Nonlinear hyperbolic balance laws coupled with ordinary differential - PowerPoint PPT Presentation

Nonlinear hyperbolic balance laws coupled with ordinary differential equations Mauro Garavello Department of Mathematics and Applications University of Milano Bicocca mauro.garavello@unimib.it Joint works with R. Borsche and R. M. Colombo June 28, 2012 HYP2012

PDE-ODE Model  ∂ t u + ∂ x f ( u ) = g ( u ) t > 0 , x > γ ( t )   b ( u ( t , γ ( t ))) = B ( t , w ( t )) t > 0     ˙ w ( t ) = F ( t , u ( t , γ ( t )) , w ( t )) t > 0    γ ( t ) = Π( w ( t )) ˙ t > 0 u ( 0 , x ) = u o ( x ) x > x o      w ( 0 ) = w o     γ ( 0 ) = x o �� t �� b ( u ) = B ( t, w ( t )) �� γ �� ∂ t u + ∂ x f ( u ) = g ( u ) �� x �� u o June 28, 2012 HYP2012

PDE-ODE Model  ∂ t u + ∂ x f ( u ) = g ( u ) t > 0 , x > γ ( t )   b ( u ( t , γ ( t ))) = B ( t , w ( t )) t > 0     ˙ w ( t ) = F ( t , u ( t , γ ( t )) , w ( t )) t > 0    γ ( t ) = Π( w ( t )) ˙ t > 0 u ( 0 , x ) = u o ( x ) x > x o      w ( 0 ) = w o     γ ( 0 ) = x o �� u ∈ Ω ⊆ R n , w ∈ R m t �� b ( u ) = B ( t, w ( t )) �� γ �� ∂ t u + ∂ x f ( u ) = g ( u ) �� x �� u o June 28, 2012 HYP2012

PDE-ODE Model  ∂ t u + ∂ x f ( u ) = g ( u ) t > 0 , x > γ ( t )   b ( u ( t , γ ( t ))) = B ( t , w ( t )) t > 0     ˙ w ( t ) = F ( t , u ( t , γ ( t )) , w ( t )) t > 0    γ ( t ) = Π( w ( t )) ˙ t > 0 u ( 0 , x ) = u o ( x ) x > x o      w ( 0 ) = w o     γ ( 0 ) = x o �� u ∈ Ω ⊆ R n , w ∈ R m t �� f smooth and Df ( u ) strictly hyperbolic b ( u ) = B ( t, w ( t )) �� ∀ u ∈ Ω �� γ �� ∂ t u + ∂ x f ( u ) = g ( u ) �� x �� u o June 28, 2012 HYP2012

PDE-ODE Model  ∂ t u + ∂ x f ( u ) = g ( u ) t > 0 , x > γ ( t )   b ( u ( t , γ ( t ))) = B ( t , w ( t )) t > 0     ˙ w ( t ) = F ( t , u ( t , γ ( t )) , w ( t )) t > 0    γ ( t ) = Π( w ( t )) ˙ t > 0 u ( 0 , x ) = u o ( x ) x > x o      w ( 0 ) = w o     γ ( 0 ) = x o �� u ∈ Ω ⊆ R n , w ∈ R m t �� f smooth and Df ( u ) strictly hyperbolic b ( u ) = B ( t, w ( t )) �� ∀ u ∈ Ω �� γ λ 1 ( u ) < · · · < λ ℓ ( u ) < inf Π − c < �� sup Π + c < λ ℓ + 1 ( u ) < · · · < λ n ( u ) for �� ∂ t u + ∂ x f ( u ) = g ( u ) �� all u ∈ Ω �� x �� u o June 28, 2012 HYP2012

Nonlinear hyperbolic balance laws coupled with ordinary differential - PowerPoint PPT Presentation

Nonlinear hyperbolic balance laws coupled with ordinary differential equations Mauro Garavello Department of Mathematics and Applications University of Milano Bicocca mauro.garavello@unimib.it Joint works with R. Borsche and R. M. Colombo

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