Existence of solutions and other properties for an internal wave - PowerPoint PPT Presentation

Existence of solutions and other properties for an internal wave model Higidio Portillo Oquendo et. al. 1 1 Department of Mathematics Federal University of Paran´ a Ufpr 2017 higidio@ufpr.br UFPR

The Nonlinear ILW System We consider the nonlinear ILW system η t = [( 1 − αη ) u ] x = η x − α uu x u t with initial data η ( x , 0 ) = η 0 ( x ) , u ( x , 0 ) = u 0 ( x ) . t denote the time variable and the spatial variable x stands in R T = [ − π , π ] (periodic functions) . or higidio@ufpr.br UFPR

Questions Existence of local solutions? 1 Blow up? or Global solutions? 2 Other properties? 3 higidio@ufpr.br UFPR

A Friendly System A system with more properties is given by η t = [( 1 − αη ) u ] x = η x − α uu x + M ( u )+ N ( u ) u t where M ( u ) denote dispersive terms and N ( u ) denote dissipative terms. Some dispersive terms are u xxt , H ( u xt ) , T ( u xt ) , u xxx , etc . Some dissipative terms are δ u xx , H ( u x ) , T ( u x ) , − u xxxx , etc . higidio@ufpr.br UFPR

Our problem We will study the following nonlinear system η t = [( 1 − αη ) u ] x = η x − α uu x + δ u xx u t with initial data η ( x , 0 ) = η 0 ( x ) , u ( x , 0 ) = u 0 ( x ) . where the spatial variable stands at x ∈ T = [ − π , π ] (periodic solutions) . higidio@ufpr.br UFPR

Questions Existence of local solutions? 1 Blow up? or Global solutions? 2 Smoothing properties? 3 Asymptotic behavior? 4 Who is the limit? 5 Speed of this asymptotic behavior? 6 higidio@ufpr.br UFPR

The nonlinear system can be written in the abstract framework d dt U = A 0 U + AF ( U ) , where � η � � ∂ x � � � ∂ x 0 0 U = , A = , A 0 = , ∂ x ∂ x δ∂ xx u 0 � − αη u � F ( U ) = . − α u 2 / 2 higidio@ufpr.br UFPR

We will work in the space H s = H s ( T ) × H s ( T ) , s ≥ 0 with the the inner product � U 1 , U 2 � = � η 1 , η 2 � H s ( T ) + � u 1 , u 2 � H s ( T ) . This operators are defined in D ( A 0 ) = { U ∈ H s : A 0 ( U ) ∈ H s } D ( A ) = H s + 1 , and D ( F ) = H s ( for s > 1 / 2 ) . higidio@ufpr.br UFPR

Note that D ( A 0 ) is characterized by { U ∈ H s : u ∈ H s + 1 ( T ) , η + δ u x ∈ H s + 1 ( T ) } . D ( A 0 ) = is easy to see that H s + 1 ( T ) × H s + 2 ( T ) ⊂ D ( A 0 ) ⊂ H s . Therefore A 0 is densely defined. higidio@ufpr.br UFPR

Some properties of the linearized system are preserved by the nonlinear system. In this case the linearized system is η t = u x , = η x + δ u xx . u t This system can be written in the abstract framework by d dt U = A 0 U . higidio@ufpr.br UFPR

Theorem A 0 is the generator of a contractions semigroup. We use the following result: Theorem (Lumer-Phillips) Let A be a operator in a Hilbert space X . Then, A is the generator of a contractions semigroup if and only if it is densely defined and m-dissipative. Definition: A is m-dissipative if Re � A U , U � X ≤ 0 Im ( λ I − A ) = X and for some λ > 0. higidio@ufpr.br UFPR

Proof We have Re � A 0 U , U � = − δ � u x � 2 < 0 , then, the operator A 0 is dissipative. Let F ∈ H s , solving the equation � ( I − A 0 ) U = � F . If µ = ˆ ( I − A 0 ) U = F is equivalent to solve η , ω = ˆ u is equivalent to solve the system µ − ik ω = f ω − ik ( µ + i δ k ω ) = g where ( f , g ) = � F . The solutions are µ = ( 1 + δ k 2 ) f + ikg g + ikf , ω = 1 + δ k 2 + k 2 . 1 + δ k 2 + k 2 after some computations we verify U ∈ D ( A 0 ) . higidio@ufpr.br UFPR

With the above theorem for each initial data U 0 ∈ D ( A 0 ) , we have a unique global solutions U ( t ) = e t A 0 U 0 for the linearized system in the space U ∈ C ([ 0 , ∞ [ , D ( A 0 )) ∩ C 1 ([ 0 , ∞ [ , H s ) . higidio@ufpr.br UFPR

Theorem The semigroup { e t A 0 } t ≥ 0 is analytic. we use the following result: Theorem (a particular case of this theorem is proved in Liu’s book) Let A the generator of a contractions semigroup { e t A } t ≥ 0 . If the following conditions ρ ( A ) ⊃ i R \{ 0 } and 1 � R ( i λ , A ) � ≤ C | λ | , for all λ ∈ R , λ � = 0 2 are satisfied, then { e t A } t ≥ 0 is an analytic semigroup. higidio@ufpr.br UFPR

Proof We will use the discrete Fourier transform to show this theorem. If ( µ , ω ) denote the Fourier transform of U = ( η , u ) . The system ( i λ I − A 0 ) U = F , for λ ∈ R is satisfied if i λ µ − ik ω = f , i λω − ik ( µ + i δ k ω ) = g . Solving this equations we have µ = − i ( λ − i δ k 2 ) f + ikg i λ g + ikf λ 2 − i δ k 2 λ + k 2 , ω = − λ 2 − i δ k 2 λ + k 2 . After some computations we have � U � ≤ C | λ |� F � , λ � = 0 . From this estimate we conclude that � λ ( i λ I − A 0 ) − 1 � ≤ C ρ ( A 0 ) ⊃ i R \{ 0 } and higidio@ufpr.br UFPR

Returning to the nonlinear system Applying the technique of parameters variations, the solution of the nonlinear system must satisfy the Duhamel’s formula � t U ( t ) = e t A 0 U 0 + e ( t − s ) A 0 AF ( U ( s )) ds 0 If we consider the operator � t ( GU )( t ) = e t A 0 U 0 + e ( t − s ) A 0 AF ( U ( s )) ds 0 we use some theorem of fixed point to find solutions of the nonlinear system. higidio@ufpr.br UFPR

Because AA 0 = A 0 A in D ( A 0 ) ∩ D ( A ) we can verify that A commute with e ( t − s ) A 0 em D ( A ) , and in this case � t e t A 0 U 0 + 0 A e ( t − s ) A 0 F ( U ( s )) ds ( GU )( t ) = � t e s A 0 U 0 + 0 A e s A 0 F ( U ( t − s )) ds . = Difficulty: unfortunately the operator A e t A 0 blow up at t = 0. I explain: it can be show that the function t → A e t A 0 are continuous in L ( H s ) for t > 0. The blow up is consequence of this fact A e t A 0 → A , t → 0 + A �∈ L ( H s ) . when and higidio@ufpr.br UFPR

Therefore we need some kind control on � A e t A 0 � near to t = 0. It is known that A 0 e t A 0 is a limited operators for t > 0 and � A 0 e t A 0 � ≤ C t , t > 0 , Because the operator A is more “weak” than A 0 , it is possible to show the same inequality, that is � A e t A 0 � ≤ C t , t > 0 , but this do not help me. higidio@ufpr.br UFPR

Theorem There exist θ ∈ ] 0 , 1 [ such that � A e t A 0 � ≤ C t θ Corollary we have the following estimate � t C 0 � A e s A 0 � ds ≤ 1 − θ t 1 − θ , ∀ t > 0 . higidio@ufpr.br UFPR

Some estimates for the nonlinear term � − αη u � � − α u � − αη F ( U ) = D F ( U ) = ⇒ − α u 2 / 2 − α u 0 Therefore � D F ( U ) � ≤ C � U � . Since � 1 � � F ( U 2 ) − F ( U 1 ) = D F U 1 + r ( U 2 − U 1 ) ( U 2 − U 1 ) dr , 0 it follows that � F ( U 2 ) − F ( U 1 ) � ≤ C ( � U 1 � + � U 2 � ) � U 2 − U 1 � . Consequently, if � U 1 − U 0 � ≤ R , � U 2 − U 0 � ≤ R for some U 0 , we have � F ( U 2 ) − F ( U 1 ) � ≤ C ( R + � U 0 � ) � U 2 − U 1 � . higidio@ufpr.br UFPR

Theorem (Local solutions) For U 0 ∈ D ( A 0 ) , the nonlinear system has a unique solution em C ([ 0 , T ] , H s ) for some T > 0 Proof Let T > 0, R > 0, we consider the subset of the space C ([ 0 , T ] , H s ) : � � U ∈ C ([ 0 , T ] , H s ) : U ( 0 ) = U 0 , U ( t ) ∈ B R ( U 0 ) M T = . We define the operator G : M T → C ([ 0 , T ] , H s ) given by � t e t A 0 U 0 + 0 A e s A 0 F ( U ( t − s )) ds . G ( U )( t ) = higidio@ufpr.br UFPR

we have � G ( U )( t ) − U 0 � � t � e t A 0 U 0 − U 0 � + 0 � A e s A 0 � ( � F ( U ( t − s )) − F ( U 0 ) � + � F ( U 0 � ) ds ≤ � � t � � e t A 0 U 0 − U 0 � + 0 � A e s A 0 � ds ≤ { CR ( R + � U 0 � )+ � F ( U 0 �} � e t A 0 U 0 − U 0 � + C ( θ ) t 1 − θ { CR ( R + � U 0 � )+ � F ( U 0 �} . ≤ Taking T small we have � G ( U )( t ) − U 0 � ≤ R , ∀ t ∈ [ 0 , T ] . This shows that G ( M T ) ⊂ M T for T small. higidio@ufpr.br UFPR

On the other hand, � G ( U 2 )( t ) − G ( U 1 )( t ) � � t 0 � A e s A 0 �� F ( U 2 ( t − s )) − F ( U 1 ( t − s )) � ds ≤ � � t � 0 � A e s A 0 � ds ≤ C ( R + � U 0 � ) � U 2 − U 1 � C ([ 0 , T ] , H s ) C ( θ ) t 1 − θ ( R + � U 0 � ) � U 2 − U 1 � C ([ 0 , T ] , H s ) . ≤ Taking T small we have that G is a contraction operator. higidio@ufpr.br UFPR

Questions Existence of local solutions? OK 1 Blow up? or Global solutions? still trying! 2 Smoothing properties? it’s possible 3 Asymptotic behavior? 4 Who is the limit? 5 Speed of this asymptotic behavior? 6 higidio@ufpr.br UFPR

Asymptotic behavior If ( η ∞ ( x ) , u ∞ ( x )) is the limit of the solutions ( η ( x , t ) , u ( x , t )) when t → ∞ , then ( η ∞ , u ∞ ) is the solution of the stationary system [( 1 − αη ) u ] x = 0 , η x − α uu x + δ u xx = 0 . But the solutions of this this system are constants. Therefore η ∞ , u ∞ are constants. On the other hand, from the preserved amounts � π � π � π � π − π η ( x , t ) dx = − π η 0 ( x ) dx , u ( x , t ) dx = u 0 ( x ) dx − π − π we conclude that � π � π η ∞ = 1 u ∞ = 1 − π η 0 ( x ) dx , u 0 ( x ) dx . 2 π 2 π − π higidio@ufpr.br UFPR