Some inverse problems for dispersive partial differential equations. - PowerPoint PPT Presentation

Some inverse problems for dispersive partial differential equations. Alberto Mercado Saucedo. Universidad T´ ecnica Federico Santa Mar´ ıa, Valpara´ ıso Chile. 29 Col´ oquio Brasileiro de Matem´ atica, IMPA, 2013 Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 1 / 28

Presentation of the problem The Korteweg-de Vries (KdV) equation y t ( t, x ) + y xxx ( t, x ) + y x ( t, x ) + y ( t, x ) y x ( t, x ) = 0 , is a nonlinear dispersive equation that serves as a mathematical model to study the propagation of long water waves in channels of relatively shallow depth and flat bottom. Here, y ( t, x ) = surface elevation of the water wave at time t and position x. The study of water waves moving over variable topography has been considered. If we denote h = h ( x ) the variations in depth of the channel, then the proposed model becomes (after scaling) 1 p y t ( t, x ) + h 2 ( x ) y xxx ( t, x ) + ( h ( x ) y ( t, x )) x + y ( t, x ) y x ( t, x ) = 0 . (1) p h ( x ) Thus, we are led to consider variable coefficients KdV equations to model the water wave propagation in non-flat channels. Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 2 / 28

Presentation of the problem The Korteweg-de Vries (KdV) equation y t ( t, x ) + y xxx ( t, x ) + y x ( t, x ) + y ( t, x ) y x ( t, x ) = 0 , is a nonlinear dispersive equation that serves as a mathematical model to study the propagation of long water waves in channels of relatively shallow depth and flat bottom. Here, y ( t, x ) = surface elevation of the water wave at time t and position x. The study of water waves moving over variable topography has been considered. If we denote h = h ( x ) the variations in depth of the channel, then the proposed model becomes (after scaling) 1 p y t ( t, x ) + h 2 ( x ) y xxx ( t, x ) + ( h ( x ) y ( t, x )) x + y ( t, x ) y x ( t, x ) = 0 . (1) p h ( x ) Thus, we are led to consider variable coefficients KdV equations to model the water wave propagation in non-flat channels. Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 2 / 28

Presentation of the problem We will deal with the KdV equation with non-constant coefficient a = a ( x ) given by 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , where the initial data y 0 , the source term g , and the functions g 0 , g 1 , g 2 are assumed to be known. In this context, the principal coefficient a = a ( x ) represents the deepness of the bottom of the channel where the water wave propagates. If a > 0 is bounded by below and above, the direct problem is well posed. Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 3 / 28

Presentation of the problem We will deal with the KdV equation with non-constant coefficient a = a ( x ) given by 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , where the initial data y 0 , the source term g , and the functions g 0 , g 1 , g 2 are assumed to be known. In this context, the principal coefficient a = a ( x ) represents the deepness of the bottom of the channel where the water wave propagates. If a > 0 is bounded by below and above, the direct problem is well posed. Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 3 / 28

Presentation of the problem We are concerned with the inverse problem of recovering the shape of the bottom of a channel, from partial knowledge of the solution of 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , Inverse Problem Can we recover a = a ( x ) from some partial knowldege of y = y ( x, t ) ? Inverse Problem (Uniqueness) Given some boundary observations Obs ( y ) , is there a unique a = a ( x ) ? Obs ( y ) = Obs (˜ y ) = ) a = ˜ a ? i.e. Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 4 / 28

Presentation of the problem We are concerned with the inverse problem of recovering the shape of the bottom of a channel, from partial knowledge of the solution of 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , Inverse Problem Can we recover a = a ( x ) from some partial knowldege of y = y ( x, t ) ? Inverse Problem (Uniqueness) Given some boundary observations Obs ( y ) , is there a unique a = a ( x ) ? Obs ( y ) = Obs (˜ y ) = ) a = ˜ a ? i.e. Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 4 / 28

Presentation of the problem We are concerned with the inverse problem of recovering the shape of the bottom of a channel, from partial knowledge of the solution of 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , Inverse Problem Can we recover a = a ( x ) from some partial knowldege of y = y ( x, t ) ? Inverse Problem (Uniqueness) Given some boundary observations Obs ( y ) , is there a unique a = a ( x ) ? Obs ( y ) = Obs (˜ y ) = ) a = ˜ a ? i.e. Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 4 / 28

Presentation of the problem 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , Inverse Problem (Stability) k a � ˜ a k X  C k Obs ( y ) � Obs (˜ y ) k Y ? Inverse Problem (Reconstruction) Given some measurement Obs ( y ) , is it possible to reconstruct the coefficient a = a ( x ) ? In this talk, we are concerned with the stability of the inverse problem. Remark: This kind of inverse problem is called a single-measurement IP Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 5 / 28

Presentation of the problem 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , Inverse Problem (Stability) k a � ˜ a k X  C k Obs ( y ) � Obs (˜ y ) k Y ? Inverse Problem (Reconstruction) Given some measurement Obs ( y ) , is it possible to reconstruct the coefficient a = a ( x ) ? In this talk, we are concerned with the stability of the inverse problem. Remark: This kind of inverse problem is called a single-measurement IP Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 5 / 28

Presentation of the problem 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , Inverse Problem (Stability) k a � ˜ a k X  C k Obs ( y ) � Obs (˜ y ) k Y ? Inverse Problem (Reconstruction) Given some measurement Obs ( y ) , is it possible to reconstruct the coefficient a = a ( x ) ? In this talk, we are concerned with the stability of the inverse problem. Remark: This kind of inverse problem is called a single-measurement IP Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 5 / 28

Recovering the main coefficient in KdV 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , Inverse Problem (Stability) k a � ˜ a k X  C k Obs ( y ) � Obs (˜ y ) k Y ? We hope to get only boundary observations: k y x ( t, 0) � ˜ y x ( t, 0) k , k y xx ( t, 0) � ˜ y xx ( t, 0) k or k y xx ( t, L ) � ˜ y xx ( t, L ) k Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 6 / 28

Recovering the main coefficient in KdV 8 y t + a ( x ) y xxx + y x + yy x = g, 8 ( x, t ) 2 (0 , L ) ⇥ (0 , T ) , > > < y ( t, 0) = g 0 ( t ) , y ( t, L ) = g 1 ( t ) , 8 t 2 (0 , T ) , y x ( t, L ) = g 2 ( t ) , 8 t 2 (0 , T ) , > > : y (0 , x ) = y 0 ( x ) , 8 x 2 (0 , L ) , Inverse Problem (Stability) k a � ˜ a k X  C k Obs ( y ) � Obs (˜ y ) k Y ? We hope to get only boundary observations: k y x ( t, 0) � ˜ y x ( t, 0) k , k y xx ( t, 0) � ˜ y xx ( t, 0) k or k y xx ( t, L ) � ˜ y xx ( t, L ) k Alberto Mercado (UTFSM) Inverse Problems for dispersive PDEs IMPA, 2013 6 / 28