2021/2023 UNC Greensboro PDE Conference,

Electronic Journal of Differential Equations, Conference 26 (2022/2025), pp. 171–178.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

DOI: 10.58997/ejde.conf.26.c2

BILINEAR ESTIMATES POSED IN FINITE DOMAINS

IN 2D AND 3D

ZHUORU CHEN, TAIGE WANG, XIANGFEI XIE

Abstract. In this article, we establish bilinear estimates for the nonlinear

term appearing in fluid mechanics models posed in bounded domains of 2D

and 3D. Also, we give an example to apply the estimates in a 2D bounded
domain.

1. Introduction

We summarize and extend the bilinear estimate posed in a Sobolev space Yτ,T for
nonlinear term (u · ∇)v which models convection effects in flow phenomena. Here
(u, v) are velocity vectors in Rn, n = 2 and 3 and they depend on time. Operator
u · ∇ =

∑n
i=1 ui∂xi , which models convection effect in fluids.

In this article, we extend the bilinear estimate by Bona, Sun and Zhang [3] on
1D finite domains to multi-dimensional finite domains. This technique has lots of
applications in handling nonlinear terms when one uses semi-group mild solution
formulation for nonlinear dispersive and dissipative equations [1, 2, 3, 4, 21]. For
instance of one application is to formulate mild solution for the linear initial-value
problem y′(t) +Ay(t) = 0, y(0) = ϕ as

y(t) = e−Atϕ.

Its nonlinear (bilinear) counterpart problem y′ + Ay + B(y, y) = 0 has a mild
solution form

y(t) = e−Atϕ+

∫ t

0

e−A(t−s)B(y(s), y(s))ds,

with the additional nonlinear term treated as a perturbation. If the former linear
problem has been solved, then bilinear estimates developed can lead to the solvabil-
ity of nonlinear one in same work function spaces such as Hs and/or Yτ,T (defined
in later context).

Via this fashion, one can even handle unbounded domains. Specifically, in
[1, 2, 4], bilinear estimates are obtained in half-line spatial domain R+ with data
prescribed on one bound (x = 0). More general development of this type of esti-
mates and applications are in wellposedness theory in low-regularity spaces with

2020 Mathematics Subject Classification. 35D05, 35K55, 34K13, 35Q93.

Key words and phrases. Bilinear estimate; wellposedness; existence.
©2025 This work is licensed under a CC BY 4.0 license.
Published ??.

171



172 Z. CHEN, T. WANG, X. XIE EJDE-2022/25/CONF/26

negative indices for dispersive equations posed in unbounded domains. These re-
lated estimates are defined in Fourier restriction spaces such as Bourgain Spaces,
and we would refer readers to content and references therein [11, 12] by Kenig et
al, and the recent work [20] on coupled KdV equations by Yang and Zhang.

In light of the application facilitating existence theory of strong or weak solutions,
Zhang et al including one of the authors considered KdV equations and dissipative
PDEs, respectively on finite domains in 1D (see e.g. [17, 18, 19, 21]).

Similar mechanism to approach existence of solution and special case: forced
oscillations in high dimensions could be pursued, which might be based on results
in this article. For high dimension fluid models, such as 3D Navier-Stokes equations,
results about forced oscillations could be found in series of works [13, 14, 9, 10]) by
Serrin, Shinbrot, Kato et al, followed with [5, 7, 8] by Hsia et al from 1D Burgers
to high dimensional models. They approached problem by using trilinear estimates
and Galerkin’s method (see Temam’s famous monographs [15, 16]). From author’s
point of view, if using the bilinear estimates and the related techniques presented
in [3, 17, 19], proof could be more compact and terse.

We would cite the bilinear estimate lemma for term (uv)x (including uvx) in [3]:

Theorem 1.1. There exists a constant C separated from T such that for any T > 0
and (u, v) ∈ Yτ,T ,∫ τ+T

τ

∥(u(s)v(s))x∥ds ≤ C(T 1/2 + T 1/3)∥u∥Yτ,T
∥v∥Yτ,T

. (1.1)

where the norm ∥ · ∥ is of L2, and the function space Yτ,T is defined on spatial
interval I = [0, 1] as

Yt,T =
{
u
∣∣ sup
t≤s≤t+T

∥u(s)∥+
(∫ t+T

t

∥ux(s)∥2ds
)1/2

< ∞
}

with norm

∥u∥Yt,T
= sup

t≤s≤t+T
∥u(s)∥+

(∫ t+T

t

∥ux(s)∥2ds
)1/2

.

In this theorem, the function space Yτ,T is L2 adding a smoothing over time
interval [τ, τ + T ]. Zhang and one of the authors invoked this version in analyzing
forced oscillation and its stability of 1D Burgers equation (see [19]). However, when
dimension increases, Sobolev imbedding differs from case in n = 1, and we need
variants for analysis on nonlinear term. We would mimic similar fashion but shift
to higher regularity in H1 to define suitable function spaces for 2D and 3D and
seek their linear estimates. We have applied some of these estimates in some fluid
models in high dimensions (see [6]).

This article is organized as follows: we present main results in Section 2, and
their proofs in Section 3. We show the usage of the inequalities in Section 4.

2. Main results

Norm ∥ · ∥X is used to denote the endowed canonical norm of a Banach space
X, and ∥ · ∥ denotes the L2 norm. Since we treat u as a vector in Rn, the L2 norm
of u is

∥u(t)∥ =
(∫

Ω

n∑
i=1

u2
i (x, t)dx

)1/2

with u = (u1, u2, . . . , un)



EJDE-2022/2025/CONF/26 BILINEAR ESTIMATES 173

on a bounded domain Ω in Rn. Also, given the context of flow following time
evolution, we may use u(t) or ∥u(t)∥X .

We will consider bilinear estimates related to this Sobolev function space

Yt,T =
{
u
∣∣ sup
t≤s≤t+T

∥∇u(s)∥+
(∫ t+T

t

∥Au(s)∥2ds
)1/2

< ∞
}

endowed with the norm

∥u∥Yt,T
= sup

t≤s≤t+T
∥∇u(s)∥+

(∫ t+T

t

∥Au(s)∥2ds
)1/2

on the bounded time interval [t, t+ T ].
Note that Yt,T is related to Hilbert space L2 or H1 relying on dimension n. ∇u is

the Jacobian. Operator A is the Stokes operator which relates to second derivatives
according to [15, 16] by Laplacian △ and Leray projection operator P:

Au = −P△u : H2 7→ L2, in the solenoidal vector field ∇ · u = 0,

which gives Au being a vector in Rn.
Our results for convection nonlinear terms are presented as follows:

Theorem 2.1. If n = 2 and 3, for given positive T and τ , there exists a positive
number C(T ) such as∫ τ+T

τ

∥u(s) · ∇v(s)∥H1ds ≤ C(T )∥u∥Yτ,T
∥v∥Yτ,T

. (2.1)

We shall prove this result sequentially in 2D and 3D, when Poicaré inequality
holds between the L2 norms of ∇u and u, Au and ∇u.

3. Proof of main results

Proof of Theorem 2.1 when n = 2. We prove when τ = 0. We first prove estimate
on ∥u · ∇v∥.

In 2D finite domains,

∥u · ∇v∥ ≤ ∥u∥L4∥∇v∥L4 ≤ C∥∇u∥∥∇v∥L4 .

From Hölder inequality and the Sobolev inequality, we have ∥u∥L4 ≲ ∥∇u∥. Owing
to the Gagliardo-Nirenburg inequality in 2D finite domains,

∥∇v∥L4 ≲ ∥∇v∥+ ∥∇v∥1/2∥Av∥1/2,

we have∫ T

0

∥u(s) · ∇v(s)∥ds ≤ C

∫ T

0

∥∇u(s)∥(∥∇v(s)∥+ ∥∇v(s)∥1/2∥Av(s)∥1/2)ds

whence ∫ T

0

∥u(s) · ∇v(s)∥ds ≤ C∥u∥Y0,T
∥v∥Y0,T

,

in that ∫ T

0

∥∇u(s)∥∥∇v(s)∥ds ≤ sup
s∈[0,T ]

∥∇v∥T 1/2
(∫ T

0

∥∇u(s)∥2ds
)1/2

≤ T 1/2∥v∥Y0,T
∥u∥Y0,T

,



174 Z. CHEN, T. WANG, X. XIE EJDE-2022/25/CONF/26∫ T

0

∥∇v(s)∥1/2∥Av(s)∥1/2∥∇u(s)∥ds

≤ sup
s∈[0,T ]

(∥∇u(s)∥∥∇v(s)∥)T 3/4
(∫ T

0

∥Av(s)∥2ds
)1/4

≤ T 3/4∥u∥Y0,T
∥v∥Y0,T

.

Second, on
∫ T

0
∥∇(u(s) · ∇v(s))∥ds, we have∫ T

0

∥∇(u(s) · ∇v(s))∥ds

≤
∫ T

0

∥∇u(s) · ∇v(s)∥ds+
∫ T

0

∥u(s) ·Av(s)∥ds

≤
∫ T

0

∥∇u(s)∥L4∥∇v(s)∥L4ds+

∫ T

0

∥u(s)∥L∞∥Av(s)∥ds

≤ C

∫ T

0

(∥∇u(s)∥+ ∥∇u(s)∥1/2∥Au(s)∥1/2)(∥∇v(s)∥+ ∥∇v(s)∥1/2∥Av(s)∥1/2)ds

+ C

∫ T

0

∥Au(s)∥∥Av(s)∥ds

≤ CT 3/4∥u∥Y0,T
∥v∥Y0,T

+ sup
s∈[0,T ]

(∥∇u(s)∥∥∇v(s)∥)1/2

× T 1/2
(∫ T

0

∥Au(s)∥2ds
)1/4(∫ T

0

∥Av(s)∥2ds
)1/4

≤ C(T 3/4 + T 1/2 + 1)∥u∥Y0,T
∥v∥Y0,T

in which we use Hölder inequality, Poicaré inequality, Gagliardo-Nirenburg inequal-
ity, Sobolev inequality ∥u∥L∞ ≲ ∥Au∥ in 2D finite domains. Combining these two
estimates, we obtain the bilinear estimate (2.1) for 2D. □

Proof of Theorem 2.1 when n = 3. With a similar manner in last proof, we consider
two terms: ∫ T

0

∥u · ∇v∥ds and

∫ T

0

∥∇(u · ∇v)∥ds.

In 3D finite domains, one can invoke Hölder inequality to reach∫ T

0

∥u(s) · ∇v(s)∥ds ≤ C

∫ T

0

∥u(s)∥L12∥∇v(s)∥L12/5ds

≤ C

∫ T

0

∥∇u(s)∥L12/5∥∇v(s)∥L12/5ds.

with Sobolev inequality ∥u∥L12 ≲ ∥∇u∥L12/5 for n = 3.
Owing to the interpolation in Lp for ∇u,

∥∇u∥L12/5 ≤ C∥∇u∥3/4∥∇u∥1/4L6 ,

and Sobolev inequality ∥∇u∥L6 ≲ ∥Au∥ for n = 3, whence∫ T

0

∥u(s) · ∇v(s)∥ds



EJDE-2022/2025/CONF/26 BILINEAR ESTIMATES 175

≤ C
(∫ T

0

∥∇u(s)∥2L12/5ds
)1/2(∫ T

0

∥∇v(s)∥2L12/5ds
)1/2

≤ C
(∫ T

0

∥∇u(s)∥3/2∥∇u(s)∥1/2L6 ds
)1/2(∫ T

0

∥∇v(s)∥3/2∥∇v(s)∥1/2L6 ds
)1/2

≤ C
(∫ T

0

∥Au(s)∥2ds
)1/2(∫ T

0

∥Av(s)∥2ds
)1/2

≤ C∥u∥Y0,T
∥v∥Y0,T

.

For the second term, since ∥∇(u · ∇v)∥ ≲ ∥∇u · ∇v∥+ ∥uAv∥, we have∫ T

0

∥∇u(s) · ∇v(s)∥ds ≲
∫ T

0

∥∇u(s)∥L4∥∇v(s)∥L4ds

≲
∫ T

0

∥∇u(s)∥L6∥∇v(s)∥L6ds

≲
(∫ T

0

∥Au(s)∥2ds
)1/2(∫ T

0

∥Av(s)∥2ds
)1/2

≲ ∥u∥Y0,T
∥v∥Y0,T

,

by ∥∇u∥L6 ≲ ∥Au∥.∫ T

0

∥u(s)Av(s)∥ds ≲
∫ T

0

∥u(s)∥L∞∥Av(s)∥ds

≲
∫ T

0

∥Au(s)∥∥Av(s)∥ds

≲
(∫ T

0

∥Au(s)∥2ds
)1/2(∫ T

0

∥Av(s)∥2ds
)1/2

≲ ∥u∥Y0,T
∥v∥Y0,T

.

By Agmon’s inequality, ∥u∥L∞ ≲ ∥u∥1/2H1 ∥u∥1/2H2 when n = 3. Therefore,∫ T

0

∥∇(u · ∇v)∥ds ≤ C∥u∥Y0,T
∥v∥Y0,T

.

Combining these two estimates for ∇(u ·∇v) and u ·∇v, we complete the proof. □

4. Application

In this section, we show a usage of bilinear estimates by looking at a simplified
model of 2D incompressible Navier-Stokes equation when external force f can be
controlled. Some of the following practice has been recently used in [6] by some of
authors.

Consider a Burgers-type equation posed in 2D finite domain Ω× [τ, τ + T ]:

∂

∂t
u+Au+ u · ∇u = f (4.1)

with homogeneous Dirichlet boundary condition (no-slip boundary condition). We
also have initial value at t = τ that u ∈ H1(Ω) (we will use H1 for this space). We
will obtain the following existence result of u.



176 Z. CHEN, T. WANG, X. XIE EJDE-2022/25/CONF/26

Theorem 4.1. If equation (4.1) with no-slip boundary conditions, H1 initial value
and f ∈ H1, then

u ∈ Yτ,T . (4.2)

Let u(x, t) = v(x, t) + z(x, t), where v satisfies linear equation

∂

∂t
v +Av = f,

v(x, τ) = u(x, τ),
(4.3)

and z satisfies the nonlinear equation

∂

∂t
z +Az = −z · ∇z − z · ∇v − v · ∇z − v · ∇v,

z(x, τ) = 0.
(4.4)

We could use mild solution form when we take gradient on both sides of (4.1), to
get the following Yτ,T -result for linear equation (4.3).

Lemma 4.2. There exists a positive constant C not relying on T , such that

∥v∥2Yτ,T
≤ C

[
∥∇v(τ)∥2 +

(∫ τ+T

τ

∥∇f(s)∥2ds
)]

. (4.5)

We can reach a result for z when obtained magnitude of v (∥v∥Yτ,T
) is small.

Lemma 4.3. If there exists M > 0 such that if ∥v∥Yτ,T
≤ M , then

∥z∥Yτ,T
≤ C∥v∥Yτ,T

. (4.6)

Proof. We write a mild solution form for ∇z,

∇z(t) = −
∫ t

τ

e∆(t−s)∇ (z · ∇z + z · ∇v + v · ∇z + v · ∇v) ds.S

for t ∈ (τ, τ + T ).
We define the map q 7→ Γ(q) referring to v ∈ Yτ,T : Yτ,T 7→ Yτ,T such that

Γ(q; v) = −
∫ t

τ

e∆(t−s)∇(q · ∇q + q · ∇v + v · ∇q + v · ∇v)ds.

We aim to prove that Γ(q; v) has a fix point in space Yτ,T to obtain the existence
of z(t).
We define a positive controlling constant M to bound the magnitude of v in Yτ,T

and a bounded set SM ∈ Yτ,T :

M = ∥v∥Yτ,T
, SM =

{
q ∈ Yτ,T

∣∣∥Q∥Yτ,T
≤ M

}
.

We will apply the Banach Fixed Point Theorem on SM . We could prove Γ(q) ∈ SM

if q ∈ SM . In fact, by the bilinear estimate

∥Γ(q; v)∥Y0,T
≤

∫ t

τ

∥(q · ∇q + q · ∇v + v · ∇q + v · ∇v∥H1ds

≤
∫ t

τ

(∥q∥2Yτ,T
+ ∥q∥Yτ,T

∥v∥Y0,T
+ ∥v∥2Yτ,T

)ds

≤ C(∥q∥2Yτ,T
+ ∥v∥2Yτ,T

)

≤ CM2,



EJDE-2022/2025/CONF/26 BILINEAR ESTIMATES 177

Let M be so small that M ≤ 1
2C , then we infer that

∥Γ(q; v)∥Yτ,T
≤ M

2
.

At this point, we would prove the Lipschitz continuity of Γ with respect to q. Pick
any two distinct q1, q2 ∈ SM , then

∥Γ(q1; v)− Γ(q2; v)∥Yτ,T

=

∫ t

τ

∥e∆(T−s)∥∥∇(q1 · ∇q1 + q1 · ∇v + v · ∇q1 − q2 · ·q1 − q2 · ∇v − v · ∇q2∥ds

≤ C∥q1 − q2∥Yτ,T
∥q1∥Y0,T

+ C∥q1 − q2∥Yτ,T
∥q2∥Y0,T

+ C∥q1 − q2∥Yτ,T
∥v∥Yτ,T

≤ 3CM∥q1 − q2∥Yτ,T

≤ 1

2
∥q1 − q2∥Yτ,T

if so small M ≤ 1
6C .

We take M < 1/(6C), hence the above two estimates hold and the fix point is
in SM . □

Proof of Theorem 4.1. Combining Lemmas 4.2 and 4.3, we obtain the sum u ∈
Yτ,T . □

In the above analysis, we obtain the existence fo small data, i.e., when u(τ) and
∇f are sufficiently small in H1.

Acknowledgments. The authors want to thank Prof. Roberto Triggianni for his
comments during the SIAM Central State Annual Meeting in Iowa State University
in 2019. Taige Wang was supported by Faculty Development Fund granted by
College of Arts and Sciences, University of Cincinnati, and by the Taft Travel
Award by Taft Research Center, University of Cincinnati.

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Zhuoru Chen

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221,
USA

Email address: 1210506001@cnu.edu.cn

Taige Wang

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221,

USA
Email address: taige.wang@uc.edu

Xiangfei Xie
Department of Computer Science, College of Engineering and Applied Science, Uni-

versity of Cincinnati, Cincinnati, OH 45221, USA

Email address: xiexf@mail.uc.edu