Asymptotic stabilization of the hyperelastic-rod wave equation Giuseppe Maria Coclite Department of Mathematics University of Bari Via E. Orabona 4 70125 Bari (Italy) EMAIL: coclitegm@dm.uniba.it URL: http://www.dm.uniba.it/Members/coclitegm/ Padova HYP 2012 joint work with Prof. F. Ancona (Padova) Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 1 / 32
The Hyperelastic-Rod Wave Equation � � ∂ t u − ∂ 3 2 ∂ x u ∂ 2 xx u + u ∂ 3 txx u + 3 u ∂ x u = γ xxx u t ≥ 0 time x ∈ R space (one-dimensional) u ( t , x ) ∈ R unknown (one-dimensional) γ > 0 is a given constant Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 2 / 32
Physics Hyperelastic-Rod Waves Finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. γ depends on the material constants and the pre-stress of the rod. Dai (1998 - 1998) Dai & Huo (2002) Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 3 / 32
Shallow Water Waves Shallow Water Waves � Depth of the water < < Length of the waves γ = 1 unidirectional shallow water waves u ≡ wave velocity above the bottom flat bottom Camassa-Holm equation Camassa & Holm (1993) Johnson (2002) Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 4 / 32
The Hyperelastic-Rod Wave Equation � � ∂ t u − ∂ 3 2 ∂ x u ∂ 2 xx u + u ∂ 3 txx u + 3 u ∂ x u = γ xxx u is formally equivalent to the elliptic-hyperbolic system xx P + P = 3 − γ u 2 + γ 2 ( ∂ x u ) 2 . − ∂ 2 ∂ t u + γ u ∂ x u + ∂ x P = 0 , 2 Since e −| x | 2 is the Green’s function of the Helmholtz operator − ∂ 2 xx + 1 � 3 − γ � � P ( t , x ) = 1 u ( t , y ) 2 + γ e −| x − y | 2 ( ∂ x u ( t , y )) 2 dy , 2 2 R � 3 − γ � � ∂ x P ( t , x ) = 1 u ( t , y ) 2 + γ e −| x − y | sign ( y − x ) 2 ( ∂ x u ( t , y )) 2 dy . 2 2 R Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 5 / 32
Asymptotic Stabilization Problem Find an operator f : H 1 ( R ) − → H − 1 ( R ) such that for every initial condition u 0 ∈ H 1 ( R ) the solution of the Cauchy problem � � ∂ t u − ∂ 3 2 ∂ x u ∂ 2 xx u + u ∂ 3 txx u + 3 u ∂ x u = γ xxx u + f [ u ] u ( 0 , x ) = u 0 ( x ) decays as t − → ∞ , i.e., lim →∞ u ( t , x ) = 0 . t − Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 6 / 32
Physics Damp the waves on hyperelastic rods f [ u ] ≡ external force H 1 Regularity Weak solutions u ( t , · ) ∈ H 1 ( R ) u ( t , · ) is bounded and continuous Blow-up of ∂ x u More regularity on the initial conditions gives more regularity of the solutions (there are several smooth solutions) Solitons and peakons are weak solutions Literature Glass (2008): source compactly sopported type feedback, H 2 solutions Perrollaz (2010): boundary feedback, H 2 solutions Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 7 / 32
Our feedback law f [ u ] = − λ ( 1 − ∂ 2 xx ) u , λ > 0 � � ∂ t u − ∂ 3 2 ∂ x u ∂ 2 xx u + u ∂ 3 − λ ( 1 − ∂ 2 txx u + 3 u ∂ x u = γ xxx u xx ) u � � ∂ t u + γ u ∂ x u + ∂ x P = − λ u 2 u 2 + γ 2 ( ∂ x u ) 2 xx P + P = 3 − γ − ∂ 2 Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 8 / 32
Formal Energy Estimate � ∂ t u + γ u ∂ x u + ∂ x P = − λ u 2 u 2 + γ xx P + P = 3 − γ 2 ( ∂ x u ) 2 − ∂ 2 ⇓ � � � γ � u 2 + ( ∂ x u ) 2 � u 2 + ( ∂ x u ) 2 � 2 u ( ∂ x u ) 2 − 1 − γ u 3 + uP + ∂ x = − λ ∂ t 2 2 The total energy � � u ( t , x ) 2 + ( ∂ x u ( t , x )) 2 � E ( t ) := � u ( t , · ) � 2 H 1 ( R ) = dx R satisfies the following ordinary differential equation d dt E ( t ) = − 2 λ E ( t ) and therefore E ( t ) = E ( 0 ) e − 2 λ t , t ≥ 0 . Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 9 / 32
Definition (Weak Solutions) A function u : [ 0 , ∞ ) × R − → R is a weak solution of the Cauchy problem � � ∂ t u − ∂ 3 2 ∂ x u ∂ 2 xx u + u ∂ 3 u − λ ( 1 − ∂ 2 txx u + 3 u ∂ x u = γ xx ) u xxx u u ( 0 , x ) = u 0 ( x ) if u is continuous; u ( t , · ) ∈ H 1 ( R ) at every t ∈ [ 0 , ∞ ) ; → u ( t , · ) is Lipschitz continuous from [ 0 , ∞ ) into L 2 ( R ) the map t − and satisfies the initial condition together with the following equality between functions in L 2 ( R ) : d dt u = − γ u ∂ x u − ∂ x P − λ u , for a.e. t ∈ [ 0 , ∞ ) . Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 10 / 32
The Main Result Let γ, λ > 0 be fixed. There exists a semigroup S : [ 0 , ∞ ) × H 1 ( R ) − → H 1 ( R ) such that the following properties hold. For every u 0 ∈ H 1 ( R ) , u ( t , x ) = S t ( u 0 )( x ) is a weak solution of � � ∂ t u − ∂ 3 2 ∂ x u ∂ 2 xx u + u ∂ 3 u − λ ( 1 − ∂ 2 txx u + 3 u ∂ x u = γ xx ) u , xxx u u ( 0 , x ) = u 0 ( x ) . E ( t ) ≤ E ( 0 ) e − 2 λ t , t ≥ 0 . For every u 0 ∈ H 1 ( R ) there exists a constant C > 0 such that � � 1 + 1 ∂ x S t ( u 0 )( x ) ≤ C t > 0 . , t For every { u 0 , n } n ⊂ H 1 ( R ) and u 0 ∈ H 1 ( R ) → u 0 in H 1 ( R ) = → S ( u 0 ) in L ∞ u 0 , n − ⇒ S ( u 0 , n ) − loc (( 0 , ∞ ) × R ) . Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 11 / 32
Remark Semigroup of solutions no uniqueness of weak solutions solitons interaction conservative and dissipative solutions Energy exponential decay Oleinik type estimate ∂ x u is bounded from above ∂ x u may go to −∞ S is not continuous as a map with values in H 1 . Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 12 / 32
Strategy We introduce a new set of variable that solves a semilinear system of ordinary differential equations. Short time existence of solutions for the semilinear system. An energy estimate gives the existence of global in time solutions for the semilinear system. Continuous dependence with respect to the initial conditions for the semilinear system. We come back to the original variables and prove our result. Bressan & Constantin (Anal. Appl. - 2007) Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 13 / 32
New variables Let u be a smooth solution of � � ∂ t u − ∂ 3 2 ∂ x u ∂ 2 xx u + u ∂ 3 u − λ ( 1 − ∂ 2 txx u + 3 u ∂ x u = γ xxx u xx ) u u ( 0 , x ) = u 0 ( x ) . Therefore ( u , P ) is a smooth solution of ∂ t u + γ u ∂ x u + ∂ x P = − λ u 2 u 2 + γ xx P + P = 3 − γ 2 ( ∂ x u ) 2 − ∂ 2 u ( 0 , x ) = u 0 ( x ) . Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 14 / 32
Energy variable ξ ∈ R . The map � y � 1 + ( ∂ x u 0 ) 2 � y ∈ R �− → dx 0 is continuous, increasing, and goes to ∞ and −∞ as y − → ∞ and y − → −∞ , respectively. So we can define implicitly the function y 0 = y 0 ( ξ ) by the relation � y 0 ( ξ ) � 1 + ( ∂ x u 0 ) 2 � dx = ξ, ξ ∈ R . 0 ξ plays the role of a Lagrangian variable (it is constant along characteristics) Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 15 / 32
Characteristic curve t �− → y ( t , ξ ) ∂ t y ( t , ξ ) = γ u ( t , y ( t , ξ )) , y ( 0 , ξ ) = y 0 ( ξ ) . Notation u ( t , ξ ) := u ( t , y ( t , ξ )) , P ( t , ξ ) := P ( t , y ( t , ξ )) . New variables v = v ( t , ξ ) and q = q ( t , ξ ) q := ( 1 + ( ∂ x u ) 2 ) ∂ ξ y . v := 2 arctan ( ∂ x u ) , v is bounded q ≥ 0 Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 16 / 32
The semilinear system for u = u ( t , ξ ) , v = v ( t , ξ ) , q = q ( t , ξ ) ∂ t u = − ∂ x P − λ u � � � − λ sin ( v ) ( 1 + cos ( v )) − γ sin 2 � v 2 u 2 − P 3 − γ ∂ t v = 2 � � sin ( v ) q − 2 λ sin 2 � v � q 2 u 2 − P + γ 3 − γ ∂ t q = 2 2 u ( 0 , ξ ) = u 0 ( y 0 ( ξ )) v ( 0 , ξ ) = 2 arctan ( ∂ x u 0 ( y 0 ( ξ ))) q ( 0 , ξ ) = 1 Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 17 / 32
The nonlocal term P = P ( t , ξ ) � � � ξ ′ cos 2 � v ( t , s ) � � � � P ( t , ξ ) = 1 − � q ( t , s ) ds � × 2 e ξ 2 R � 3 − γ � v ( t , ξ ′ ) � � v ( t , ξ ′ ) �� + γ u ( t , ξ ′ ) 2 cos 2 2 sin 2 × × 2 2 2 × q ( t , ξ ′ ) d ξ ′ , � � � ξ ′ cos 2 � v ( t , s ) � � � � ∂ x P ( t , ξ ) = 1 − � q ( t , s ) ds � × 2 ξ e 2 R � ξ − ξ ′ � × × sign � 3 − γ � v ( t , ξ ′ ) � � v ( t , ξ ′ ) �� + γ u ( t , ξ ′ ) 2 cos 2 2 sin 2 × × 2 2 2 × q ( t , ξ ′ ) d ξ ′ . Giuseppe Maria Coclite (Bari) Asymptotic stabilization Padova HYP 2012 18 / 32
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