(ρ) ∼ ργ .


The difficulty

One has the following a priori estimatesEnergy estimate: For P(ρ) s.t. P ′ ρ− P = p(ρ),∫ (

P(ρ(t, x)) +1

2ρ u2

)dx +

∫ t

0

∫|∇u|2 = const.

Note that if C−1ργ ≤ p ≤ C ργ then P(ρ) ∼ ργ .Pressure estimates∫ t

0

∫ρa p(ρ) dx dt ≤ C , a <

2

dγ − 1.

−→ The issue is to obtain compactness on ρ, i.e.Control possible oscillations in the density.


The Lions-Feireisl theory

By using the monotonicity of the pressure law, Lions obtained that

Theorem P.L. LionsAssume p′(ρ) ∼ ργ−1 with γ > 9/5 and p monotone.Then there exists a weak solution to the compressibleNavier-Stokes system.

With refined techniques, this was improved to

Theorem E. FeireislAssume p′(ρ) ∼ ργ−1 with γ > 3/2 and p monotone.Then there exists a weak solution to compressible Navier-Stokes.


Which notion of solutions?

• Strong/classical solutions are of course the most convenient.They provide uniqueness and they preserve the most physicalproperties such as conservation of energy...

• However strong solutions only exist for short times, even indimension 2 (vacuum problem), or for small initialperturbations of an equilibrium in some cases.

• Weak solutions can be global in time and also allow to workwith non smooth initial data with only a bound on the energy.


What should the pressure law be?

• Ideal gas (Clapeyron 1834):

p = ρ θ.

• Van der Waals law (1873):

(p + a ρ2) (1− b ρ) = c ρ θ.

• Polynomial barotropic flows:

p = p(ρ), with often p = ργ .

• Virial equation of state (H. Kamerlingh Onnes 1901):

p = ρ θ(1 + B(θ) ρ+ C (θ) ρ2 + . . .

).


What should the pressure law be?• Ideal gas (Clapeyron 1834):

p = ρ θ.

• Van der Waals law (1873):

(p + a ρ2) (1− b ρ) = c ρ θ.

• Polynomial barotropic flows:

p = p(ρ), with often p = ργ .

• Variants of the Virial such as the Benedict-Webb-Rubin:

p =ρR θ +

(B0 R θ − A0 −

C0

θ2

)ρ2 + (b R θ − a) ρ3 + α a ρ6

+c ρ3

θ2(1 + γ ρ2) e−γ ρ

2.


Monotone vs non-monotone pressure laws

• Thermodynamically the stability of the equilibrium is directlyconnected to the monotonicity of p.

• Monotone laws are also required for hyperbolicity.

• However, many physical models have p non monotone.

• It is not clear why a thermodynamical assumption shouldcontrol the stability of solutions over bounded times.

• The same type of questions may be asked about the stresstensors. For instance in some geophysical flows, one needs totake

∂t(ρ u) + div (ρ u ⊗ u)− µ∆u − µz ∂zzu = −∇p(ρ),

with µz 6= 0 (see for instance Temam-Ziane).


Our new result

Main improvements:No monotonicity assumption on p, explicit regularity.

Theorem(Bresch-Jabin 2018) Assume that for γ > 9/5

ργ

C≤ p(ρ) ≤ C ργ , |p′(ρ)| ≤ C ργ−1.

Then there exists a weak solution to compressible Navier-Stokes,with d the space dimension∫

|ρ(x)− ρ(y)|(|x − y |+ h)d

dx dy ≤ C (E 0)| log h|

log | log h|.

This is a log log scale vs the usual log scale obtained by Crippa andDeLellis but for nearly incompressible flows instead of compressible.


Advection equations with anelastic constraints

Consider the following advective equation

a (∂tφ+ u · ∇φ) = 0 in (0,T )× Ω

where a = a(x) is given. and with a velocity field u s.t.

div(au) = 0 in (0,T )× Ω, a u · n|(0,T )×Ω = 0.

−→ Many examples of such systems

• The so-called lake equation (see Bresch-Metivier,Lacave-Nguyen-Pausader, Levermore-Oliver-Titi, Masmoudi,Munteanu...) where a is the bathymetry and u is coupled to φthrough curl u = aφ.

• Meteorology (Duran, Klein, Lipps-Hemler, Wilhelmon-Ogura),congestion (Perrin-Zatorska), floating objects (see Lannes)...


The difficulty

a(x)

Main issue: The bathymetry a may be singular, and vanishes on apotentially complex set, where the equations degenerate.


The difficulty

Main issue: The bathymetry a may be singular, and vanishes on apotentially complex set, where the equations degenerate.

• First observe that a priori, φ ∈ L∞ trivially and in factφ ∈ L∞t Lpx (a dx) for any p ≥ 1. But div u 6∈ L∞ and it is notpossible to reduce the problem to an incompressible equation.

• For given a and u, obtaining weak solutions is straightforwardand require only minimal assumptions on a. This does notprovide uniqueness, even in such a linear setting.

• Strong solutions require much more regularity on a and u.

• Obtaining renormalized solutions should still allow for apotentially singular bathymetry a but maintains the physicalproperties: Energy, uniqueness in the linear case...

−→ Compactness and hence additional regularity on φ is required.


Our result

TheoremAssume that for some p > 1 and 1

q + 1p < 1, there exists

0 ≤ α ≤ a s.t.∫Ω

(a(x) | logα(x)|+ |a(x)1/p∗∇ logα(x)|q)

)dx <∞,

and that

‖u‖L∞t Lpa+

∫ T

0

∫Ωa(x) |∇u(t, x)| log

(e+a(x)|∇(u(t, x))|

)dx dt <∞.

Then there exists a renormalized solution to the system.

−→ Having α 6= a allows a wide range of possible a: For exampleif a = 0 on Oc for some Lipschitz domain O and on O

(d(x , ∂O))k . a(x) . (d(x , ∂O))l =⇒ α(x) = (d(x , ∂O))θ, θ large.


The idea

Propagate some explicit regularity on ρ by computing∫|ρ(t, x)− ρ(t, y)|

(|x − y |+ h)kdx dy ,

for some k ≥ d .However this corresponds to a Sobolev like regularity on ρ whichcannot work (see Alberti-Crippa-Mazzucato, J.). So instead...


The idea


(|x − y |+ h)kW (t, x , y) dx dy ,

for some k ≥ d .Where the weight W solves the same transport equation

∂tW + u(t, x) · ∇xW + u(t, y) · ∇yW = −D,

for a well chosen penalization D.


The idea


(|x − y |+ h)kW (t, x , y) dx dy ,

for some k ≥ d .Where the weight W solves the same transport equation

∂tW + u(t, x) · ∇xW + u(t, y) · ∇yW = −D,

for a well chosen penalization D.Then explain that W cannot be too small, too often to bound∫

|ρ(x)− ρ(y)|(|x − y |+ h)k

dx dy ,

in terms of h.Because of the compressible flow, this will force k = d .


Sketch of the estimates: Compressible N-S

Define w(t, x) solution to

∂tw + u · ∇w = −λM |∇u|w ,

with M the maximal operator.Actually to obtain realistic conditions on γ, we take

∂tw + u · ∇w = −λ (ρ |div u|+ M |∇u|+ ργ)w .

But we will keep the simplest version here...


Sketch of the estimates: Compressible N-S

Define w(t, x) solution to

∂tw + u · ∇w = −λM |∇u|w ,

with M the maximal operator.One can simply show that∫

| logw | ρ(t, x) dx ≤ λ ‖u‖L2tH

1x‖ρ‖L2

t,x.

This forces the choice

W (t, x , y) = w(t, x) + w(t, y),

so that W = 0 only if both w(x) = 0 and w(y) = 0, i.e. ifρ(x) = 0 and ρ(y) = 0.This would not be needed for nearly incompressible flows where wecould take W = w(x)w(y) and it creates difficulties...


Sketch of the estimates: Compressible N-S, Part 2Denote δρ = ρ(t, x)− ρ(t, y) and per Kruzkov’ doubling ofvariables

∂t |δρ|2 + div x(u(t, x) |δρ|2) + div y (u(t, y) |δρ|2)

= −1

2(div u(x)− div u(y)) δρ (ρ(x) + ρ(y)).

Recall

∂tW + u(t, x) · ∇xW + u(t, y) · ∇yW = −D,

and calculate

d

dt

∫|δρ|2

(|x − y |+ h)dW

= − 1

2

∫div u(x)− div u(y)

(|x − y |+ h)dδρ (ρ(x) + ρ(y))W

+

∫(u(x)− u(y)) · (x − y)

|x − y | (|x − y |+ h)d+1|δρ|2 W −

∫|δρ|2

(|x − y |+ h)dD.


Sketch of the estimates: Compressible N-S, Part 2

d

dt

∫|δρ|2

(|x − y |+ h)dW

= − 1

2


(|x − y |+ h)dδρ (ρ(x) + ρ(y))W

+

∫(u(x)− u(y)) · (x − y)

|x − y | (|x − y |+ h)d+1|δρ|2 W −

∫|δρ|2

(|x − y |+ h)dD.

The first term in the r.h.s. logically involves the regularity ofdiv u. For compressible Navier-Stokes, it is possible to couple thisback to ρ through the pointwise relation

div u = p(ρ(x)) + ∆−1 div (∂t(ρ u) + div(ρ u ⊗ u))︸︷︷︸OK

.



d

dt

∫|δρ|2

(|x − y |+ h)dW

= − 1

2


(|x − y |+ h)dδρ (ρ(x) + ρ(y))W

+

∫(u(x)− u(y)) · (x − y)

|x − y | (|x − y |+ h)d+1|δρ|2 W −

∫|δρ|2

(|x − y |+ h)dD.

The third term will help bound the others∫|δρ|2

(|x − y |+ h)dD = 2

∫|δρ|2

(|x − y |+ h)dM |∇u|(x)w(t, x).



d

dt

∫|δρ|2

(|x − y |+ h)dW

= − 1

2


(|x − y |+ h)dδρ (ρ(x) + ρ(y))W

+

∫(u(x)− u(y)) · (x − y)

|x − y | (|x − y |+ h)d+1|δρ|2 W −

∫|δρ|2

(|x − y |+ h)dD.

The second term is a commutator estimate but cannot besimply controlled...


Commutator estimate: The problem

We need to control the commutator by the dissipation∫|u(x)− u(y)|

(|x − y |+ h)d+1|δρ|2 W .

∫|δρ|2

|(|x − y |+ h)dM |∇u|(x)w(x)+sym.

Following Crippa-DeLellis, one could try to use

|u(x)− u(y)| ≤ C (M |∇u|(x) + M |∇u|(y)) |x − y |,

to get∫|u(x)− u(y)|

(|x − y |+ h)d+1|δρ|2 W

≤ C

∫|δρ|2

|(|x − y |+ h)d(M |∇u|(x)w(x)︸︷︷︸

OK

+M |∇u|(y)w(x)︸︷︷︸???

) + sym.

−→ Due to choosing W (x , y) = w(x) + w(y), not w(x)w(y).


Commutator estimate: Lagrangian viewLarge values of |grad u|

Fluid trajectories

Good trajectory, w(t,X (t, x)) > 0:∫ t

0 M |∇u|(X (s, x)) ds ∼ 1,

Bad trajectory, w(t,X (t, x)) << 1:∫ t

0 M |∇u|(X (s, x)) ds >> 1.

Nearly compressible case: Only compare good vs. goodOur compressible case: Also compare good vs. bad

−→ Cannot work if good and bad trajectories intertwine...


Commutator estimate: The solution

Instead one uses the more precise estimate

|u(x)− u(y)| ≤ C

∫|z−x |≤2 |x−y |

|∇u(z)||z − x |d−1

dz︸︷︷︸=L|x−y|?|∇u|(x)

+sym.

The control is achieved through square function∫|u(x)− u(y)|

(|x − y |+ h)d+1|δρ|2 (w(x) + w(y))

≤ C

∫|δρ|2

(|x − y |+ h)dM |∇u|(x)w(x)

+ C

∫|δρ|2

(|x − y |+ h)d|L|x−y | ? |∇u|(x)− L|x−y | ? |∇u|(y)|w(x)︸︷︷︸

term to bound

.


Commutator estimate: The solutionInstead one uses the more precise estimate

|u(x)− u(y)| ≤ C

∫|z−x |≤2 |x−y |

|∇u(z)||z − x |d−1

dz︸︷︷︸=L|x−y|?|∇u|(x)

+sym.

The control is achieved through square function∫|δρ|2

(|x − y |+ h)d∣∣L|x−y | ? |∇u|(x)− L|x−y | ? |∇u|(y)

∣∣ w(x) dx dy

=

∫Sd−1

∫ 1

h

∫|δρ|2 |Lr ? |∇u|(x)− Lr ? |∇u|(x + r ω)| w(x) dx

dr

rdω,

with the property that for a normalized convolution kernel Lr∫ 1

h

dr

r‖Lr ? f − Lr ? f (.+ r ω)‖Lp ≤ C | log h|1/2 ‖f ‖Lp .


Compressible N-S: Conclusion

Summing up, one finds

d

dt

∫|δρ|

(|x − y |+ h)d(w(x) + w(y)) ≤ C | log h|θ ‖∇u‖Lp ‖ρ‖2

L2 p∗ ,

for some θ < 1.Coupled with∫

| logw | ρ(t, x) dx ≤ λ ‖u‖L2tH

1x‖ρ‖L2

t,x,

this leads to ∫|δρ|2

(|x − y |+ h)d≤ C

| log h|log | log h|

.


Sketch of the estimates: Anelastic constraintsLagrangian approach: Requires a to belong to Muckenhoupt spaceto bound

∫a |M f |2 by

∫a f 2.

−→ Introduce a three-level weight procedure with first a weightcontrolling large velocities

∂twu + u · ∇wu = −wu |u(t, x)|α(x) +

∫ t0 |∇(α(x)u(s, x))| ds

1 +∫ t

0 |α(x)u(s, x)| ds,

which is used on the weight controlling shallow and fast-varyingbathymetry

∂twa + u · ∇wa = −γ |u · ∇α|α

wa, wa|t=0 = (α(x))γ ,

through an estimate on

d

dt

∫a(x)wu(t, x) | logwa(t, x)| dx .


Sketch of the estimates: Anelastic constraints, Part 2The weight wa is used in turn to bound the final weight penalizingoscillations in u

∂tw +u ·∇w = λ[ M |∇(α u)|

α+(M |∇α|(x))θ |u(x)|θ+ |α(x)|−θ∗

],

again through an estimate on

d

dt

∫a(x)wa(t, x) | logw(t, x)| dx .

Finally all three weights are used to obtain the regularity of a f (φ)for any f by estimating∫

a(x) a(y)|φ(x)− φ(y)|(|x − y |+ h)d

wu(x)wu(y)wa(x)wa(y)w(x)w(y).

=⇒∫

Ω2

|φ(x)− φ(y)|(h + |x − y |)2

a(x) a(y) dx dy ≤ C| log h|

log | log h|.


Conclusion

We have developed a method for advection equations wherecompressibility is an issue. The method is flexible, allowing forstraightforward penalization of appropriately defined bad regions inthe fluid.This allows to handle for the first time physical phenomena thatare a priori unstable.

Thank you!

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Quantitative estimates for advective equations with