Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 129, pp. 1–25.

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

POINTWISE ESTIMATES OF SOLUTIONS TO CONSERVATION

LAWS WITH NONLOCAL DISSIPATION-TYPE TERMS

FENGBAI LI, WEIKE WANG, YUTONG WANG

Abstract. This article concerns the Cauchy problem of conservation laws

with nonlocal dissipation-type terms in R3. By using Green’s function and the
time-frequency decomposition method, we study global classical solutions and

their long time behavior including pointwise estimates for large initial data,

for solutions near the nontrivial equilibrium state.

1. Introduction

In this article, we study the existence of global solutions and their decay estimates
to conservation laws with nonlocal dissipation-type terms in R3. We consider the
solution, near the equilibrium state u∗ 6= 0, of the equation

∂tu− γ14ut − γ24u = div h(u), (1.1)

subject to the initial value

u(x, 0) = u0(x), (t, x) ∈ R+ × R3. (1.2)

Here, γ1, γ2 > 0, and h(u) = (h1(u), h2(u), h3(u)) satisfies the condition

hk(u) = O(u2), k = 1, 2, 3.

This model is motivated by physical considerations from fluid dynamics. It
describes a variety of important physical processes, such as the aggregation of pop-
ulations in which u describes the population density [12]. It can also be used in
the analysis of a seepage of homogeneous fluids through a fissured rock and the
unidirectional propagation of nonlinear dispersive long waves [1, 14]. There are
many equations similar to problem (1.1). For instance, the equation

∂tu− k4ut −4u = div h(u),

which is (1.1) with γ2 = 1, can be used to describe the nonstationary processes
in semiconductors in the presence of sources [9, 10]. Here k∆ut − ut indicates the
rate of change in free charge’s density, ∆u indicates the electric current of linear
dissipative free charge and div h(u) describes a source of free electron current [9].
Ting, Showalter and Golpala Rao studied the initial-boundary value problem and
the Cauchy problem for the linear equation when γ2 = 1 and proved the existence
and uniqueness of the solution [6, 13, 15]. Afterwards, considerable attention has

2010 Mathematics Subject Classification. 35A01, 35A22, 35A25.
Key words and phrases. Conservation law; dissipation; Cauchy problem; equilibrium state;

Green’s function; time-frequency decomposition method; pointwise estimate.
c©2020 Texas State University.

Submitted February 12, 2020. Published December 22, 2020.

1



2 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

been paid to the study of the nonlinear equation, and the large time behavior of
the solutions [7, 8, 21].

Cao, Yin and Wang [2] studied the equation

∂tu− k4ut −4u = up.

They focused on the parameter p and sufficiently small initial data. They obtained
two critical indices about p and then considered the global existence or the blow
up of the solutions.

In [16], the authors considered the following equation under the condition 0 ≤
s ≤ 1/2:

ut −
∆u

(1−∆)s
= div h(u). (1.3)

They got the global classical solution and long-time behavior for arbitrary large
initial data. In [17], Wang proved that in the case s = 1, the solution is global with
sufficiently small initial data, but will blow up with some large initial data.

Wang and Zhang [20] investigated the problem

∂tu− γ14ut − γ24u+ γ3∆2u =

n∑
j=1

hj(u)xj , (1.4)

which is what we call the Benjamin-Bona-Mahony (BBM) equation. They obtained
ta global solution and the optimal decay estimates with large initial data when
γ3 6= 0.

In our case, i.e. γ3 = 0, there are some new difficulties. First of all, the condition
γ3 6= 0 in (1.4) gives a dissipation term ∆2u, which allows using the energy method
to get the L∞ estimate, and it also gives a better decay. Without this term, we need
to use some new methods such as the time-frequency decomposition. Moreover,
there is no maximum principle like for the viscous Burgers equation, which increases
the difficulty.

Another motivation for us to consider the solution near the nontrivial equilibrium
state is that the equation in our paper actually has some hyperbolic properties which
however will vanish if we just consider the trivial equilibrium state. From the main
results, we will see from the solution that the phenomenon of a propagating wave
appears, and the propagation direction and speed of the wave are decided by the
initial data. Obviously, if we only consider the solution near zero, the speed and
direction will both be zero and will not be observed. So here, to get the optimal
decay estimate and the hyperbolic property of the solution, we first do the pointwise
estimate of the Green’s function, then combining it with the variable substitution
and Green’s method to obtain the L1 and L∞ bounded estimate. Based on this,
we obtain the pointwise estimate of the solution.

Since we consider the solution near the equilibrium state u∗, using u−u∗ as the
new u, we can rewrite equation (1.1) as

∂tu− γ14ut − γ24u+

3∑
j=1

bjuxj = div f(u), (1.5)

with initial condition

u(x, 0) = u0(x)− u∗. (1.6)



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 3

Here, γ1, γ2 > 0, b = (C1u
∗, C2u

∗, C3u
∗) and f(u) = (f1(u), f2(u), f3(u)) satisfies

the condition
fk(u) = O(u2), (k = 1, 2, 3.

The main results of this article read as follows.

Theorem 1.1. Suppose that the initial data v0 = u0−u∗ ∈ L1(R3)∩Hs(R3), with
s > 1 + [ 3

2 ]. Then, the Cauchy problem (1.1)-(1.2) has a global solution

(u− u∗) ∈ L∞(0,∞;Hs(R3)).

Theorem 1.2. Assume that the initial data u0 satisfies the conditions in Theorem
1.1, and u is a solution of (1.1)-(1.2) in L∞([1,∞), Hs(R3)) with initial data u0.
Then

‖(u− u∗)‖Ḣs(R3) ≤ C(1 + t)−
3
4−

s
2 ,

for any t ≥ 1, where C only depends on the initial data.

Theorem 1.3. Assume that the initial data u0 satisfies the conditions in Theorem
1.1, and v0 satisfies

|Dβ
xv0| = |Dβ

x(u0 − u∗)| ≤ C(1 + |x|2)−r, r >
3

2
,

with |β| ≤ s− 1. Then the solution satisfies the pointwise estimate

|Dα
x (u− u∗)| ≤ C(1 + t)−

3+|α|
2 Br(|x− bt|, t).

Where Br(|x|, t) is defined by (2.1), t ≥ 1, and |α| ≤ s− 1.

From the above results, we see that the solution mainly follows the decay of heat
equation and, near the equilibrium state u∗ 6= 0, the equation has some hyperbolic
properties with speed b.

This article is organized as follows. In Section 2, we first recall some important
lemmas that will be used in this paper. In Section 3, we introduce the properties of
the Green’s function, mainly focusing on the pointwise estimate and the Lp estimate
of the Green’s function. Then in Section 4, using Green’s function and the variable
substitution, we obtain the Lp boundedness of the solution u to equation (1.5)
with initial value (1.6). In Section 5, we obtain the optimal decay estimates of

the solution in Ḣs by using the time-frequency decomposition method. And, as a
corollary of Theorem 1.2, the optimal L∞ convergence rate can be obtained easily.
Finally, in Section 6, based on the Lp boundedness and the decay of the solution,
the pointwise estimate of the solution is established.

2. Preliminaries

In this section, we first introduce some notation, and then give some lemmas
which will be used later.

For a multi-index α = (α1, · · · , αn), we write Dα
x = ∂αnxn · · · ∂

α2
x2
∂α1
x1

and |α| =∑n
i=1 |αi|. As usual, the Fourier transform of f(t, x) with respect to the variable

x ∈ Rn is

f̂(t, ξ) = (F(f))(t, ξ) =

∫
Rn
f(t, x)e−ix·ξdx,

and the inverse Fourier transform is

f(t, x) = (F−1f̂)(t, x) = (2π)−n
∫
Rn
f̂(t, ξ)eix·ξdξ.



4 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

Here, n denotes the space dimension. The space W s,p(Rn), with s a positive integer
and p ∈ [1,+∞], is the usual Sobolev space with the norm

‖f‖W s,p(Rn) =

s∑
|α|=0

‖Dα
xf‖Lp(Rn).

In particular, when p = 2, W s,2(Rn) = Hs(Rn).

We denote by Ẇ s,p(Rn) the Sobolev space equipped with the norm

‖ · ‖Ẇ s,p(Rn) = ‖Λs · ‖Lp(Rn),

where Λs is defined by

Λ̂sf(ξ) = |ξ|sf̂(ξ).

When p = 2, Ẇ s,2(Rn) = Ḣs(Rn) is the homogeneous Sobolev space. Next, we
recall some useful lemmas. The first lemma gives some estimate of f(u), its proof
can be found in [11].

Lemma 2.1. Assume that f(u) is smooth enough and f(u) = O(|u|1+α0) when
|u| ≤ ν0, where α0 ≥ 1 is an integer. For each integer s ≥ 0, if u satisfies
‖u‖L∞(Rn) ≤ ν0, then

‖f(u)‖Ẇ s,r(Rn) ≤ C‖u‖Ẇ s,q(Rn)‖u‖Lp(Rn)‖u‖α0−1
L∞(Rn),

‖f(u)‖L1(Rn) ≤ C‖u‖2L2(Rn)‖u‖
α0−1
L∞(Rn),

where C is a constant and only depends on s and ν0, and 1
r = 1

p + 1
q , with 1 ≤

p, q, r ≤ ∞.

The following two lemmas are very useful for obtaining the pointwise estimate
of the solutions. Their proofs can be found in [19].

Lemma 2.2. If f̂(ξ, t) has a compact support about ξ, and there exists b > 0 such
that

|Dβ
ξ ξ
αf̂(ξ, t)| ≤ C(|ξ|(|α|+k−|β|)+ + |ξ|(|α|+k)t

|β|
2 )(1 + t|ξ|2)

m
e−b|ξ|

2t,

for all |β| ≤ 2N , with N a positive integer, then

|Dα
xf(x, t)| ≤ CN (1 + t)−(|α|+k+n)/2BN (t, |x|),

where k and m are positive integers, and

BN (t, |x|) =
(

1 +
|x|2

1 + t

)−N
, (a)+ =

{
a, a ≥ 0,

0, a < 0.
(2.1)

Lemma 2.3. If supp f̂(ξ) ⊂ OR =: {ξ, |ξ| > R}, and f̂(ξ) satisfies

|Dβ
ξ f̂(ξ)| ≤ C|ξ|−1−|β|,

then there exist distributions f1(x), f2(x) and a constant C0 such that

f(x) = f1(x) + f2(x) + C0δ(x),

where δ(x) is the Dirac function. Furthermore, for positive integers N with 2N >
3+ |α|, we have |Dα

xf1(x)| ≤ C(1+ |x|2)−N , ‖f2‖L1 ≤ C and supp f2(x) ⊂ {x; |x| <
2ε0}, with ε0 sufficiently small.



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 5

Lemma 2.4 ([18]). For |y| ≤ M , t ≥ 4M2 and N > 0, there exists CN > 0 such
that (

1 +
|y − x|2

1 + t

)−N
≤ CN

(
1 +

|x|2

1 + t

)−N
.

Finally, note that for any g(u) ∈ Lp(Rn) and p ≥ 1, we have

(1− γ1∆)−1g(u) = Kγ1(·) ∗ g(u(·, t)),
where

Kγ1(x) = (4π)−n/2
∫ t

0

e−γ1s−
|x|2
4s s−n/2ds,

‖Kγ1‖L1(Rn) = γ1.

Thus, we have

‖(1− γ1∆)−1g(u)‖Lp(Rn) ≤ C‖Kγ1‖L1(Rn)‖g(u(·))‖Lp(Rn)

≤ C‖g(u(·))‖Lp(Rn).
(2.2)

3. Existence of a local solution

In this section, we construct a convergent Cauchy sequence to obtain the local
solution. Let vm(x, t) be the solution of the Cauchy problem

∂tv
m − γ14vmt − γ24vm = div h(vm−1),

vm(x, 0) = v0(x),
(3.1)

where m ≥ 1, v0(x, t) = 0 and v0(x) ∈ Hs(R3).
For any given integer s ≥ 1 + [3/2], define the function space

X = {v(x, t) : ‖v‖X < E},
where ‖v‖X = sup0≤t≤T0

‖v‖Hs , T0 is an undetermined constant and E = C0v0‖v‖Hs ,
with C0 > 2. One readily checks that X is a complete metric space. Next we prove
that the function sequence vm(x, t) converges in the space X.

Lemma 3.1. For any m ≥ 1, there exists a sufficiently small constant T0, such
that vm(x, t) ∈ X.

Proof. We use induction. For m = 1, we know that

∂tv
1 − γ1∆v1

t − γ2∆v1 = div h(v0).

Taking derivatives with respect to x, α times, on both sides of the above equation,
where |α| ≤ s− 1, we have

Dα
xv

1
t − γ1D

α
x∆v1

t − γ2D
α
x∆v1 = 0.

Multiplying Dα
xv

1 on both sides, and integrating with respect to x and t on R3 ×
[0, t], we have

‖v1‖2Hs−1 + γ1‖∇v1‖2Hs−1 + 2γ2

∫ t

0

‖∇v1‖2Hs−1dτ = ‖v0‖2Hs−1 + γ1‖∇v0‖2Hs−1 .

Since γ1 > 0 is a constant, it implies that

sup
0≤t≤T0

‖v1‖Hs−1 ≤ ‖v0‖Hs .

Thus for T0 > 0, we have v1(x, t) ∈ X.



6 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

Assume that there is a sufficiently small T0 such that for each j ≤ m, vj(x, t) ∈ X.
We shall show that vm+1(x, t) ∈ X. In fact, from (3.1) we have

1

2

d

dt
‖Dα

xv
m+1‖2L2 +

γ1

2

d

dt
‖∇(Dα

xv
m+1)‖2L2 + γ2‖∇(Dα

xv
m+1)‖2L2

=

∫
(Dα

xv
m+1)Ḋα

x div h(vm)dx

≤ C‖∇(Dα
xv

m+1)‖2L2‖vm‖L∞‖Dα
xv

m‖2L2

≤ C‖∇(Dα
xv

m+1)‖2L2‖vm‖Hs‖Dα
xv

m‖2L2

≤ C‖vm|2Hs‖Dα
xv

m‖2L2 + ε‖∇(Dα
xv

m+1)‖2L2 .

(3.2)

Here, we have used the inequalities ‖vm‖L∞ ≤ C‖vm‖Hs and s ≥ 1 + [ 2
3 ]. In-

tegrating with respect to t and taking the summation of |α| from 0 to s − 1, we
have

‖vm+1‖2Hs−1 + γ1‖∇vm+1‖2Hs−1 + γ2

∫ t

0

‖∇vm+1‖2Hs−1dτ

≤
∫ t

0

‖vm|2Hs‖vm‖2Hs−1dτ + ‖v0‖2Hs−1 + γ1‖∇v0‖2Hs−1 .

(3.3)

By (3.3), we have

‖vm+1‖2Hs−1 + γ1‖∇vm+1‖2Hs−1 ≤ CE4T0 + ‖v0‖2Hs−1 + γ1‖∇v0‖2Hs−1 .

Since γ1 > 0 is an arbitrary constant,

‖vm+1‖2Hs ≤ CE4T0 + ‖v0‖2Hs .
Thus for T0 small enough, we obtain vm+1(x, t) ∈ X. �

Lemma 3.2. There is a sufficiently small constant T0, such that vm(x, t) is a
Cauchy sequence in the complete metric space X.

Proof. We are going to show that for some 0 < κ < 1,

‖vm+1 − vm‖X ≤ κ‖vm − vm+1‖X.
Similar to the proof of (3.2), we have

‖vm+1 − vm‖X + γ1‖∇(vm+1 − vm)‖2Hs + γ2

∫ t

0

‖∇(vm+1 − vm)‖2Hsdτ

≤
∫ t

0

‖vm+1 − vm‖2Hs(‖vm‖2Hs−1 + ‖vm‖2Hs)dτ

Since vm(x, t) ∈ X, we have sup0≤t≤T0
‖vm‖Hs ≤ E. Thus,

sup
0≤t≤T0

‖vm+1 − vm‖Hs ≤ (CE2T0)1/2 sup
0≤t≤T0

‖vm − vm−1‖Hs .

Choosing sufficiently small T0 such that CE2T0 ≤ 1
4 , we obtain

sup
0≤t≤T0

‖vm+1 − vm‖Hs ≤
1

2
sup

0≤t≤T0

‖vm − vm−1‖Hs .

Thus vm(x, t) is a Cauchy sequence in X. �

Since X is a complete metric space, combining Lemmas 3.1 and 3.2, the limit
function v(x, t) ∈ X satisfies the Cauchy problem (3.1). Hence we obtain the
existence of a local solution.



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 7

4. Lp decay estimate of Green’s function

In this section, we consider the behavior of the solution to the linear form of
(1.5). Consider the following Cauchy problem

∂tG− γ14Gt − γ24G+ 2

3∑
j=1

bj ·Gxj = 0, x ∈ R3, t > 0,

G
∣∣
t=0

= δ(x).

(4.1)

Here, b = (b1, b2, b3), bi = ciu
∗, δ(x) is the Dirac function. G is called Green’s

function.
Let G(t, x) be the inverse Fourier transform corresponding to Ĝ(t, ξ). Taking the

Fourier transform with respect to x and solving the ordinary differential equation
directly, we obtain

Ĝ(t, ξ) = eµ(ξ)t+η(ξ)t,

with

µ(ξ) = − γ2|ξ|2

1 + γ1|ξ|2
, η(ξ) = −

√
−1b · ξ

1 + γ1|ξ|2
.

Note that the solution of the Cauchy problem (1.5) with initial data (1.6) has the
integral representation

u = G ∗ u0 +

∫ t

0

G(t− τ, ·) ∗ div

1− γ1∆
f(u)(τ, ·)dτ.

Next, we use the frequency decomposition to get an estimate for the Green’s func-
tion G. We divide the solution into the low frequency part and the high frequency
part and discuss them separately to obtain the decay property of the solution. Let

χ1(ξ) =

{
1, |ξ| ≤ ε,
0, |ξ| > 2ε,

χ3(ξ) =

{
1, |ξ| ≥ R,
0, |ξ| < R− 1,

χ2(ξ) = 1− χ1(ξ)− χ3(ξ)

be smooth cut-off functions in C∞(R3), where ε and R are positive constants satis-
fying 2ε < R−1. We define three pseudo-differential operators χ1(D), χ2(D), χ3(D)
with the symbols χ1(ξ), χ2(ξ), χ3(ξ) respectively and set

Ĝi(t, ξ) = χi(ξ)Ĝ(t, ξ), i = 1, 2, 3.

Now we estimate the three parts separately. Using Lemma 2.2, we have the following
estimate on G1(t, x).

Proposition 4.1. For sufficiently small ε, there exists a constant CN > 0 such
that

|Dα
xG1(t, x)| ≤ CN t−

3+|α|
2 BN (t, |x− bt|).

Proof. For ξ sufficiently small, the Taylor expansion gives

µ(ξ) + η(ξ) = −(γ2|ξ|2 +
√
−1b · ξ) 1

1 + γ1|ξ|2
= −(γ2|ξ|2 +

√
−1b · ξ) +O(|ξ|3).

Therefore,

Ĝ(t, ξ) = e−γ2|ξ|
2t · e−

√
−1b·ξt(1 +O(|ξ|3)t).



8 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

Now we take the Fourier transform and set

Ĝ1(t, ξ) = χ1(ξ)e−γ2|ξ|
2t(1 +O(|ξ|3)t)e−

√
−1b·ξ =: Ĥ(t, ξ)e−

√
−1b·ξ.

By properties of the Fourier transform, we have

G1(t, ξ) = H(t, x) ∗ F−1(e−
√
−1b·ξ) = H(t, x− bt),

Dα
xG1(t, ξ) = H(t, x) ∗Dα

xF−1(e−
√
−1b·ξ) = Dα

xH(t, x− bt).

Thus it suffices to show that

|Dα
xH(t, x)| ≤ CN t−

3+|α|
2 BN (t, |x|).

Since Ĥ(t, ξ) is smooth in the variable ξ near |ξ| = 0, for |β| ≤ 2N , we obtain

|Dβ
ξ (ξαĤ(t, ξ))| ≤ C(|ξ|(|α|−|β|)+ + |ξ||α|t

|β|
2 )(1 + t|ξ|2)

|β|
2 +1

e−γ2|ξ|
2t.

Thus, by Lemma 2.2, we have

|Dα
xH(t, x)| ≤ CN (1 + t)−

3+|α|
2 BN (t, |x|).

This completes the proof. �

Next, we consider G2(t, x).

Proposition 4.2. For fixed ε and R, there exist positive numbers m0 and C such
that

|Dα
xG2(t, x)| ≤ Ce−

t
2m0BN (t, |x|).

Proof. Choose m > 1/(2ε). When ε ≤ |ξ| ≤ R, we have

Re(µ(ξ) + η(ξ)) ≤ − 1

2m
.

This implies

|Ĝ2| = |χ2(ξ)eµ(ξ)t| ≤ Ce− t
2m . (4.2)

Thus,

|G2(t, x)| ≤ C
∣∣ ∫

R3

eix·ξĜ2(t, ξ)dξ
∣∣ ≤ Ce− t

2m . (4.3)

Next, we give an estimate for xβG2(t, x) using induction on β. Assume that for
any |β| ≤ l − 1, it holds

|Dβ
ξ Ĝ2(t, ξ)| ≤ Ct|β|e− t

2m , (4.4)

for |β| = 0 by (4.2).
Taking the Fourier transform of (4.1) with respect to x and then multiplying by

χ2(ξ) we have

∂tĜ2(t, ξ)− (µ(ξ) + η(ξ))Ĝ2(t, ξ) = 0,

Ĝ2(0, ξ) = χ2(ξ).
(4.5)

Applying the operator Dβ
ξ on (4.5), we obtain

∂tD
β
ξ Ĝ2(t, ξ)− (µ(ξ) + η(ξ))Dβ

ξ Ĝ2(t, ξ) = F (ξ),

Ĝ2(0, ξ) = a0,



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 9

where a0 is a polynomial of |ξ|, and

F (ξ) =
∑

β1+β2=β,|β1|6=0

β!

β1!β2!
(Dβ1

ξ (µ(ξ) + η(ξ)))Dβ2

ξ Ĝ2(t, ξ).

Then for |β| = l, by the ODE theory, we have

Dβ
ξ Ĝ2(t, ξ) = a0Ĝ(t, ξ) +

∫ t

0

Ĝ(t− s, ξ)F (ξ)ds.

Using the induction hypothesis, we obtain

|Dβ
ξ Ĝ2(t, ξ)|ε≤|ξ|≤R ≤ Ce−

t
2m + C

∫ t

0

e−
t−s
2m t|β|−1e−

s
2m ds ≤ Ct|β|e− t

2m ,

which implies that (4.4) is also valid for |β| = l. Then for 1 ≤ |β| ≤ l, we have

|xβDα
xG2(t, x)| ≤ C|

∫
R3

e
√
−1xξDβ

ξ (ξαĜ2(t, ξ))dξ|

≤ Ce− t
2m

∫
ε≤|ξ|≤R

(|ξ||α| + |ξ|(|α|−|β|)+)t|β|dξ

≤ Ct|β|e− t
2m

≤ Ct
|β|
2 e−

t
2m0 , m0 < m.

(4.6)

Using (4.3) when |x|2 ≤ t and using (4.6) when |x|2 > t, we obtain (|β| = 2N)

|Dα
xG2(t, x)| ≤ Ce−

t
2m0 min{1, tN

|x|2N
}.

Since

1 +
|x|2

t
≤

{
2, |x|2 ≤ t,
2 |x|

2

t , |x|
2 ≥ t,

we have

|Dα
xG2(t, x)| ≤ Ce−

t
2m0BN (t, |x|),

which completes the proof. �

Finally, we consider G3(t, x) by using Lemma 2.3.

Proposition 4.3. For sufficiently large R, there exist positive numbers C0 and C
such that

|Dα
x (G3(t, x)−Gfα)| ≤ Ce−C0tBN (t, |x|).

Here Gfα is the distribution of the form

Gfα = χ3(D)
[
e−γ2t/γ1

(
δ(x) +

[
|α|+3

2 ]∑
j=1

pj(t)∆
−j+ |α|2

)]
, (4.7)

where pj(t)s are polynomials of degree j.

Proof. Since R is large enough,

|Dα
xG3| ≤ Ce−C1teµ(ξ)t.

We set

ρ =
1

|ξ|2
, h(ρ) = e

− γ2t

|ρ|+γ1 .



10 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

Then, it is easy to see that

h(ρ) = e−γ2t/γ1(1 +O(|ρ|)), |ρ| → 0.

Thus,

eµ(ξ)t = χ3(ξ)e−γ2t/γ1
(

1 +

[
|α|+3

2 ]∑
j=1

pj(t)(|ξ|−2)j
)

+ R̂(t, ξ).

Using Lemma 2.3, we obtain

| |ξ|αR̂(t, ξ)| ≤ Ce−γ2T/γ1
+∞∑

j=[
|α|+3

2 ]+1

pj(t)(|ξ|−2j),

|Dα
xR(t, x)| ≤ Ce−γ2t/γ1(1 + |x|2)−N ,

which implies

|xβDα
xR(t, x)| ≤ Ce−γ2t/γ1

∫
|ξ|>R

|ξ|−2(
|α|+3

2 )−|β|dξ ≤ Ce−γ2t/γ1 .

Hence,

|Dα
x (G3 −Gfα)| ≤ Ce−γ2t/γ1BN (t, |x|).

�

In conclusion, we obtain the following estimate for the regular part of G.

Theorem 4.4. Assume that G is the solution of the linear form of equation (1.5).
Then

|Dα
x (G−Gfα)| ≤ Ct−

3+|α|
2 BN (t, |x− bt|),

with Gfα defined by (4.7).

Proof. Since

|Dα
x (G−Gfα)(t, x)| ≤ |Dα

xG1(t, x)|+ |Dα
xG2(t, x)|+ |Dα

x (G3 −Gfα)(t, x)|,
the desired estimate follows from Propositions 4.1, 4.2 and 4.3. �

Noting that ∫
Rn
BN (t, |x|)dx ≤ C(1 + t)3/2,

we can use the Hausdorff-Young’s inequality to obtain the following theorems about
the Green’s function, which are essential for the proof of the existence and the decay
estimate of the Cauchy problem (1.5)-(1.6).

Theorem 4.5. For any multi-index α and p ∈ [1,∞], we have

‖Dα
x (G−Gfα)‖Lp(R3) ≤ Ct−

3
2 (1− 1

p )− |α|2 .

Theorem 4.6. For all ϕ and N ≥ 0 integer, we have

‖(G−Gfα) ∗ ϕ‖WN,∞(R3) ≤ C0(1 + t)−3/2‖ϕ‖WN+4,1(R3),

‖(G−Gfα) ∗ ϕ‖WN,1(R3) ≤ ‖ϕ‖WN,1(R3).

Theorem 4.7. For f ∈ Lp(R3) with p ∈ [1,∞], we have

‖G ∗ f‖Lp(R3) ≤ Cp‖f‖Lp(R3).

Here Cp depends only on p.



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 11

5. Lp bound estimate of solutions

Now we prove an Lp (p ∈ [0,∞]) estimate of the solution u of (1.5)-(1.6). First,
by the fundamental energy estimate, we have

d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) + 2γ2‖∇u‖2L2 = 0. (5.1)

Integrating with respect to t, we obtain

(‖u‖2L2 + γ1‖∇u‖2L2) + 2γ2

∫ t

0

‖∇u‖2L2dτ = ‖u0‖2L2 + γ1‖∇u0‖2L2 .

Therefore, we have the following basic bounds for u:

‖u‖L2 ≤ C, ‖∇u‖L2 ≤ C,
∫ t

0

‖∇u(τ, ·)‖2L2dτ ≤ C.

5.1. L1 bounds. Let

χ0(η) =

{
1, |η| ≤ 1,

0, |η| > 2,

be a smooth cut-off function in C∞(R3). Define the time-frequency cut-off oper-
ator χ(t,D) with symbol χ(t, ξ) = χ0

(
1+t
µ |ξ|

2
)
, where µ is a constant and µ >

max{ 3
2γ1

, 3
2γ2
}. A solution u of (1.5) can be decomposed it into two parts: the low

frequency part uL and the high frequency part uH , where uL(t, x) = χ(t,D)u(t, x)
and uH(t, x) = (1− χ(t,D))u(t, x).

Lemma 5.1. Assume that u0 ∈ L1(R3). Then

‖uL‖L2 ≤ C(1 + t)−3/4,

where C is a positive constant depending only on u0.

Proof. Using Green’s function, we have

uL = χ(t,D)(G ∗ u0) +

∫ t

0

χ(t,D)
(G(t− τ, ·)

1− γ1∆
∗ div f(u)

)
dτ.

Using Minkowski’s inequality, we obtain

‖uL‖L2

= ‖χ(t,D)(G ∗ u0)‖L2 +
(∫ t

0

∥∥χ(t,D)
(G(t− τ, ·)

1− γ1∆
∗ div f(u)

)
)
∥∥2

L2dτ
)1/2

= S1 + S2.

(5.2)

For S1, by Hausdorff-Young’s inequality and Theorem 4.4, we have

S1 ≤ ‖χ(t,D)G‖L2 · ‖u0‖L1 ≤ C(1 + t)−3/4. (5.3)



12 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

For S2, by Plancherel’s theorem, we obtain

|S2|2 =

∫ t

0

∥∥χ(t,D)
(G(t− τ, ·)

1− γ1∆
∗ div f(u)

)∥∥2

L2dτ

=

∫ t

0

∥∥χ(t, ξ)
Ĝ

1 + γ1|ξ|2
|ξ|f̂(u)

∥∥2

L2dτ

≤ C
∫ t

0

∫
R3
ξ

χ2(t, ξ)|ξ|2 |f̂(u)|2

(1 + γ1|ξ|2)
dξdτ

≤ C
∫ t

0

∥∥ f̂(u)

1 + γ1|ξ|2
∥∥2

L∞

∫
R3
ξ

χ2(t, ξ)|ξ|2dξdτ

≤ C
∫ t

0

∥∥ f(u)

1− γ1∆

∥∥2

L1(1 + t)−
3
2−1dτ

≤ C
∫ t

0

‖f(u)‖2L1(1 + t)−
3
2−1dτ

≤ C(1 + t)−3/2.

(5.4)

Hence, by (5.2), (5.3) and (5.4), we obtain the desired estimate. �

From the decay estimate of ‖uL‖L2 , we have the following interesting relation
between the decay rates of uL and u.

Lemma 5.2. If u0 ∈ H1(R3) and ‖uL‖L2 ≤ C(1 + t)−σ, then

‖u‖2L2 + γ1‖∇u‖2L2 ≤ C(1 + t)−2σ,

where C is a positive constant depending only on u0.

Proof. We already have the basic energy estimate (5.1), that is

d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) + 2γ2‖∇u‖2L2 = 0.

Set ε2(t) = µ/(1 + t). Then

‖∇u‖L2 ≥
∫
|ξ|>ε(t)

|ξ|2|û|2dξ

≥ ε2(t)

∫
|ξ|>ε(t)

|û|2dξ

≥ ε2(t)
(
‖u‖2L2 −

∫
|ξ|≤ε(t)

|û|2dξ
)
.

From the above inequality, we deduce that

d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) +

γ2µ

1 + t
‖u‖2L2 + γ2‖∇u‖2L2 ≤ ε2(t)

∫
|ξ|≤ε(t)

γ2|û|2dξ

≤ Cε2(t)‖uL‖2L2 .

(5.5)

By the assumption, (5.5) becomes

d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) + C0

µ

1 + t
(‖u‖2L2 + γ1‖∇u‖2L2)

≤ d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) +

γ2µ

1 + t
‖u‖2L2 + γ2‖∇u‖2L2



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 13

≤ C(1 + t)−2σ−1.

Multiplying the above inequality by e
∫ t
0

C0µ
1+τ dτ = (1 + t)C0µ and integrating from 0

to t, we obtain

(1 + t)C0µ(‖u‖2L2 + γ1‖∇u‖2L2) ≤ ‖u0‖2L2 + γ1‖∇u0‖2L2 + C(1 + t)C0µ−2σ.

Thus

‖u‖2L2 + γ1‖∇u‖2L2 ≤ (‖u0‖L2 + γ1‖∇u0‖2L2)(1 + t)−C0µ + C(1 + t)−2σ.

Since C0µ > 2σ, we have

‖u‖2L2 + γ1‖∇u‖2L2 ≤ C(1 + t)−2σ.

�

Remark 5.3. Combining Lemmas 5.2 with 5.1, we obtain

‖u‖2L2 + γ1‖∇u‖2L2 ≤ C(1 + t)−3/2.

Proposition 5.4. If u0 ∈ L1(R3) ∩H1(R3), then we have ‖u‖L1 ≤ C, where C is
a positive constant depending only on u0.

Proof. Using Green’s function, we have

‖u‖L1 ≤ ‖G ∗ u0‖L1 +
∥∥∫ t

0

G(t− τ, ·)
1− γ1∆

∗ div f(u)dτ
∥∥
L1

≤ ‖G ∗ u0‖L1 +

∫ t

0

‖G(t− τ, ·) ∗ div f(u)‖L1dτ

= Q1 +Q2.

(5.6)

For Q1, by Theorem 4.7, we have

Q1 ≤ C‖u0‖L1 ≤ C. (5.7)

For Q2, notice that

‖f(u)‖L1 ≤ C‖u‖2L2 ≤ C(1 + t)−3/2.

Again, by Theorem 4.7, we have

Q2 =

∫ t

0

‖G(t− τ, ·) ∗ div f(u)‖L1dτ

=

∫ t

0

‖(G−Gf1 +Gf1) ∗ div f(u)‖L1dτ

≤
∫ t

0

‖(G−Gf1) ∗ div f(u)‖L1dτ +

∫ t

0

‖Gf1 ∗ div f(u)‖L1dτ

= Q2,1 +Q2,2.

(5.8)



14 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

For Q2,1, by Young’s inequality, we have

Q2,1 =

∫ t

0

‖(G−Gf1) ∗ div f(u)‖L1dτ

=

∫ t

0

‖∇(G−Gf1) ∗ f(u)‖L1dτ

≤ C
∫ t

0

‖∇(G−Gf1)‖L1‖f(u)‖L1dτ

≤ C
∫ t

0

(1 + t− τ)−
1
2 (1 + τ)−3/2dτ ≤ C.

(5.9)

For Q2,2, we have

Q2,2 =

∫ t

0

‖Gf1 ∗ div f(u)‖L1dτ

≤
∫ t

0

e−
γ2
γ1

(t−τ)‖∇u‖L2‖∇u‖L2dτ

≤
∫ t

0

e−
γ2
γ1

(t−τ)(1 + τ)−3/2dτ ≤ C.

(5.10)

Now (5.6), (5.7), (5.8), (5.9) and (5.10) imply ‖u‖L1 ≤ C, as desired. �

5.2. L∞ bounds. We now estimate ‖u‖L∞ . Unlike the method in [20], the energy
method is not enough. However, using the variable substitution and Green’s func-
tion method, we can first obtain the bounded estimate of ‖u‖H2 and then apply
the embedding theorem to get the L∞ bounded estimate. Set

w = (1− γ1∆)u, w0 = (1− γ1∆)u0.

Then (1.5) can be rewritten as

∂tw −
γ2∆

1− γ1∆
w + b

γ2∇
1− γ1∆

w = div f(u).

By Duhamel Principle, we know that the solution will be of the form

w = G ∗ w0 +

∫ t

0

G(t− τ, ·) ∗ div f(u)(τ, ·)dτ. (5.11)

Now we use Green’s function to estimate w.

Proposition 5.5. Suppose that u0 ∈ L1(R3) ∩Hs(R3), s ≥ 2. Then we have

‖w‖L2 ≤ C,

where C is a positive constant depending only on u0.



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 15

Proof. Applying Theorem 4.7 to (5.11) and using the Hölder inequality, we obtain

‖w‖L3/2 = ‖G ∗ w0‖L3/2 + ‖
∫ t

0

G(t− τ, ·) ∗ div f(u)(τ, ·)dτ‖L3/2

≤ ‖G ∗ w0‖L3/2 +

∫ t

0

‖G(t− τ, ·) ∗ div f(u)‖L3/2dτ

≤ C‖w0‖L3/2 + C

∫ t

0

‖u · ∇u‖L3/2dτ

≤ C + C

∫ t

0

‖u‖L6‖∇u‖L2dτ

≤ C + C

∫ t

0

‖∇u‖2L2dτ ≤ C.

(5.12)

Note that we have used the Sobolev inequality ‖u‖L6(R3) ≤ C‖∇u‖L2(R3) above.
Combining (5.12) with (2.2), we obtain

‖u‖L3/2 = ‖(1− γ1∆)−1w‖L3/2 ≤ C.

Thus

‖∆u‖L3/2 = ‖u+ w‖L3/2 ≤ ‖u‖L3/2 + ‖w‖L3/2 ≤ C.
Combining these with the Green’s function again, we obtain

‖w‖L2 = ‖G ∗ w0‖L2 + ‖
∫ t

0

G(t− τ, ·) ∗ div f(u)(τ, ·)dτ‖L2

≤ C‖w0‖L2 +

∫ t

0

‖(G−Gf1 +Gf1) ∗ div f(u)(τ, ·)‖L2dτ

≤ C +

∫ t

0

(‖(G−Gf1) ∗ div f(u)‖L2 + ‖Gf1 ∗ div f(u)‖L2)dτ

≤ C + I1 + I2.

(5.13)

For I1, we have

I1 =

∫ t

0

‖(G−Gf1) ∗ div f(u)‖L2dτ

≤
∫ t

0

‖div(G−Gf1)‖L2‖f(u)‖L1dτ

≤ C
∫ t

0

(1 + t− τ)−
3
2 (1− 1

2 )− 1
2 ‖u‖2L2dτ ≤ C.

(5.14)

For I2, we have

I2 =

∫ t

0

‖Gf1 ∗ div f(u)‖L2dτ

≤
∫ t

0

e−
γ2
γ1

(t−τ)‖u‖L6‖∇u‖L3dτ

≤
∫ t

0

e−
γ2
γ1

(t−τ)‖∇u‖L2‖∆u‖
L

3
2

dτ ≤ C.

(5.15)

In the last inequality we used that ‖∇u‖L3(R3) ≤ C‖∆u‖L3/2(R3).

From (5.13), (5.14) and (5.15), we obtain ‖w‖L2 ≤ C + I1 + I2 ≤ C. �



16 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

Remark 5.6. By Proposition 5.5, we can easily obtain

‖u‖H2(R)3 ≤ C.

Then by the Sobolev embedding theorem H2(R3) ↪→ L∞(R3), we can get the L∞

boundedness of u,

‖u‖L∞(R3) ≤ C.

5.3. Lp bounds. Since we already have L1 and L∞ bounds, by the interpolation
inequality we easily get

‖u‖Lp(R3) ≤ C, p ∈ [1,∞].

Here C is a positive constant depending only on the initial data u0.

5.4. Hs bounds. With the L∞ boundedness, we can improve the regularity of u
as follows.

Lemma 5.7. If u0 ∈ Hs(R3), 3 ≤ s ∈ Z, then ‖u‖Hs ≤ C, where C is a positive
constant depending only on u0.

Proof. Multiplying Λ2ju to both sides of (1.5) and integrating with respect to x,
we obtain

d

dt
(‖u‖2

Ḣj
+ γ1‖u‖2Ḣj+1) + 2γ2‖u‖2Ḣj = −2

∫
R3

Λ2judiv f(u)dx.

By Hölder’s inequality, we have∣∣2 ∫
R3

Λ2judiv f(u)dx
∣∣ ≤ γ2‖u‖2Ḣj + C‖f(u)‖2

Ḣj
≤ γ2‖u‖2Ḣj + Cjγ2‖u‖2Ḣj .

Thus, for any j ≥ 1, we obtain

d

dt
(‖u‖2

Ḣj
+ γ1‖u‖2Ḣj+1) + γ2‖u‖2Ḣj ≤ Cjγ2‖u‖2Ḣj .

By summation, we have

d

dt

s∑
j=0

Cs−j(‖u‖2
Ḣj

+γ1‖u‖2Ḣj+1)+

s∑
j=0

Cs−jγ2‖u‖2Ḣj ≤
s∑
j=0

(j+1)Cs−j+1Cjγ2‖u‖2Ḣj .

One readily checks that this implies

d

dt

s∑
j=0

Cs−j(‖u‖2
Ḣj

+ γ1‖u‖2Ḣj+1) ≤ 0.

Integrating the above inequality with respect to t, we obtain

s∑
j=0

Cs−j(‖u‖2
Ḣj

+ γ1‖u‖2Ḣj+1) ≤
s∑
j=0

Cs−j(‖u0‖2Ḣj + γ1‖u0‖2Ḣj+1).

Then, it follows that ‖u‖Hs ≤ C. �

From the Hs boundedness and the existence of local solutions, we obtain the
existence of global solutions to (1.5), by the continuity method. Moreover,

(u− u∗) ∈ L∞(0,∞;Hs(R3)), s ≥ 1 +
[3
2

]
.



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 17

6. Decay estimate in Ḣs and L∞

In this section, we establish the optimal decay rates of the solution u in the
homogeneous Sobolev spaces.

6.1. Decay rate of the low frequency part in Ḣs. Based on the boundedness
of ‖u‖L1 , Plancherel’s theorem, and Hausdorff-Young’s inequality, we can directly
obtain the following estimate for the low frequency part,

‖ΛsuL‖L2 = ‖χ(t,D)Λsu‖L2

= ‖χ(t, ξ)|ξ|sû‖L2

≤ C
∥∥χ0

(1 + t

µ
|ξ|2
)
|ξ|s
∥∥
L2‖û‖L∞

≤ C‖u‖L1

∥∥χ0

(1 + t

µ
|ξ|2
)
|ξ|s
∥∥
L2

≤ C(1 + t)−
3
4−

s
2 .

(6.1)

Here, the constant C depends on the choice of µ, which is the same as in the last
section.

6.2. Decay rate of the high frequency part in Ḣs. According to the definition
of uH , uH satisfies the equation

(uH)t − γ1∆(uH)t − γ2∆uH + 2b · ∇uH
= div(f(u))H − χt(t,D)u+ γ1χt(t,D)∆u.

(6.2)

Lemma 6.1. If u0 ∈ H1(R3), then we have the estimate∫ t

0

(1 + t)C0µ‖u‖Ḣ1dτ ≤ C(1 + t)C0µ− 3
2 .

Proof. As in Lemma 5.2, using (5.1), we obtain

d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) + γ2‖∇u‖2L2 = −γ2‖∇u‖2L2

≤ −ε2(t)γ2

(
‖u‖2L2 −

∫
|ξ|≤ε(t)

|û|2dξ
)
.

This leads to

d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) +

γ2µ

1 + t
‖u‖2L2 + γ2‖∇u‖2L2 ≤ ε2(t)

∫
|ξ|≤ε(t)

γ1û
2dξ

≤ Cε2(t)‖uL‖2L2

≤ C(1 + t)−
3
2−1.

The last inequality uses (6.1). Therefore,

d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) +

C0µ

1 + t
(‖u‖2L2 + γ1‖∇u‖2L2) + C1‖∇u‖2L2

≤ d

dt
(‖u‖2L2 + γ1‖∇u‖2L2) +

γ2µ

1 + t
‖u‖2L2 + γ2‖∇u‖2L2

≤ C(1 + t)−
3
2−1.



18 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

Multiplying the inequality above by e
∫ t
0

C0µ
1+τ dτ = (1 + t)C0µ and integrating from 0

to t, we obtain

(1 + t)C0µ(‖u‖2L2 + γ1‖∇u‖2L2) + C1

∫ t

0

(1 + t)C0µ‖∇u‖2L2dτ

≤ ‖u0‖2L2 + γ1‖∇u0‖2L2 + C

∫ t

0

(1 + τ)C0µ− 3
2−1dτ

≤ C(1 + t)C0µ− 3
2 .

This completes the proof. �

The next result gives the optimal decay estimate for ‖uH‖Ḣs .

Theorem 6.2. If u0 ∈ L1(R3)∩Hs(R3) with s ≥ 1 + [ 3
2 ] a given integer, and u is

a solution of (1.5) in L∞(0,∞;Hs(R3)) with initial data u0, then

‖uH‖Ḣs ≤ C(1 + t)−
3
4−

s
2 ,

for any t ≥ 1, where C only depends on the initial data u0.

Proof. Multiplying (6.2) by Λ2suH and integrating it in x, we obtain

1

2

d

dt
(‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1) + γ2‖uH‖2Ḣs+1 − 2

∫
R3

Λ2sb · uH∇uHdx

=

∫
R3

Λ2suH div(f(u))Hdx−
∫
R3

ûûH |ξ|2s∂tχ(t, ξ)dξ

+ γ1

∫
R3

ûûH |ξ|2s+2∂tχ(t, ξ)dξ

= R1 +R2 +R3.

(6.3)

For R1, by Hölder’s inequality and Lemma 2.1, we have

|R1| =
∣∣ ∫

R3

D(u2)HD
2suHdx

∣∣
=
∣∣ ∫

R3

(iξ)(û2)H |ξ|2sûHdξ
∣∣

≤
∥∥|ξ|s∣∣f̂(u)H

∣∣ · |ξ|s+1û
∥∥
L1

≤ C‖f(u)H‖Ḣs‖u‖Ḣs+1

≤ C(‖u‖L∞‖u‖Ḣs)‖u‖Ḣs+1

≤ C‖u‖Ḣs‖u‖Ḣs+1 .

By the Galgliardo-Nirenberg-Sobolev inequality, we have

‖u‖Ḣs ≤ C‖u‖
s−1
s

Ḣs+1
‖u‖1/s

Ḣ1
.

Then, using the Young’s inequality, we obtain

|R1| ≤ C‖u‖
s−1
s +1

Ḣs+1
‖u‖1/s

Ḣ1

≤ γ2

4
‖u‖2

Ḣs+1 +
C

2
‖u‖2

Ḣ1

≤ γ2

4
‖uL‖2Ḣs+1 +

γ2

4
‖uH‖2Ḣs+1 +

C

2
‖u‖2

Ḣ1 .

(6.4)



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 19

The same calculation can be done to
∫
R3 Λ2suH∇uHdx.∣∣2 ∫

R3

Λ2sb · uH∇uHdx
∣∣ = 2b

∣∣ ∫
R3

(iξ)(û)H |ξ|2sûHdξ
∣∣

≤ C‖|ξ|s|ûH | · |ξ|s+1û‖L1

≤ C‖uH‖Ḣs‖u‖Ḣs+1

≤ γ2

4
‖uL‖2Ḣs+1 +

γ2

4
‖uH‖2Ḣs+1 +

C

2
‖u‖2

Ḣ1 .

(6.5)

For R2, by Proposition 5.4, we have

|R2| =
∫
R3

ûûH |ξ|2s∂tχ(t, ξ)dξ

≤
∫
R3

|ûû||ξ|2s+2 1

µ
χ′0

(1 + t

µ
|ξ|2
)∣∣1− χ0

(1 + t

µ
|ξ|2
)∣∣dξ

≤ C‖û2‖L∞
∫
R3

(1 + t)−(s+1)|η|2s+2χ′0(|η|2)|1− χ0(|η|2)|(1 + t)−3/2dη

≤ C‖u2‖L1(1 + t)−
3
2−s−1

≤ C(1 + t)−
3
2−s−1.

(6.6)

For R3, similar to R2, we obtain

|R3| ≤ C(1 + t)−
3
2−s−2. (6.7)

Combining (6.3)-(6.7), we obtain

d

dt
(‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1) + γ2‖uH‖2Ḣs+1

≤ γ2

2
‖uL‖2Ḣs+1 + C‖u‖2

Ḣ1 + C(1 + t)−
3
2−s−1 + C(1 + t)−

3
2−s−2

≤ C‖u‖2
Ḣ1 + C(1 + t)−

3
2−s−1.

We used (6.1) in the last inequality above. Also, according to the definition of the
cut-off operator, we find that

‖uH‖2Ḣs+1 ≥
µ

1 + t
‖uH‖2Ḣs .

Therefore,

d

dt
(‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1) + C0

µ

1 + t
(‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1)

≤ d

dt
(‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1) +

γ2

2
‖uH‖2Ḣs+1

≤ C‖u‖2
Ḣ1 + C(1 + t)−

3
2−s−1.

Using Lemma 6.1 and multiplying the inequality above by e
∫ t
0

C0µ
1+τ dτ = (1 + t)C0µ

and integrating from 0 to t, we have

(1 + t)C0µ(‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1)

≤ ‖u0H‖2Ḣs + γ1‖u0H‖2Ḣs+1 + C

∫ t

0

(1 + τ)C0µ‖u‖2
Ḣ1dτ

+ C

∫ t

0

(1 + τ)C0µ− 3
2−s−1dτ



20 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

≤ ‖u0H‖2Ḣs + γ1‖u0H‖2Ḣs+1 + C(1 + t)C0µ− 3
2 + C(1 + t)C0µ− 3

2−s.

Since C0µ > 3/2, we have

‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1 ≤ C(1 + t)−3/2.

Combining the above with (6.1) and setting s = 3, we obtain

‖u‖2
Ḣ3 ≤ C(1 + t)−3/4.

Therefore, by the Gagliardo-Nirenberg-Sobolev inequality, we have

‖u‖L∞ ≤ C‖u‖1/2Ḣ3
‖u‖1/2L2 ≤ C(1 + t)−3/4.

Hence, we can use the estimate of ‖u‖L∞ to get a better decay as follows.

|R1| ≤ C(‖u‖L∞‖u‖Ḣs)‖u‖Ḣs+1

≤ γ2

2
‖u‖2

Ḣs+1 + C‖u‖2
Ḣs
‖u‖2L∞

≤ γ2

2
‖u‖2

Ḣs+1 + C(1 + t)−3/2(‖uL‖2Ḣs + ‖uH‖2Ḣs).

(6.8)

By (6.3) and (6.6)-(6.8), we have

d

dt
(‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1) + γ2‖uH‖2Ḣs+1

≤ C(1 + t)−3/2(‖uH‖2Ḣs) + C(1 + t)−
3
2−s−1.

Similar to the proof above and note that C0µ >
3
2 , we have

(1 + t)C0µ(‖uH‖2Ḣs + γ1‖uH‖2Ḣs+1)

≤ ‖u0H‖2Ḣs + γ1‖u0H‖2Ḣs+1 + C

∫ t

0

(1 + τ)C0µ− 3
2 (‖uH‖2Ḣs)dτ

+ C

∫ t

0

(1 + τ)C0µ− 3
2−s−1dτ

≤ ‖u0H‖2Ḣs + γ1‖u0H‖2Ḣs+1 + C(1 + t)C0µ− 3
2−s + C

∫ t

0

(1 + τ)C0µ− 3
2 (‖uH‖2Ḣs)dτ.

By Gronwall’s inequality [3], we have

(1 + t)C0µ‖uH‖2Ḣs ≤ C(1 + (1 + t)C0µ− 3
2−s)e

∫ t
0

(1+τ)−3/2dτ

≤ C + C(1 + t)C0µ− 3
2−s.

Since C0µ > 3/2, we obtain

‖uH‖Ḣs ≤ C(1 + t)−
3
4−

s
2 ,

which is the optimal decay estimates of uH . This completes the proof. �

Corollary 6.3. Suppose that u0 satisfies the same assumptions as in Theorem 6.1
and u is a solution of (1.5) in L∞([1,∞), Hs(R3)). Then

‖u‖L∞ ≤ C(1 + t)−3/2,

for t ≥ 1, where C only depends on ‖u0‖L1 .



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 21

Proof. By Theorem 1.2, we have

‖u‖Ḣs ≤ C(1 + t)−
3
4−

s
2 .

In particular, for s = 3, we have ‖u‖Ḣ3 ≤ C(1+ t)−
9
4 . Therefore, by the Gagliardo-

Nirenberg-Sobolev inequality, we immediately obtain

‖u‖L∞ ≤ C‖u‖1/2Ḣ3
‖u‖1/2L2 ≤ C(1 + t)−3/2,

which completes the proof. �

7. Pointwise estimate of the solution

Finally, we come to the pointwise estimate of the solution u. Our goal is to prove
the following theorem.

Theorem 7.1. For Cauchy problem (1.1), if the initial data v0 = u0 − u∗ ∈
L1(R3) ∩Hs(R3), and satisfies

|Dβ
xv0| = |Dβ

x(u0 − u∗)| ≤ C(1 + |x|2)−r, r >
3

2
,

then the solution has the pointwise estimate

|Dα̃
x (u− u∗)| ≤ C(1 + t)−

3+|α̃|
2 Br(t, |x− bt|),

where |β| ≤ s− 1, |α̃| ≤ s− 1, t ≥ 1 and Br(t, |x|) is given in (2.1).

Proof. From (5.11), we have

Dα̃
xu = Dα̃

xG ∗ u0 +

∫ t

0

Dα̃
xG(t− τ, ·) ∗ div

1− γ1∆
f(u)dτ = Π1 + Π2. (7.1)

Obviously Π1 means the initial term while Π2 represents the nonlinear term.

7.1. Estimate of the initial term. For Π1 = Dα̃
xG∗u0, since G has a distribution

Gfα̃, we need to separate it into two parts:

|Π1| = |Dα̃
xG ∗ u0|

≤ |Dα̃
x (G−Gfα̃) ∗ u0|+ |Gfα̃ ∗Dα

xu0|
= Π1,1 + Π1,2.

For Π1,1, since |u0| ≤ (1 + |y|2)−N and supp u0 ⊂ {|y| ≤ M}, if t is large enough,
then applying Lemma 2.4 we obtain

|Dα̃
x (G−Gfα̃) ∗ u0| ≤ Ct−

3
2−
|α̃|
2

∫
Rn
Br(t, |x− bt− y|)(1 + |y|2)−rdy

≤ Ct− 3
2−
|α̃|
2 Br(t, |x− bt|).

(7.2)

For Π1,2, by the assumption of the initial data and [18], we have

|Gfα̃ ∗Dα̃
xu0| ≤ C

∫
Rn
e−

γ2
γ1
t(1 + |y|2)−rdy

≤ Ct− 3
2−
|α̃|
2 Br(t, |x− bt|).

(7.3)

Combining (7.2) and (7.3), we obtain

|Dα̃
xG ∗ u0| ≤ Ct−

3
2−
|α̃|
2 Br(t, |x− bt|). (7.4)



22 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

7.2. The estimate of the nonlinear term. Set

Φ = (1 + t)−3/2Br(t, |x− bt|),

Φα̃ = (1 + t)−
3
2−
|α̃|
2 Br(t, |x− bt|),

M(t) = sup
0≤s≤t,x∈R

|Dα̃
xu(x, s)|Φ−1

α̃ (x, s).

Then

|u| ≤M · Φ, |Dα̃
xu| ≤M · Φα̃.

For the nonlinear term, we again separate it into two parts:

Π2 =

∫ t

0

Dα̃
xG(t− τ, ·) ∗ div f(u)dτ.

≤
∫ t

0

Dα̃+1
x (G−Gfα̃) ∗ f(u)dτ +

∫ t

0

Gfα̃ ∗Dα̃
x div f(u)dτ

= Π2,1 + Π2,2.

For Π2,1, let Ω = [0, t]× Rn, Ω1 = Ω ∩ { t2 < τ ≤ t}, Ω2 = Ω ∩ {0 ≤ τ ≤ t
2}. Then

Π2,1 =

∫ t

0

Dα̃
x (G−Gfα̃) ∗ div f(u)dτ

≤ C
∫

Ω1

(G−Gfα̃) ∗Dα̃
x div f(u)dτ + C

∫
Ω2

Dα̃+1
x (G−Gfα̃) ∗ f(u)dτ.

There are two cases:

Case 1: |x− bt|2 < t. Then

Π2,1 = C

∫
Ω1

(t− τ)−
3
2Br(t− τ, |x− y − b(t− τ)|)|Dα̃+1u2|dydτ

+ C

∫
Ω2

(t− τ)−
3
2−
|α̃|+1

2 Br(t− τ, |x− y − b(t− τ)|)‖u‖L∞ |u|dydτ

≤ CM(t)(1 + t)−
3+|α̃|

2 − 1
8

∫ t

t
2

(t− τ)−
3
2 (1 + τ)−

3
2−

3
8 (t− τ)

3
2 dτ

+ CM(t)t−
4+|α̃|

2

∫ t/2

0

(1 + τ)−3τ
3
2 dτ

≤ CM(t)t−
3+|α̃|

2 − 1
8 .

When |x− bt|2 < t, we have

1 ≤ 2r
(

1 +
|x− bt|2

t

)−r
= 2rBr(t, |x− bt|).

Thus, we obtain

Π2,1 ≤ CM(t)t−
3+|α̃|

2 − 1
8Br(t, |x− bt|). (7.5)

Case 2: |x− bt|2 ≥ t. We set

P =
(

1 +
|x− y − b(t− τ)|2

t− τ

)−r(
1 +
|y − bτ |2

τ

)−r
.



EJDE-2020/129 POINTWISE ESTIMATES FOR CONSERVATION LAWS 23

Then

P ≤

C
(
1 + |x−bt|2

t−τ
)−r(

1 + |y−bτ |2
τ

)−r
, |x− bt| ≥ |y−bτ |2 ,

C
(
1 + |x−y−b(t−τ)|2

t−τ
)−r(

1 + |x−bτ |2
τ

)−r
, |x− bt| ≤ |y−bτ |2 .

We separate the region into {y : |x− bt| ≥ |y−bτ |2 } and {y : |x− bt| ≤ |y−bτ |2 }. Then

Π2,1

= C

∫
Ω1

(t− τ)−3/2Br(t− τ, |x− y − b(t− τ)|)(1 + τ)−3− |α̃|+1
2 Br(τ, |y − bτ |)dydτ

+ C

∫
Ω2

(t− τ)−
3
2−
|α̃|+1

2 Br(t− τ, |x− y − b(t− τ)|)(1 + τ)−3Br(τ, |y − bτ |)dydτ

≤ CM(t)Br(t, |x− bt|)
∫

Ω1

(t− τ)−
4+|α̃|

2 (1 + τ)−3 τ
3/2

t3/2
Br(t− τ, |x− y − bτ |)dydτ

+ CM(t)Br(t, |x− bt|)
∫

Ω1

(t− τ)−3/2(1 + τ)−3− |α̃|+1
2

τ3/2

t3/2
Br(t− τ, |x− y − bτ |)dydτ

+ CM(t)Br(t, |x− bt|)
∫

Ω2

(t− τ)−
3+|α̃|+1

2 (1 + τ)−3 (t− τ)3/2

t3/2
Br(τ, |y − bτ |)dydτ

+ CM(t)Br(t, |x− bt|)
∫

Ω2

(t− τ)−3/2(1 + τ)−3− |α̃|+1
2

(t− τ)3/2

t3/2
Br(τ, |y − bτ |)dydτ

≤ CM(t)Br(t, |x− bt|)
∫ t

t
2

(t− τ)−
4+|α̃|

2 (1 + τ)−3 τ
3/2

t3/2
(t− τ)3/2dτ

+ CM(t)Br(t, |x− bt|)
∫ t

t
2

(t− τ)−3/2(1 + τ)−3− |α̃|+1
2

τ3/2

t3/2
(t− τ)3/2dτ

+ CM(t)Br(t, |x− bt|)
∫ t

2

0

(t− τ)−
3+|α̃|+1

2 (1 + τ)−3 (t− τ)3/2

t3/2
τ3/2dτ

+ CM(t)Br(t, |x− bt|)
∫ t

2

0

(t− τ)−3/2(1 + τ)−3− |α̃|+1
2

(t− τ)3/2

t3/2
τ3/2dτ

≤ CM(t)Br(t, |x− bt|)t−
4+|α̃|

2

∫ t

t
2

(1 + τ)−3/2dτ

+ CM(t)Br(t, |x− bt|)t−
4+|α̃|

2

∫ t
2

0

(1 + τ)−3/2dτ

≤ CM(t)t−
3+|α̃|

2 − 1
8Br(t, |x− bt|).

(7.6)

Then, by (7.5) and (7.6), we obtain

Π2,1 ≤ CM(t)t−
3+|α̃|

2 − 1
8Br(t, |x− bt|) = CM(t)Φα̃(1 + t)−1/8. (7.7)



24 F. B. LI, W. K. WANG, Y. T. WANG EJDE-2020/129

Next, we consider Π2,2. From [19], we have

Π2,2 =

∫ t

0

Gfα̃ ∗Dα̃
x div f(u)dτ

≤ C
∫ t

0

∫
Rn
Gfα̃(x− y, τ)Φα̃(y, τ)Φα̃+1dydτ

≤ C
∫ t

0

e−
γ2
γ1

(t−τ)Φα̃Φα̃+1dτ

≤ CΦα̃.

(7.8)

Combining (7.1), (7.4), (7.7) and (7.8), and noticing that t ≥ 1, we have

|Dα̃
xu| ≤ CΦα̃ + CM(t)Φα̃(1 + t)−1/8.

Thus we obtain

M(t) ≤ C + CM(t)(1 + t)−1/8.

Since there exists T > 0 such that when t > T ,

C(1 + t)−1/8 ≤ 1

2
,

we conclude that M(t) ≤ 2C, t > T , which implies that

‖Dβu‖L∞ ≤ C, |β| = |α|+ 2.

Then, by the structure of the equation, we have

‖∂tDβu‖L∞ ≤ C.
By the continuity of M(t) on [1, T ], M(1) ≤ C0 and M(t) ≤ C0 for t > T , we have
M(t) ≤ C0 for t ≥ 1. By the definition of M(t), this gives us

|Dα̃
xu| ≤ C(1 + t)−

3+|α̃|
2 Br(|x− bt|, t).

This completes the proof. �

Acknowledgments. F. Li was supported by the National Natural Science Foun-
dation of China (grant nos. 11501347 and 11971290). W. Wang was supported
by the National Natural Science Foundation of China (grant nos. 11771284 and
11831011).

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Fengbai Li

School of Mathematics, Shanghai University of Finance and Economics, 777 Guo Ding

Road, Shanghai 200433, China
Email address: li.fengbai@mail.shufe.edu.cn

Weike Wang
School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao

Tong University, 800 Dong Chuan Road, Shanghai 200240, China

Email address: wkwang@sjtu.edu.cn

Yutong Wang

School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dong Chuan
Road, Shanghai 200240, China

Email address: wyt0521@sjtu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Existence of a local solution
	4. Lp decay estimate of Green's function
	5. Lp bound estimate of solutions
	5.1. L1 bounds
	5.2. L bounds
	5.3. Lp bounds
	5.4. Hs bounds

	6. Decay estimate in s and L
	6.1. Decay rate of the low frequency part in s
	6.2. Decay rate of the high frequency part in s

	7. Pointwise estimate of the solution
	7.1. Estimate of the initial term
	7.2. The estimate of the nonlinear term
	Acknowledgments

	References