High regularity results of solutions tomodified p-Navier-Stokes equations
Francesca Crispo
joint work with P. Maremonti
Department of Mathematics and Physics
Second University of Naples

The 9th Japanese-German International Workshop
on Mathematical Fluid Dynamics
November 5 - 8, 2013
Waseda University, Tokyo

Francesca Crispo (University of Naples II)

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Introduction

We consider the following problem:∇ · T (u, π) = F + (u · ∇)u ,∇ · u = 0 in R3 ,(1)

where u = (u1 , u2 , u3 ) is a vector field, π is a scalar field,
T (u, π) is a special tensor of the kind
T (u, π) = −πI + |∇u|p−2 ∇u ,(2)

with I identity operator, p ∈ (1, 2).
The system is nonlinear and singular.
It was studied for the first time by J.L.Lions (1969), through the monotone
operators theory.

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Introduction

e of non-Newtonian fluids,
Tensor T is “close” to the well known stress tensor T
in the singular case,e (u, π) = −πI + |Du|p−2 Du ,T
T

Du = ∇u+(∇u), which gives rise to the so called p-Navier-Stokes problem
2

∇ · |Du|p−2 Du − ∇π = F + (u · ∇)u ,∇ · u = 0 in R3 .

(3)

Hence, we call, on the contrary, system (1) modified p-Navier-Stokes system.

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Introduction
e makes our results not interesting from a fluid
The difference between T and T
dynamic point of view.
Nevertheless the results gain a special interest from the analytical point ofview.
Indeed, the literature concerning the “high regularity” of solutions to the
singular p-(Navier-)Stokes problem is not satisfactory. We are just aware of
the following results:
Naumann & Wolf (JMFM 2005)

2, 3p

0

Ω bounded, p ∈ (9/5, 2), (u, π) ∈ W`ocp+1 (Ω) × Lp`oc (Ω),
Berselli, Diening & Růžička (JMFM 2010)3pΩ space-periodic, p ∈ (9/5, 2), u ∈ W 2, p+1 (]0, 1[3 ),Ebmeyer (MMAS 2006)3pΩ bounded, slip boundary conditions, p ∈ (9/5, 2), u ∈ W 2, p+1 (Ω) .
No global regularity result for the pressure field π.
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Introduction

The corresponding elliptic system is the well known p-Laplacian, whose study
has a wide literature.
We quote in particular the results due to:
Acerbi & Fusco (’89), DiBenedetto (’93), Ebmeyer (’06), Iwaniec & Manfredi (’89), Liu & Barrett (’93), Málek & Rajagopal & Růžička (’95), Acerbi &Mingione & Seregin (’04), Tolksdorf (’05), Mingione (06);high regularity with D 2 u ∈ Lq (Ω):
Beirão da Veiga & Crispo (’12) and (’13),
Crispo & Maremonti (forthcoming).
These results lead us to wonder if it is possible to deduce analogous results
related to the p-(Navier-)Stokes problem.

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Introduction

Our task is to understand if the couple of the vector field u and the scalar field
π can give a smooth solution of (1) at least with ∇u in place of Du.
In other words, we investigate if, in spite of the presence of the scalar field π,
the regularity of the field u can be compared with the one of solutions to thecorresponding elliptic system.
The results I’m going to introduce are part of the papers
F.C. and P. Maremonti, A high regularity result of solutions to modified p-Stokes
equations, submitted;
F.C. and P.Maremonti, A high regularity result of solutions to modified p-Navier-Stokes equations, forthcoming.

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“high regularity” for the modified p-Navier-Stokes system (1)Definition 1 (High regular solution)
A pair (u, π) is a high regular solution of (1) if:0for some q ∈ (3, +∞), D 2 u, ∇π ∈ Lq (R3 ), ∇u ∈ Lp (R3 ), π ∈ Lp (R3 ),∇ · u = 0, a. e. in R3 ,(|∇u|p−2 ∇u, ∇ϕ) = (∇π, ϕ) + ((u · ∇)u, ϕ) + (F , ϕ), for all ϕ ∈ C0∞ (R3 ).

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We setM(r ) := 1 − (2 − p)H(r 0 )(5 + H(r )),andM(2) := 2p − 3 − (2 − p)(1 + H(2)),
where H(s) is the Ls -singular norm of Calderón-Zygmund type, r 0 coniugate
exponent of r .
Our first result is the following existence result of high regular solution of themodified p-Stokes problem.

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Theorem 1:“high regularity” for the modified p-StokesLetp ∈ ( 32 , 2];
3p
M(2) > 0, M(q1 ) > 0 with q1 = 3+p;q ∈ (3, +∞), F ∈ Lq (R3 ) ∩ Lq1 (R3 ).
If M(q) > 0, then there exists a solution (u, π) in the sense of Definition 1 of
the modified p-Stokes system, and1

kD 2 ukq + kD 2 ukq1 ≤ c (kF kq + kF kq1 ) p−1 .
k∇πkq ≤ c kF kq ,k∇πkq1 ≤ c kF kq1 .

(4)
(5)

Moreover, the solution (u, π) is unique in the class of solutions with∇u ∈ Lp (R3 ).
The regularity results contained in the above theorem are the first highregularity results for solutions to a modified p-Stokes system,in the sense of D 2 u ∈ Lq (R3 ) and also of C 1,α -regularity, obtained by
Sobolev embedding (α = 1 − q3 )Francesca Crispo (University of Naples II)

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Approximating solutions

Before giving some comments on the statement, to better explain, we pointout that we develop our arguments starting from a nonsingular problem:
i
h
p−2
(6)
∇ · u = 0, in Rn , ∇ · (µ + |∇u|2 ) 2 ∇u −∇π = F , in R3 ,where µ > 0.
The idea is to work on this problem and to to obtain all the estimates uniformly
with respect to the parameter µ.

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Some comments on the statement of the theorem

3p
The choice of the exponent q1 = 3+p
has been made in order to get ∇u ∈p
3
L (R ) by Sobolev embedding. ∇u ∈ Lp (R3 ) enables us to discuss uniqueness. Hence in particular to compare our solution with weak solutions.
n(Note that q1 > 1 if and only if p > n−1, which excludes the value n = 2.)

The requests on the constants M(q1 ) > 0 and M(q) > 0 translate a conditionof proximity of p to 2, which is a sufficient condition in order to get the following
kind of estimate
D2u2−p(µ + |∇u|2 ) 2

Lr (R3 )≤ c kF kLr (R3 ) , for all µ > 0,(7)

for exponents r = q1 , q .

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Some comments on the statement of the theorem

3p
The choice of the exponent q1 = 3+p
has been made in order to get ∇u ∈p
3
L (R ) by Sobolev embedding. ∇u ∈ Lp (R3 ) enables us to discuss uniqueness. Hence in particular to compare our solution with weak solutions.
n(Note that q1 > 1 if and only if p > n−1, which excludes the value n = 2.)

The requests on the constants M(q1 ) > 0 and M(q) > 0 translate a conditionof proximity of p to 2, which is a sufficient condition in order to get the following
kind of estimate
D2u2−p(µ + |∇u|2 ) 2

Lr (R3 )≤ c kF kLr (R3 ) , for all µ > 0,(7)

for exponents r = q1 , q .

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Why does the theorem not work under the assumptione (u, π) = −πI + |Du|p−2 Du ? Hydrodynamic case!
T
The reason is connected with the fact that we employ the pointwise estimate|Dxi uj (x)|2≤ 1,µ + |∇u(x)|2that clearly does not hold, in general, withµ + |Du(x)|2 in place of µ + |∇u(x)|2 .
e (u, π) if we were able to show the
However, we could obtain the results for T
following crucial kind of estimates:
Z
Z|Dxh ui Dxi |Du|2 |r|Du|r |D 2 u|rdx ≤ c3−p4−pr dx ,2Ω (µ + |Du|2 ) 2 rΩ (µ + |Du| ) 2
for r = 2 and bounded Ω, u ∈ W01,2 (Ω) ∩ W 2,2 (Ω), and for r 6= 2 and Ω =R3 , ∇u ∈ Lp (R3 ) and D 2 u ∈ Lr (R3 ).
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Why does our proof work? What is new?

It is new in the approach
there is the advantage of a domain like R3

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Our approach

The new idea is to investigate the existence and the regularity looking at the
problem as a perturbed elliptic problem.
Since we work in R3 , we are able to arrive at a representation formula of thepressure field of this kindZ
Dy us (y)Dys |∇u|2
π := (2 − p)
Dyi E(x − y ) i
4−p dy ,R3(µ + |∇u(y)|2 ) 2E(x − y) being the fundamental solution of the Laplace equation.
This representation is crucial for our aims.

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Our approach

The new idea is to investigate the existence and the regularity looking at the
problem as a perturbed elliptic problem.
Since we work in R3 , we are able to arrive at a representation formula of thepressure field of this kindZ
Dy us (y)Dys |∇u|2
π := (2 − p)
Dyi E(x − y ) i
4−p dy ,R3(µ + |∇u(y)|2 ) 2E(x − y) being the fundamental solution of the Laplace equation.
This representation is crucial for our aims.

Francesca Crispo (University of Naples II)

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Setting K [∇u] := π(x), where π(x) is the pressure field already defined, thenwe study the problem∇ · (µ + |∇u|)p−2 ∇u − ∇K [∇u] = F , for all µ > 0.
(8)
Here the perturbation is the term ∇K [∇u], for p near 2 it can be small insuitable norms.
The advantage to handle system (8) is that now we can employ the methodsof the elliptic problems, the eigenfunctions of the Laplace operator as special
Galerkin basis (on a bounded domain), and all the estimates up to the second
derivatives are made in Lq spaces and not in the spaces of the hydrodynamic(which intrinsically contain the Helmohltz-Weyl orthogonality).

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Setting K [∇u] := π(x), where π(x) is the pressure field already defined, thenwe study the problem∇ · (µ + |∇u|)p−2 ∇u − ∇K [∇u] = F , for all µ > 0.
(8)
Here the perturbation is the term ∇K [∇u], for p near 2 it can be small insuitable norms.
The advantage to handle system (8) is that now we can employ the methodsof the elliptic problems, the eigenfunctions of the Laplace operator as special
Galerkin basis (on a bounded domain), and all the estimates up to the second
derivatives are made in Lq spaces and not in the spaces of the hydrodynamic(which intrinsically contain the Helmohltz-Weyl orthogonality).

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In particular, our estimates on second derivatives have, as a starting point, theestimate of the kind:D2u2−p(µ + |∇u|2 ) 2

Lr (R3 )≤ c kF kLr (R3 ) , for all µ > 0,

(9)for exponents r = q1 , q .
A crucial tool for the above estimate is the following one
D2u2−p(µ + |∇u|2 ) 2

Lr (R3 )≤ c(r , p)∆u2−p(µ + |∇u|2 ) 2

Lr (R3 )which is a generalization of the well known inequality||D 2 u||Lr (R3 ) ≤ c(r )||∆u||Lr (R3 ) .
Estimate (9) plays the same role as the estimate of kP∆ukr for the classicalStokes system.

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We gain, roughly speaking, a solution of the problem∇ · T (u, π, µ) = F ,in R3 , for all µ > 0 ,Z

Dyi us Dys |∇u|2with
π = (2 − p)R3

Dyi E(x − y )4−p(µ + |∇u|2 ) 2

dy .

(10)

(11)

So the divergence free equation∇·u =0(12)

seems missing.
But our pressure field has the representation formula of a pressure fieldarisen from the equations of the pressure:∆π = ∇ · ∇ · (µ + |∇u|)p−2 ∇u , joint with the condition ∇ · u = 0.
Therefore equation (12) is satisfied by our solution as a compatibility condition
between the equations (10) and the representation formula (11) and finally by
the condition ∇ · u → 0 for |x| → ∞.
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Indeed, the compatibility between the equation (10) and (11) formally gives,for ∇ · u, the equation∆(∇ · u) +(2 − p) ∇(∇ · u) · ∇|∇u|2= 0 , in R3 ,2(µ + |∇u|2 )we also can append the condition∇ · u → 0 as |x| → ∞.
Therefore the maximum principle ensures that ∇ · u = 0.
So, the solution of (10) is divergence free.
Hence u is a solution to the modified p-Stokes problem (1).

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Regularity under more general assumptions

In order to obtain a high regularity result for the modified p-Navier-Stokes
system, it is crucial to generalize the result obtained for the modified p-Stokes
system, by making different assumptions on the integrability of the force
term F .
This is the aim of the following theorem, whose proof relies on Theorem 1.

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Theorem 2: “high regularity” for the modified p-StokesLetp ∈ ( 95 , 2];
3p
M(2) > 0, M(q1 ) > 0 with q1 = 3+p;0q ∈ (3, +∞), F ∈ Lq (R3 ) ∩ D0−1,p (R3 ).
If M(q) > 0, then there exists a solution (u, π) in the sense of Definition 1 of
the modified p-Stokes system, and(1−a)(2−p)
p−1
kD 2 ukq ≤ c (kF kq kF k−1,p01) 1−a(2−p) ,

(13)

3q
a = 3(q−p)+pq. Moreover, the solution (u, π) is unique in the class of solutions
with ∇u ∈ Lp (R3 ).

Estimate (13) comes from the fundamental estimate
kD 2 ukq ≤ c (kF kq + k∇πkq )k∇uk2−p∞ .
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Theorem 2: “high regularity” for the modified p-StokesLetp ∈ ( 95 , 2];
3p
M(2) > 0, M(q1 ) > 0 with q1 = 3+p;0q ∈ (3, +∞), F ∈ Lq (R3 ) ∩ D0−1,p (R3 ).
If M(q) > 0, then there exists a solution (u, π) in the sense of Definition 1 of
the modified p-Stokes system, and(1−a)(2−p)
p−1
kD 2 ukq ≤ c (kF kq kF k−1,p01) 1−a(2−p) ,

(13)

3q
a = 3(q−p)+pq. Moreover, the solution (u, π) is unique in the class of solutions
with ∇u ∈ Lp (R3 ).

Estimate (13) comes from the fundamental estimate
kD 2 ukq ≤ c (kF kq + k∇πkq )k∇uk2−p∞ .
Francesca Crispo (University of Naples II)

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Theorem 3: “high regularity” for the modified p-Navier-StokesLetp ∈ ( 95 , 2];
3p
M(2) > 0, M(q1 ) > 0 with q1 = 3+p;0q ∈ (3, +∞), F ∈ Lq (R3 ) ∩ D0−1,p (R3 ).
If M(q) > 0, then there exists a solution (u, π) in the sense of Definition 1 of
the modified p-Navier- Stokes system.
The high regularity obtained in the above theorem is completely similar to the
one given for the p-Stokes system, as it happens for the classical NavierStokes equations.
Likewise, no uniqueness result is at disposal.

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The p-Navier-Stokes equations

The leading ideas behind this result are the following.
We consider the ”regularized” nonlinear system∇· |∇v |p−2 ∇v )−∇πv = χρ Jε (v ·∇)Jε (v χρ ) +f ,∇·v = 0 in R3 . (15)Using the Galerkin method and the monotone operator theory, for0f ∈ D0−1,p (R3 ) there exists a weak solution of (15) satisfying
1p−1
k∇v kp ≤ ckf k−1,p0 , ∀ε > 0, ∀ρ > 0 .

(16)

Note that (χρ Jε (v · ∇)Jε (v χρ ) , v ) = ((v · ∇)Jε (v χρ ), Jε (v χρ )) = 0 .

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We consider the modified p-Stokes system∇ · |∇u|p−2 ∇u − ∇πu = F , ∇ · u = 0 in R3 .

(17)

with

0
F := f + Fε,ρ := f + χρ Jε (v · ∇)Jε (v χρ ) ∈ Lq (R3 ) ∩ D0−1,p (R3 ) , (18)for which our Theorem 2 holds. Hence there exists a high regularsolution u.
By uniqueness in the class of weak solutions, u ≡ v .
Now we need to get estimates on kD 2 ukq , uniformly in ε and ρ, in order touse weak sequential compactness results, and pass to the limit on ε → 0 andρ → ∞.

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This is possible thanks to estimate (14), that we repreoduce herekD 2 ukq ≤ c (kF kq + k∇πkq )k∇uk2−p
∞ .

(19)

Indeed, from Gagliardo-Nirenberg’s inequality, Sobolev’s inequality and Theorem 1,k∇πkq ≤ ckF kq ≤ c(kf kq + kFε,ρ kq ),a(2−p)

2k∇uk2−p∞ ≤ ckD ukq(1−a)(2−p)

k∇ukpa(2−p)≤ ckD 2 ukq(1−a)(2−p)
p−1kf k−1,p0,

andkFε,ρ kq ≤ kuk∞ k∇(uχρ )kq ≤ ckD 2 ukbq k∇ukp1−b (k∇ukq +≤ ckD 2 ukbq k∇uk1−b(kD 2 ukdq k∇uk1−d+ppk∇uk2−b−d+≤ ckD 2 ukb+dqp1
kukLq (ρ≤|x|≤2ρ) )ρ

1
3 kuk∞ )ρ 1− qc2(1−b)
22b
3 kD uk.
q k∇ukpρ 1− qfor suitable constants a, b, d.
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Some remarks

All the previous estimates on the convective term, more precisely onFε,ρ , are not formal, as we already know that the norms on the right-hand side are finite.
Further the regularity result doesn’t require any iterative procedure.
The ideas of the proof are not dependent on the modified problem weare dealing with, neither on the domain R3 . Actually we could consider
the p-Stokes problem (also for p = 2) in a bounded domain. In this case:
If u is a high regular solution of the p-Stokes problem (in R3 , in Ω
bounded domain with homogeneous Dirichlet boundary conditions)which is unique in the class of weak solution, then if p > 59 there exists ahigh regular solution of the p-Navier-Stokes equations enjoying thesame regularity properties.

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Some remarks

All the previous estimates on the convective term, more precisely onFε,ρ , are not formal, as we already know that the norms on the right-hand side are finite.
Further the regularity result doesn’t require any iterative procedure.
The ideas of the proof are not dependent on the modified problem weare dealing with, neither on the domain R3 . Actually we could consider
the p-Stokes problem (also for p = 2) in a bounded domain. In this case:
If u is a high regular solution of the p-Stokes problem (in R3 , in Ω
bounded domain with homogeneous Dirichlet boundary conditions)which is unique in the class of weak solution, then if p > 59 there exists ahigh regular solution of the p-Navier-Stokes equations enjoying thesame regularity properties.

Francesca Crispo (University of Naples II)

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THANK YOU FOR YOUR ATTENTIONFrancesca Crispo (University of Naples II)

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