Analytic Solutions for a Nonlinear Transport Equation

Department of Mathematics, Kazakhstan

Math 499 Capstone Project

Analytic Solutions for a Nonlinear Transport Equation

Magzhan Biyar

Research Supervisor:

Dr. Daniel Oliveira da Silva

Second Reader:

Dr. Alejandro Javier Castro Castilla

April, 2019

Abstract

We prove that the Cauchy problem for a transport equation with algebraic nonlinearity of degreepwith initial data in Gevrey spaces is locally well-posed. In particular, we show that the analyticity of solutions persists for a short time and we derive a sufficient condition for solutions to be analytic for all times.

Keywords: Transport equation, well-posedness, analytic spaces, Gevrey spaces MSC 2010: 35F25

1 Introduction

Solutions to partial differential equations often behave in unexpected ways. An interesting example of this is the Korteweg-de Vries equation u_t+u_xxx+uu_x = 0, which has solutions which are analytic, even for initial conditions which are not differentiable. Observations such as this have led many mathematicians to consider the following question: suppose we have a partial differential equation, and an initial condition u(x,0) = f(x) which has an analytic continuation on the subset of the complex plane

S_σ =

x+iy ∈C: |y|< σ .

What will happen to the analyticity of the solution u as time progresses? Many authors have considered this question for many different equations [1, 2, 3]. In almost all cases, the authors obtained estimates on how quickly the widthσcan shrink over time. In this project, we will consider this question for the initial value problem

(u_t−u_x =u^p,

u(x,0) =f(x). (1.1)

where p > 1 is an integer, the unknown u(x, t), and the datum f(x) are real-valued. In particular, we will try to obtain a simple condition on the solution to ensure that the width σ does not decay at all over time, as was done in [4, 5].

Next, we introduce some notations and definitions. We use the Lebesgue norms kfk_L^q_(R) =

Z

R

|f(x)|^qdx 1/q

,

for1≤q <∞, with the usual convention

kfk_L^∞_(R) = ess sup

x∈R

|f(x)|.

We define the mixed Lebesgue normsL^q_tL^r_x(I×R)for any time interval I as the space of all functionsu(x, t) with norm

kuk_L^q

tL^r_x(I×R)= Z

I

ku(t)k^q_Lr x(R)dt

1/q

= Z

I

Z

R

|u(x, t)|^rdx q/r

dt

!1/q

with the usual modifications when q = ∞ or r = ∞. Then the vector space C(I) can be defined by the norm

kuk_C(I) = sup

x∈I

|u(x)|.

Suppose we have a function u(x, t). Then the combination of L^∞(R) and H^s(R) spaces is L^∞_t H_x^s(R×R) with the norm

kuk_L^∞_{t Hx}^s_(R×R)= sup

t∈I

ku(·, t)k_Hx^s_(R).

The Fourier transform of f, denoted by f, is given byˆ (Ff)(ξ) = ˆf(ξ) =

Z ∞

−∞

f(x)e^−iξxdx.

Here,i=√

−1, andξis the frequency variable. The inverse Fourier transform ofg, denoted byg, is given byˇ

(F⁻¹g)(x) = ˇg(x) = Z ∞

−∞

g(ξ)e^ixξdξ.

The Sobolev space H^s(R) for s∈R is defined by the norm kfk_H^s_(R) =khξi^sfˆ(ξ)k_L²

ξ(R), wherehξi= (1 +ξ²)^1/2.

Definition 1.1. Let u(x) be a function with Fourier transform u(ξ). Letˆ p(ξ) be some locally integrable function ofξ. Then the expression

p(∇/i)u=F⁻¹(p(ξ)ˆu(ξ))

is called a Fourier multiplier. The functionp(ξ)is called the symbol of the multiplier.

We define the operatorp(∇/i) = |D_x|=|d/dx|as the Fourier multiplier with symbol|ξ|.

In the present work, we will consider the problem (1.1) with initial data in Gevrey spaces G^σ,s =G^σ,s(R), with norm defined by

kfk_G^σ,s_(R)=ke^σ|Dx|hD_xi^sfk_L²_x(R)=ke^σ|ξ|hξi^sf(ξ)kˆ _L²

ξ(R),

where D_x = −i∂_x, ξ is the Fourier variable corresponding to x. These spaces have the following property

Theorem 1.2 (Paley-Wiener theorem (see [6], p. 209)). Let σ > 0 and s ∈ R. Then, the following statements are equivalent:

(i) f ∈G^σ,s(R).

(ii) f is the restriction to the real line of a function F which is holomorphic in the strip S_σ ={x+iy: x, y ∈R, |y|< σ}

and satisfies

sup

|y|<σ

kF(x+iy)kH_x^s(R) <∞.

A function f :R−→C is said to be rapidly decreasing if we have khxi^Nf(x)k_L^∞_(R)<∞

for all N ≥ 0, where hxi = (1 +x²)^1/2. We say that a function is a Schwartz function if it is smooth and all of its derivatives are rapidly decreasing. We use S(R) to denote the space of all Schwartz functions. The space S(R) has a dual S⁰(R), the space of tempered distributions.

Definition 1.3. A tempered distribution onR is a continuous linear functional on S(R).

Definition 1.4. Let q∈ [1,∞]. The Fourier Lebesgue space FL^q(R) is the inverse Fourier image ofL^q(R), i.e., FL^q(R)consists of all f ∈S⁰(R)such that

kfkFL^q(R) =kfˆ(ξ)k_L^q_(R) is finite ([7], p. 378).

Thus, any function f inG^σ,s(R) is analytic in a strip whose width is at leastσ. We will consider the following question: will the solution of the problem (1.1) also belong to the spaceG^σ,s(R) at a time t >0? Similar issues were considered in [4, 5, 8].

Our main results in this paper are the following theorems

Theorem 1.5. Let σ > 0 and s > 1/2. Then the Cauchy problem (1.1) is locally well- posed for initial data f in G^σ,s(R). That is, there exists T > 0 such that there exists a unique solutionu(x, t) of problem (1.1) inC([0, T];G^σ,s(R)). Moreover, the solution depends continuously on the initial data. Thus, a solution which is analytic in S_σ at t = 0 will continue to be analytic in S_σ at least in [0, T].

This first main result states that analycity persists, at least for a short time. The proof of Theorem 1.5 is contained in Section 3.

Theorem 1.6. Letu be a solution of the Cauchy problem (1.1) for initial data f ∈G^σ,s(R), σ >0 and s >1/2. Define

T^∗ = sup{T >0 : ku(·, t)k_G^σ,s <∞ for t∈[0, T]}.

Then either T^∗ =∞ or there exists at least one k such that 1≤k≤p−1 and e^σ|Dx|u^k ∈/ L^∞_t FL¹_x [0, T^∗]×R

.

This second main result gives us a sufficient condition to determine if analyticity persists for all time. The proof is contained in Section 4.

2 Preliminaries

We will need the following inequality later (see [9, Lemma A8, p. 338]) Lemma 2.1. If f, g ∈H^s(R)∩L^∞(R) and s≥0, then

kf gk_H^s_(R) ≤C_s(kfk_H^s_(R)kgk_L^∞_(R)+kfk_L^∞_(R)kgk_H^s_(R)), (2.1) if f, g∈H^s(R) and s >1/2, then

kf gk_H^s_(R) ≤Ckfk_H^s_(R)kgk_H^s_(R). (2.2) Next we will prove the following product estimates

Lemma 2.2. If f, g ∈G^σ,s(R) and s >1/2, then

kf gk_G^σ,s ≤Ckfk_G^σ,skgk_G^σ,s, (2.3) if e^σ|D|f, e^σ|D|g ∈H^s(R)∩ FL¹(R), and s≥0 then

kf gk_G^σ,s ≤C(kfk_G^σ,ske^σ|D|gk_FL¹_(R)+ke^σ|D|fk_FL¹_(R)kgk_G^σ,s), (2.4)

Proof. Indeed, if s >1/2, then by (2.2) and Plancherel’s theorem [6, p.156], we have kf gkG^σ,s =ke^σ|D|hDi^sf gk_L²_(R)=ke^σ|ξ|hξi^s(cf g)(ξ)k_L²_(R)

=Z

R

e^2σ|ξ|hξi^2s Z

R

f(ξˆ −η)ˆg(η)dη

2

dξ1/2

≤Z

R

hξi^2shZ

R

e^σ|ξ−η||fˆ(ξ−η)|

e^σ|η||ˆg(η)|

dηi2

dξ1/2

=

hDi^s e^σ|D|F⁻¹(|f(ξ)|)ˆ

e^σ|D|F⁻¹(|ˆg(ξ)|) L²(R)

=

e^σ|D|F⁻¹(|fˆ(ξ)|)

e^σ|D|F⁻¹(|ˆg(ξ)|) H^s(R)

≤C

e^σ|D|F⁻¹(|fˆ(ξ)|) H^s(R)

e^σ|D|F⁻¹(|ˆg(ξ)|) H^s(R)

=CZ

R

hξi^2se^2σ|ξ||fˆ(ξ)|²dξ1/2Z

R

hξi^2se^2σ|ξ||ˆg(ξ)|²dξ1/2

=Ckfk_G^σ,skgk_G^σ,s,

if e^σ|D|f, e^σ|D|g ∈H^s(R)∩ FL¹(R), and s≥0 then by (2.1) kf gk_G^σ,s ≤

e^σ|D|F⁻¹(|fˆ(ξ)|)

e^σ|D|F⁻¹(|ˆg(ξ)|) H^s(R)

≤C

e^σ|D|F⁻¹(|f(ξ)|)ˆ H^s(R)

e^σ|D|F⁻¹(|ˆg(ξ)|) L^∞(R)

+

e^σ|D|F⁻¹(|fˆ(ξ)|) L^∞(R)

e^σ|D|F⁻¹(|ˆg(ξ)|) H^s(R)

≤C

kfk_G^σ,s Z

R

e^σ|ξ||ˆg(ξ)|dξ+ Z

R

e^σ|ξ||fˆ(ξ)|dξkgk_G^σ,s

=C kfkG^σ,ske^σ|D|gkFL¹(R)+ke^σ|D|fkFL¹(R)kgkG^σ,s

. in the last inequality we used the following

e^σ|D|F⁻¹(|fˆ(ξ)|) _L∞(

R)≤ ke^σ|D|fk_FL¹_(R). (2.5) Indeed,

|e^σ|D|F⁻¹(|fˆ(ξ)|)|=|F⁻¹(e^σ|ξ||fˆ(ξ)|)|

=

Z

R

e^iξxe^σ|ξ||f(ξ)|dξˆ ≤

Z

R

e^σ|ξ||f(ξ)|dξˆ

=ke^σ|ξ|f(ξ)kˆ _L¹_(R) =ke^σ|D|fkFL¹(R). Hence we have (2.5). Thus, Lemma 2.2 is proved.

3 Proof of Theorem 1.5

For the proof of Theorem 1.5, we will use Picard iteration. Define a sequence of functions {u_n}^∞_n=0 according to

∂_tu₀−∂_xu₀ = 0, u₀(x,0) =f(x), f ∈G^σ,s(R), and

∂_tu_n−∂_xu_n =u^p_n−1, u_n(x,0) =f(x), n≥1.

Using Duhamel’s principle [9, p. 67], it is easy to see that u0(x, t) = f(x+t),

u_n(x, t) =f(x+t) + Z t

0

u^p_n−1(x+ (t−τ), τ)dτ. (3.1) We must first show thatu_n ∈C([0, T]; G^σ,s)for alln ≥0. We will give a proof by induction onn. Indeed, first observe that

ku₀k_C([0,T];G^σ,s₎ = sup

t∈[0,T]

ku₀(·, t)k_G^σ,s

= sup

t∈[0,T]

Z

R

e^2σ|ξ|hξi^2s|f(x\+t)|²dξ 1/2

= sup

t∈[0,T]

Z

R

e^2σ|ξ|hξi^2s|e^iξtf(ξ)|ˆ ²dξ 1/2

=kfkG^σ,s ≤2kfkG^σ,s. Now assume thatun−1 ∈C([0, T];G^σ,s), with

kun−1k_C([0,T];G^σ,s₎≤2kfk_G^σ,s. We will show that the same is true foru_n. By (3.1), we have

ku_nk_G^σ,s ≤ kfk_G^σ,s +k Z t

0

u^p_n−1(x+ (t−τ), τ)dτk_G^σ,s.

By the formula (3.1), Minkowski inequality [6, p. 273], and using inequality (2.3)

Z t

0

u^p_n−1(x+ (t−τ), τ)dτ G^σ,s

≤ Z t

0

ku^p_n−1(x+ (t−τ), τ)k_G^σ,sdτ

= Z t

0

ku^p_n−1(·, τ)k_G^σ,sdτ . Z t

0

ku^p−1_n−1(·, τ)k_G^σ,skun−1(·, τ)k_G^σ,sdτ .

Z t

0

ku_n−1(·, τ)k^p_Gσ,sdτ ≤C Z t

0

ku_n−1k^p_C([0,T];Gσ,s)dτ ≤2CTkfk^p_Gσ,s. Then we have

ku_nk_C([0,T];G^σ,s₎ ≤(1 + 2CTkfk^p−1_Gσ,s)kfk_G^σ,s ≤2kfk_G^σ,s, if

T ≤ 1

2Ckfk^p−1_Gσ,s

.

We will now use this bound on the norms of theu_n to show convergence. Note that it suffices to show that

ku_n−un−1k_C([0,T];G^σ,s₎ <kun−1−un−2k_C([0,T];G^σ,s₎,

for all n. Using the inequality (2.3) it follows that ku_n−u_n−1k_G^σ,s =

Z t

0

u^p_n−1(x+ (t−τ), τ)−u^p_n−2(x+ (t−τ), τ) dτ

G^σ,s

≤ Z t

0

ku^p_n−1(·, τ)−u^p_n−2(·, τ)k_G^σ,sdτ .

Z t

0

ku^p−1_n−1(·, τ) +u^p−2_n−1(·, τ)u_n−2(·, τ) +u^p−3_n−1(·, τ)u²_n−2(·, τ) + . . . +u²_n−1(·, τ)u^p−3_n−2(·, τ) +un−1(·, τ)u^p−2_n−2(·, τ)

+u^p−1_n−2(·, τ)k_G^σ,skun−1(·, τ)−un−2(·, τ)k_G^σ,sdτ .

Z t

0

kun−1(·, τ)k^p−1_Gσ,s +kun−1(·, τ)k^p−2_Gσ,skun−2(·, τ)k_G^σ,s+ . . . +kun−1(·, τ)k_G^σ,skun−2(·, τ)k^p−2_Gσ,s

+kun−2(·, τ)k^p−1_Gσ,s

kun−1(·, τ)−un−2(·, τ)k_G^σ,sdτ

≤Cp2^p−1kfk^p−1_Gσ,s

Z t

0

ku_n−1(·, τ)−u_n−2(·, τ)k_G^σ,sdτ

≤Cp2^p−1kfk^p−1_Gσ,s

Z t

0

kun−1−un−2k_C([0,T];G^σ,s₎dτ

≤Cp2^p−1kfk^p−1_Gσ,sku_n−1−u_n−2k_C([0,T];G^σ,s₎<ku_n−1−u_n−2k_C([0,T];G^σ,s₎, as

T < 1 Cp2^p−1kfk^p−1_Gσ,s

.

Thus, it was proved that the sequence {u_n}^∞_n=0 as the Cauchy sequence in Banach space C([0, T]; G^σ,s) converges to u(x, t) inC([0, T]; G^σ,s). By passing to the limit as n −→ ∞ in the equations (3.1), we see thatu(x, t) is the solution of the problem (1.1).

We will prove that the map f −→ u is continuous as a mapping from G^σ,s(R) to C([0, T];G^σ,s). Suppose that for two initial conditions f, g ∈ G^σ,s, we have correspond- ing solutions u and v, respectively. Let us estimate the difference between the solutions u and v in the spaceC([0, T]; G^σ,s).

ku−vk_G^σ,s

≤ kf(x+t)−g(x+t)kG^σ,s+ Z t

0

ku^p(x+ (t−τ), τ)−v^p(x+ (t−τ), τ)kG^σ,sdτ

=kf −gk_G^σ,s+ Z t

0

ku^p(·, τ)−v^p(·, τ)k_G^σ,sdτ

≤ kf−gkG^σ,s+C Z t

0

ku^p−1(·, τ) +u^p−2(·, τ)v(·, τ) +. . .

+u(·, τ)v^p−2(·, τ) +v^p−1(·, τ)k_C([0,T];G^σ,s₎ku(·, τ)−v(·, τ)k_C([0,T];G^σ,s₎dτ

≤ kf −gkG^σ,s+C Z t

0

ku(·, τ)k^p−1_Gσ,s +ku(·, τ)k^p−2_Gσ,skv(·, τ)kG^σ,s+. . . +ku(·, τ)k_G^σ,skv(·, τ)k^p−2_Gσ,s+kv(·, τ)k^p−1_Gσ,s

ku(·, τ)−v(·, τ)k_C([0,T];G^σ,s₎dτ

≤ kf −gk_G^σ,s+ 2^p−1CT kfk^p−1_Gσ,s+kfk^p−2_Gσ,skgk_G^σ,s+. . . +kfk_G^σ,skgk^p−2_Gσ,s+kgk^p−1_Gσ,s

ku−vk_C([0,T];G^σ,s₎. Then

ku−vk_C([0,T];G^σ,s₎ ≤ kf−gk_G^σ,s+ 2^p−1CT kfk^p−1_Gσ,s +kfk^p−2_Gσ,skgk_G^σ,s+. . . +kfk_G^σ,skgk^p−2_Gσ,s+kgk^p−1_Gσ,s

ku−vk_C([0,T];G^σ,s₎. If

T < 1

2^p−1C kfk^p−1_Gσ,s+kfk^p−2_Gσ,skgk_G^σ,s+. . .+kfk_G^σ,skgk^p−2_Gσ,s +kgk^p−1_Gσ,s

, then

ku−vk_C([0,T];G^σ,s₎

≤ 1

1−2^p−1CT kfk^p−1_Gσ,s+kfk^p−2_Gσ,skgk_G^σ,s+. . .+kfk_G^σ,skgk^p−2_Gσ,s+kgk^p−1_Gσ,s

kf−gkG^σ,s.

Thus, we have proven Theorem 1.5.

4 Proof of Theorem 1.6

For the proof of Theorem 1.6, we need to estimate the normkuk_G^σ,s. We apply the operator e^σ|Dx|hD_xi^s to the both sides of the equation (1.1) to get

e^σ|Dx|hD_xi^su_t−e^σ|Dx|hD_xi^su_x =e^σ|Dx|hD_xi^su^p. If we define

U =e^σ|Dx|hD_xi^su,

then the commutativity of Fourier multipliers lets us rewrite the result as 1

2∂_tU²− 1

2∂_xU² =U e^σ|Dx|hD_xi^su^p.

We now integrate both sides of this equation in space. Then by the equality Z

R

∂tU²(x, t)dx= d dt

Z

R

U²(x, t)dx, we get

d dt

Z

R

U²dx− Z

R

∂_xU²dx= 2 Z

R

U e^σ|Dx|hD_xi^su^pdx.

The first integral on the left-hand side of the equation is nothing more than ku(·, t)k²_Gσ,s. The second integral on the left-hand side of the equation vanishes, since U must vanish at

infinity [10, Corollary 7.9.4, p. 243]. On the right-hand side the integral can be estimated by using H¨older’s inequality [11, Proposition 1.1, p 1] and the inequality (2.4)

2

Z

R

U e^σ|Dx|hD_xi^su^pdx

≤2Z

R

U²dx1/2Z

R

e^σ|Dx|hD_xi^su^p

2dx1/2

= 2kukG^σ,sku^pkG^σ,s

.2kuk_G^σ,s kuk_G^σ,ske^σ|Dx|u^p−1k_FL¹_(R)+ke^σ|Dx|uk_FL¹_(R)ku^p−1k_G^σ,s

.2kuk_G^σ,s kuk_G^σ,ske^σ|Dx|u^p−1k_FL¹_(R)+kuk_G^σ,ske^σ|Dx|uk_FL¹_(R)ke^σ|Dx|u^p−2k_FL¹_(R) +ke^σ|Dx|uk_FL¹_(R)ke^σ|Dx|uk_FL¹_(R)ku^p−2k_G^σ,s

≤2Ckuk²_Gσ,s ke^σ|Dx|u^p−1k_FL¹_(R)+ke^σ|Dx|uk_FL¹_(R)ke^σ|Dx|u^p−2k_FL¹_(R)

+ke^σ|Dx|uk²_FL1(R)ke^σ|Dx|u^p−3kFL¹(R)+. . .+ke^σ|Dx|uk^p−2_FL1(R)ke^σ|Dx|ukFL¹(R)

= 2Ckuk²_Gσ,s

p−2

P

j=0

ke^σ|Dx|uk^j_FL1(R)ke^σ|Dx|u^p−1−jkFL¹(R)

.

If we define

E(t) = kuk²_Gσ,s, we have the inequality

d

dtE(t)≤2C

p−2

X

j=0

ke^σ|Dx|uk^j_FL1(R)ke^σ|Dx|u^p−1−jk_FL¹_(R) E(t).

By Gr¨onwall’s inequality [9, Theorem 1.10, p. 11], we have that

E(t)≤E(0) exp 2C

Z t

0 p−2

X

j=0

ke^σ|Dx|uk^j_FL1(R)ke^σ|Dx|u^p−1−jkFL¹(R)

dτ

≤ kfk²_Gσ,sexp 2C

p−2

P

j=0

ke^σ|Dx|uk^j_L∞

t FL¹_x(R)ke^σ|Dx|u^p−1−jk_L^∞

t FL¹_x(R)

t

.

IfT^∗ =∞, then we have proven Theorem 1.6. If T^∗ <∞, then it must be the case that ku(·, t)k_G^σ,s −→ ∞, as t→T^∗.

Otherwise, we would be able to apply Theorem 1.5 again to extend our solution to a slightly larger interval [0, T^∗+ε]. But this is a contradiction, as it would violate the definition of T^∗. Then from the resulting inequality

kuk²_Gσ,s ≤ kfk²_Gσ,sexp 2C

p−2

X

j=0

ke^σ|Dx|uk^j_L∞

t FL¹_x(R)ke^σ|Dx|u^p−1−jk_L^∞

t FL¹_x(R)

t

,

it follows that there exists at least one k such that 1≤k≤p−1 and e^σ|Dx|u^k∈/ L^∞_t FL¹_x(R) [0, T^∗]×R

. Thus, Theorem 1.6 is proved.

5 Conclusion

It was proven that the Cauchy problem for a transport equation with algebraic nonlinearity of degreepwith initial data in Gevrey spaces is locally well-posed. In particular, the persistence of the analyticity of solutions for a short time was proven and a sufficient condition for solutions to be analytic for all times was derived.

Acknowledgements

I would like to express my deep gratitude to my research supervisor Dr. Daniel Oliveira da Silva for his guidance, cordial support, valuable advice and constant encouragement throughout the course of this project. I also thank Dr. Alejandro Javier Castro Castilla as the second reader of this project for critical remarks and valuable advice.

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