Higher Order Dissipative-Dispersive System and Application

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Application

Samer Israwi

To cite this version:

Samer Israwi. Higher Order Dissipative-Dispersive System and Application. Journal of Partial Dif-

ferential Equations, Global Science Press, 2020, 33 (3), �10.4208/jpde.v33.n3.6�. �hal-03168607�

Higher Order Dissipative-Dispersive System and Ap- plication

ISRAWI Samer

,

Department of Mathematics, Faculty of Sciences 1 and Laboratory of Mathematics, Doctoral School of Sciences and Technology, Lebanese University Hadat, Lebanon.

Received 20 November 2019; Accepted 14 May 2020

Abstract. In this paper, a generalized nonlinear dissipative and dispersive equation with time and space-dependent coefficients is considered. We show that the control of the higher order term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a Picard iterative method. As an application to this general Theorem, we prove the well-posedness of the higher order Camassa-Holm type equation as a scalar model which approximates the Euler system with some accuracy for water wave problem.

AMS Subject Classifications: 35C07, 47G30, 35L05, 35A01

Key Words: Cauchy problem, Euler system, Dispersive wave equation, Camassa-Holm equation.

1 Introduction

1.1 Presentation of the problem

In this paper, we study the Cauchy problem for the general nonlinear higher order dissipative- dispersive equation:

 

 

 

 

 

 



 

 

 

 

 

 

( 1 − m∂

) _u

+ _a

( _t,x,u ) _u

+ _a

( _{t, x,u,u}

) _u

+ a

( t,x,u ) u

+ a

( t, x ) u

+ a

( t, x ) u

= f ,

for ( _t,x ) ∈ ( 0, T ] × R , u

= _u

.

(1.1)

∗Corresponding author.Email addresses:[email protected](S. Israwi)

http://www.global-sci.org/jpde/ 1

Where u = _u ( _t,x ) , from [ 0, T ] × R into R , is the unknown function of the problem, m > 0, a

, 1 ≤ i ≤ ^{5 and} f are real-valued smooth given functions which exact regularities will be pre- cised later. This equation covers several important unidirectional models for the water wave problem at different regimes which take into account the variations of the bottom and the surface tension. We have in view in particular the example of the Camassa-Holm equation which was first derived by Camassa and Holm in [1] (see also [2], [3], [4]), which is more nonlinear than the KdV and BBM equations (see for instance [5] [17], [6, 9], [10], [11], [12], [13], [14], [15], [16].). The presence of the fifth order derivative term is very important, so that the equation describes both nonlinear and dispersive effects as does the Camassa-Holm equation in the case of special tension surface values (see [8, 18] page 230 the Kawahara approximation).

Looking for solutions of (1.1) plays an important and significant role in the study of uni- directional limits for water wave problems with variable depth and topographies. To the best of our knowledge the problem (1.1) has not been analyzed previously. In the present paper, we prove the local well-posedness of the initial value problem (1.1) by a standard Picard iterative scheme and the use of adequate energy estimates under a condition of nondegeneracy of the higher dispersive coefficient a

. Therefore we apply this general theorem, to prove the well-posedness of the higher order Camassa-Holm-type equation.

1.2 Notations

In the following, C

denotes any nonnegative constant different than zero whose exact expression is of no importance. The notation a ≲ b means that a ≤ C

b.

We denote by C ( _λ

, λ

,... ) a nonnegative constant depending on the parameters λ

, λ

,. . . and whose dependence on the λ

is always assumed to be nondecreasing.

For any s ∈ R, we denote [ _s ] the integer part of s.

Let p be any constant with 1 ≤ p < _∞ and denote L

= L

( _R ) the space of all Lebesgue- measurable functions f with the standard norm

| f |

= ⁽

∫

R

| f ( _x ) |

dx )

< _∞.

The real inner product of any two functions f

and f

in the Hilbert space L

( _R ) is de- noted by

( _f

, f

) =

∫

R

f

( _x ) _f

( _x ) _dx.

The space L

= L

( _R ) consists of all essentially bounded and Lebesgue-measurable func- tions f with the norm

| f |

= sup | f ( _x ) | < _∞.

We denote by W

( _R ) = { f , s.t. f , ∂

f ∈ L

( _R ) } endowed with its canonical norm.

For any real constant s ≥ ^0, H

= _H

( _R ) denotes the Sobolev space of all tempered dis-

tributions f with the norm | f |

= | Λ

f |

< _{∞, where Λ} is the pseudo-differential op-

erator Λ = ( 1 − ∂

)

. For any two functions u = _u ( _{t, x} ) and v ( _t,x ) defined on [ 0, T ) × R

with T > 0, we denote the inner product, the L

-norm and especially the L

-norm, as well as the Sobolev norm, with respect to the spatial variable x, by ( _u,v ) = ( _u ( _t, · ) _,v ( _t, · )) ,

| u |

= | u ( _t, · ) |

, | u |

= | u ( _t, · ) |

and | u |

= | u ( _t, · ) |

, respectively.

Let E be a given normed space we denote L

([ _0,T ) _;E ) the space of functions such that u ( _t, · ) is controlled in E, uniformly for t ∈ [ 0, T ) :

u

L^∞([0,T);E)

= _ess sup

t∈[0,T)

| u ( _t, · ) |

< _∞ .

Let E be a given normed space we denote C ([ _0,T ) _;E ) the space of functions such that u ( _t, · ) is controlled in E, uniformly for t ∈ [ 0, T ) :

u

L^∞([0,T);E)

= sup

t∈[0,T)

| u ( _t, · ) |

< _∞.

Let X be a given space, we denote C ([ _0,T ) ; X ) the space of functions such that u ( _t, · ) is in X.

Finally, C

( _R

) , i ≥ 1 denote the space of k-times continuously differentiable functions over R

.

For any closed operator T defined on a Banach space X of functions, the commutator [ _{T, f} ] is defined by [ _{T, f} ] _g = _T ( _{f g} ) − f T ( _g ) with f , g and f g belonging to the domain of T.

The same notation is used for f as an operator mapping the domain of T into itself.

Actually, we admit without proof this lemma that presents some properties for the com- mutator operator.

1.3 Product and commutator estimates in Sobolev spaces.

Let us recall here some product as well as commutator estimates in Sobolev spaces, used throughout the present paper (see [18]).

Lemma 1.1 (product estimates).

Let s ≥ ^{0, one has} ∀ f , g ∈ H

( _R )

_L

( _R ) , one has f g

H^s

≲ f

L^∞

g

H^s

+ _f

H^s

g

L^∞

.

If s ≥ s

> 1/2, one deduces thanks to continuous embedding of Sobolev spaces, f g

H^s

≲ f

H^s

g

H^s

.

More generally, for s ≥ ^{0 and} s

> 1/2, one has ∀ f ∈ H

( _R )

_H

( _R ) , g ∈ H

( _R ) , f g

H^s

≲ f

H^s0

g

H^s

+ ^⟨ _f

H^s

g

H^s0

⟩

s>s₀

,

Let F ∈ C

( _R ) be a smooth function such that F ( 0 ) = 0. If g ∈ H

( _R )

_L

( _R ) with s ≥ ^0, one has F ( _g ) ∈ H

( _R ) and

F ( _g )

H^s

≤ C ( _g

L^∞

,F ) _g

H^s

.

We know recall commutator estimate, mainly due to the Kato-Ponce [19], and recently improved by Lannes [18] (see Theorems 3 and 6):

Lemma 1.2 (commutator estimates).

For any s ≥ 0, and ∂

f , g ∈ L

( _R )

H

( _R ) , one has [ _Λ

, f ] _g

L²

≲ ∂

f

H^s−1

g

L^∞

+ _∂

_f

L^∞

g

H^s−1

.

Thanks to continuous embedding of Sobolev spaces, one has for s ≥ s

+ 1, s

>

, [ _Λ

, f ] _g

L²

≲ ∂

f

H^s−1

g

H^s−1

.

More generally, for any s ≥ ^{0 and} s

> 1/2, ∂

f ,g ∈ H

( _R )

_H

( _R ) , one has [ _Λ

, f ] g

L²

≲ ∂

f

H^s0

g

H^s−1

+ ^⟨ _∂

f

H^s−1

g

H^s0

⟩

s>s0+1

.

We conclude this section with the following remark Also, let us remark these contin- uous embedding.

Remark 1.1. Let s >

, then:

• H

( _R ) , → W

( _R )

• H

( _R ) , → L

( _R )

• H

( _R ) , → H

( _R ) .

Moreover, we define the following operators for s > 0: Λ

= ( 1 − m∂

)

and its inverse Λ

such that:

Λ \

( _u ) = ( 1 + _mξ

)

u. ˆ

Finally, we will study the local well-posedness of the initial value problem (1.1) in H

( _R ) endowed with canonical norm.

1.4 Main results

Let us now state our main result:

Theorem 1.1. Let s >

_{and f} ∈ C ([ _0,T ] ; H

( _R )) . We suppose that:

• a

, a

,a

are smooth mappings such that a

,a

in C ([ _0,T ] _,C

( _R

)) _{and a}

_{in C} ([ _0,T ] _,C

( _R

)) _.

• a

∈ C ([ 0, T ] ; H

( _R )) _{, ∂}

_a

∈ L

( 0, T,L

( _R )) _,

• a

∈ C ([ 0, T ] , L

( _R )) _{, ∂}

_a

∈ C ([ 0, T ] ; H

( _R )) _{, with ∂}

_a

∈ L

( 0, T; L

( _R )) _,

• F ( _{t, x} ) : = ∫

0 a4

a5

dy ∈ C ([ 0, T ] ; L

( _R )) _{and ∂}

_F ∈ L

( _{0,T; L}

( _R )) _,

Assume moreover that there is a positive constant c

> 0 such that c

≤| a

( _{t, x} ) |∀ ( _t,x ) ∈ [ 0, T ] × R. Then for all u

∈ H

( _R ) , there exist a time T

> 0 and a unique solution u to (1.1) in C ([ _0,T

] ; H

( _R )) _.

Remark 1.2. There is no restriction on the signs of the coefficients a

and a

; this means that our result handles also the case of the anti-diffusive terms, in which case these terms are controlled by dispersion.

2 Proof of the Main results

Before we start the proof, we give the following useful lemma:

Lemma 2.1. Let m > _{0, s} > 0 then the linear operator

Λ

: H

( _R ) → H

( _R ) is well defined, continuous, one-to-one and onto. If we suppose that u = _Λ

_{f for f} ∈ H

( _R ) _then:

| u |

≤ ¹

m | f |

if 0 < m ≤ 1 (2.1)

| u |

≤ | f |

if m ≥ 1. (2.2) Moreover,

Λ

Λ

= _Λ

_Λ

= _Λ

_Λ

,

where Λ

: H

( _R ) → H

( _R ) is linear continuous one-to-one and onto operator defined by Λ \

u ( _ξ ) = ( 1 + _ξ

)( 1 + _mξ

)

u ˆ ( _ξ ) ,

with

| Λ

|

≤ max ( ¹

m ,1 ) , (2.3)

| ( _Λ

)

|

≤ ^max ( _m,1 ) . (2.4) Proof. We have ∥ Λ

f ∥

= ∥ ( 1 + _ξ

)

( 1 + _mξ

)

_f ^ˆ ∥

. If m ≥ ^{1, then 1} + _mξ

≥ ¹ + _ξ

and

≤ 1 ,therefore

∥ ( 1 + _ξ

)

( 1 + _mξ

)

_f ^ˆ ∥

= ∥ ( 1 + _ξ

)

( 1 + _ξ

)( 1 + _mξ

)

_f ^ˆ ∥

≤∥ ( 1 + _ξ

)

_f ^ˆ ∥

.

If 0 < _m < 1, we have

= 1 +( 1 − m )

1+_mξ²

≤ ¹ +

=

, then

∥ Λ

f ∥

≤ ¹

m ∥ f ∥

.

Now we have

∥ Λ

f ∥

= ∥ Λ

Λ

f ∥

= ∥ Λ

f ∥

≤ ^max ( 1, 1

m ) ∥ f ∥

. and

∥ ( _Λ

)

f ∥

= ∥ ( 1 + _mξ

)( 1 + _ξ

)

( 1 + _ξ

)

f ^ˆ ∥

. If m ≥ ^{1, then} ( 1 + _mξ

)( 1 + _ξ

)

= 1 +( m − ¹ )

≤ m, therefore

∥ ( _Λ

)

f ∥

≤ m ∥ f ∥

.

If 0 < _m < 1, ( 1 + _mξ

)( 1 + _ξ

)

≤ ^{1, then}

∥ ( _Λ

)

_f ∥

≤ ∥ f ∥

. Finally ∥ ( _Λ

)

_f ∥

≤ ^max ( 1, m ) ∥ f ∥

.

We will start the proof of Theorem 1.1 by studying a linearized problem associated to (1.1).

2.1 Linear analysis:

For any smooth enough v, we define the “linearized” operator:

L ( _v,∂ ) = _Λ

_∂

+ _a

( _{t, x, v} ) _∂

+ _a

( _{t,x,v, v}

) _∂

+ _a

( _{t, x,v} ) _∂

+ _a

( _{t, x} ) _∂

+ _a

( _{t, x} ) _∂

. and the following initial value problem:

{ L ( _{v, ∂} ) _u = _f ,

u

= u

. (2.5)

Equation (2.5) is a linear equation which can be solved by a standard method (see [20]) in any time interval in which its coefficients are defined and regular enough. We first establish some precise energy-type estimates of the solution. We define the “energy”

norm,

E

( _u )

= | wΛ

u |

^,

where w is a weight function that will be chosen later. For the moment, we just require that there exists two positive numbers w

,w

such that for all ( _t,x ) in ( 0, T ] × R,

w

≤ w ( _{t, x} ) ≤ w

,

so that E

( _u ) is uniformly equivalent to the standard H

-norm. Differentiating

e

E

( _u )

with respect to time, one gets using (2.5)

1

2 e

∂

( _e

_E

( _u )

) = − ^λ

2 E

( _u )

− ⁽ Λ

Λ

( _a

_u

) _,w

_Λ

_u ⁾

− ⁽ Λ

Λ

( _a

_u

) _,w

_Λ

_u ⁾ − ⁽ Λ

Λ

( _a

_u

) _,w

_Λ

_u ⁾

− ⁽ Λ

Λ

( _a

_u

) , w

Λ

u )

− ⁽ Λ

Λ

( _a

_u

) _,w

_Λ

_u ⁾ + ⁽ _Λ

_Λ

_f , w

Λ

u )

+ ⁽ _ww

_Λ

_{u, Λ}

_u ⁾ .

We now turn to estimating the different terms of the r.h.s of the previous identity by using the needed estimates provided from section 1.3

• Estimate of (

Λ

( _a

_u

) _,Λ

_w

_Λ

_u ⁾ . By the Cauchy-Schwarz inequality and the section 1.3 on the composite functions we have

| ⁽ Λ

( _a

_u

) , Λ

w

Λ

u )

|≤ ¹

m | a

( _{t, x, v} ) − a

( _{t, x,0} ) |

| a

( _t,x,0 ) _u

|

| w

Λ

u |

≤ C ( _m

, a

, ∥ v ∥

, | w |

) _E

( _u )

.

• Estimate of (

Λ

( a

u

) _,Λ

w

Λ

u )

. Similarly as the above estimation, we have

| ⁽ Λ

( _a

_u

) , Λ

w

Λ

u )

|≤ C ( _m

_,a

, ∥ v ∥

, | w |

) _E

( _u )

.

• Estimate of (

Λ

( _a

_u

) , Λ

w

Λ

u )

. Since we have more than s derivative on u, we remark that one can write:

a

u

= _∂

( _a

_∂

_u ) − ∂

a

∂

u − 2a

∂

u, then

Λ

( _a

_u

) = _Λ

( _∂

( _a

_∂

_u )) − Λ

( _∂

_a

_∂

_u ) − ² Λ

( _∂

_a

_∂

_u ) . Now use the identity Λ

= 1 − ∂

to get that

Λ

( _∂

( a

∂

u ))= _Λ

⁽ ( 1 − Λ

)( a

∂

u ) ⁾ = _Λ

( a

∂

u ) − Λ

( a

∂

u )= _Λ

( a

∂

u ) − [ _Λ

, a

] _∂

u − a

Λ

∂

u, then we obtain:

( Λ

( _a

_u

) _,Λ

_w

_Λ

_u ⁾ = ⁽ _Λ

( _a

_∂

_u ) _,Λ

_w

_Λ

_u ⁾ − ⁽ [ _Λ

_,a

] _∂

_u,Λ

_w

_Λ

_u ⁾

− ⁽ a

Λ

∂

u, Λ

w

Λ

u )

− ⁽ Λ

( _∂

a

∂

u ) , Λ

w

Λ

u )

− ^{2 (} Λ

( _∂

a

∂

u ) , Λ

w

Λ

u ) .

By integration by parts, the third term of the last equality becomes:

( a

Λ

∂

u, Λ

w

Λ

u )

= − ¹ 2

( ∂

( _Λ

_w

_a

) , ( _Λ

_u )

⁾ ,

Now by Cauchy Schwarz we have:

| ⁽ Λ

( _a

_u

) _,Λ

_w

_Λ

_u ⁾ | ≤ ¹ m

( ∥ a

∂

u ∥

E

( _u )+ ∥ ∂

a

∥

∥ ∂

u ∥

E

( _u )

+ ∥ w

a

∥

E

( _u )

+ ∥ ∂

a

∂

u ∥

E

( _u )+ ∥ a

∂

u ∥

E

( _u ) ⁾

≤ C ( _m

, a

, ∥ v ∥

, ∥ w ∥

) _E

( _u )

.

• Estimate of (

[ _Λ

,a

] _∂

_u,Λ

w

Λ

u )

+ ⁽ a

Λ

∂

u,Λ

w

Λ

u ) :

a

Λ

∂

u = _a

_Λ

( 1 − Λ

) _∂

_u = _a

( _Λ

− Λ

) _∂

_u = _a

_Λ

_∂

_u − a

Λ

∂

u, then:

( a

Λ

∂

u,Λ

w

Λ

u )

= ⁽ _a

_Λ

_∂

_u,Λ

_w

_Λ

_u ⁾ − ⁽ a

Λ

∂

u,Λ

w

Λ

u ) By Cauchy Schwarz, the first term of the last equality is controlled by:

| ⁽ a

Λ

∂

u,Λ

w

Λ

u )

|≤ ¹

m | a

Λ

∂

u |

E

( _u ) ≤ C ( _m

, | a

|

) _E

( _u )

. ( a

Λ

∂

u, Λ

w

Λ

u )

= − ⁽ a

Λ

w

, ( _∂

_Λ

_u )

⁾ + _Q

, where

| Q

|≤ C ( _{m, s,} | w |

^, | ∂

a

|

) _E

( _u )

.

Now, using the first order Poisson brackets : (see [21] for more details) { Λ

, a

}

= − ( _s − ² ) _∂

( _a

) _Λ

_∂

,

we get:

([ _Λ

_,a

] _∂

_u,Λ

_w

_Λ

_u ) = ( _s − ² )( _∂

( _a

) _Λ

_∂

_u,Λ

_w

_Λ

_u )+ _Q

, Where

| Q

|≤ C ( _m,s, | w |

, | a

|

) _E

( _u )

. Now, by integration by parts we have:

( _s − 2 )( _∂

( _a

) _Λ

_∂

_u,Λ

_w

_Λ

_u ) = − ( _s − 2 )

2 ( _∂

( _∂

( _a

) _Λ

_w

) _Λ

_u,Λ

_u ) , then

| ([ _Λ

_,a

] _∂

_u,Λ

_w

_Λ

_u ) | ≤ C ( _m,s, | w |

, | a

|

) _E

( _u )

.

• Estimate of (

[ _Λ

_,a

] _∂

_{u, Λ}

_w

_Λ

_u ⁾ + ⁽ _a

_Λ

_∂

_{u, Λ}

_w

_Λ

_u ⁾ :

a

Λ

∂

u = _a

_Λ

( 1 − Λ

) _∂

_u = _a

_Λ

_∂

_u − a

Λ

∂

u = _a

_Λ

_∂

_u − a

Λ

∂

u − a

Λ

∂

u.