Nonuniqeness in a free boundary problem from combustion Arshak Petrosyan joint with Aaron Yip Department of Mathematics Purdue University West Lafayette, IN 47907, USA SIAM PD07 A. Petrosyan Nonuniqeness in a free boundary problem from combustion
A free boundary problem from combustion One-phase parabolic free boundary problem with fixed gradient condition ▸ Given u ∈ C ( � n ) , u ≥ A. Petrosyan Nonuniqeness in a free boundary problem from combustion
A free boundary problem from combustion One-phase parabolic free boundary problem with fixed gradient condition ▸ Given u ∈ C ( � n ) , u ≥ ▸ Find u ∶ � n × [ , ∞ ) → � , u ≥ : ∆ u − ∂ t u = in { u > } ∣ ∇ u ∣ = ( P ) on ∂ { u > } u ( ⋅ , ) = u on � n A. Petrosyan Nonuniqeness in a free boundary problem from combustion
A free boundary problem from combustion One-phase parabolic free boundary problem with fixed gradient condition ▸ Given u ∈ C ( � n ) , u ≥ ▸ Find u ∶ � n × [ , ∞ ) → � , u ≥ : ∆ u − ∂ t u = in { u > } ∣ ∇ u ∣ = ( P ) on ∂ { u > } u ( ⋅ , ) = u on � n ∣∇ u ∣ = u ≡ ✁ ✁ ☛ u > ∆ u − ∂ t u = A. Petrosyan Nonuniqeness in a free boundary problem from combustion
A free boundary problem from combustion One-phase parabolic free boundary problem with fixed gradient condition ▸ Given u ∈ C ( � n ) , u ≥ ▸ Find u ∶ � n × [ , ∞ ) → � , u ≥ : ∆ u − ∂ t u = in { u > } ∣ ∇ u ∣ = ( P ) on ∂ { u > } u ( ⋅ , ) = u on � n flame front ✁ ✁ ☛ unburnt zone A. Petrosyan Nonuniqeness in a free boundary problem from combustion
A free boundary problem from combustion One-phase parabolic free boundary problem with fixed gradient condition ▸ Appears as the limit as ε → + of the singular perturbation problem ∆ u ε − ∂ t u ε = β ε ( u ε ) in � n × ( , ∞) ( P ε ) u ε (⋅ , ) = u ε on � n , ≈ u and β ε ∈ Lip ( � ) satisfies where u ε ε β ε ( s ) ds = ∫ β ε ≥ , supp β ε = [ , ε ] , A. Petrosyan Nonuniqeness in a free boundary problem from combustion
A free boundary problem from combustion One-phase parabolic free boundary problem with fixed gradient condition ▸ Appears as the limit as ε → + of the singular perturbation problem ∆ u ε − ∂ t u ε = β ε ( u ε ) in � n × ( , ∞) ( P ε ) u ε (⋅ , ) = u ε on � n , ≈ u and β ε ∈ Lip ( � ) satisfies where u ε ε β ε ( s ) ds = ∫ β ε ≥ , supp β ε = [ , ε ] , ▸ Describes the evolution of equidiffusional flames in the limit of high activation energy . A. Petrosyan Nonuniqeness in a free boundary problem from combustion
A free boundary problem from combustion One-phase parabolic free boundary problem with fixed gradient condition ▸ Appears as the limit as ε → + of the singular perturbation problem ∆ u ε − ∂ t u ε = β ε ( u ε ) in � n × ( , ∞) ( P ε ) u ε (⋅ , ) = u ε on � n , ≈ u and β ε ∈ Lip ( � ) satisfies where u ε ε β ε ( s ) ds = ∫ β ε ≥ , supp β ε = [ , ε ] , ▸ Describes the evolution of equidiffusional flames in the limit of high activation energy . ▸ Goes back to Z eldovich and F rank -K amenetski in 1930’s A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Classical solutions How to understand the condition ∣ ∇ u ∣ = on the free boundary ∂ { u > } ? A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Classical solutions How to understand the condition ∣ ∇ u ∣ = on the free boundary ∂ { u > } ? ▸ Supersolutions : ∣ ∇ u ( x , t )∣ ≤ for any ( x , t ) ∈ ∂ { u > } limsup { u > }∋( x , t )→( x , t ) A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Classical solutions How to understand the condition ∣ ∇ u ∣ = on the free boundary ∂ { u > } ? ▸ Supersolutions : ∣ ∇ u ( x , t )∣ ≤ for any ( x , t ) ∈ ∂ { u > } limsup { u > }∋( x , t )→( x , t ) ▸ Subsolutions : { u > }∋( x , t )→( x , t ) ∣ ∇ u ( x , t )∣ ≥ for any ( x , t ) ∈ ∂ { u > } . liminf A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Classical solutions How to understand the condition ∣ ∇ u ∣ = on the free boundary ∂ { u > } ? ▸ Supersolutions : ∣ ∇ u ( x , t )∣ ≤ for any ( x , t ) ∈ ∂ { u > } limsup { u > }∋( x , t )→( x , t ) ▸ Subsolutions : { u > }∋( x , t )→( x , t ) ∣ ∇ u ( x , t )∣ ≥ for any ( x , t ) ∈ ∂ { u > } . liminf ▸ Short time existence and uniqueness, provided ∂ { u > } and u are smooth [ B aconneau -L unardi 04 ] A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Classical solutions How to understand the condition ∣ ∇ u ∣ = on the free boundary ∂ { u > } ? ▸ Supersolutions : ∣ ∇ u ( x , t )∣ ≤ for any ( x , t ) ∈ ∂ { u > } limsup { u > }∋( x , t )→( x , t ) ▸ Subsolutions : { u > }∋( x , t )→( x , t ) ∣ ∇ u ( x , t )∣ ≥ for any ( x , t ) ∈ ∂ { u > } . liminf ▸ Short time existence and uniqueness, provided ∂ { u > } and u are smooth [ B aconneau -L unardi 04 ] ▸ Generally classical solutions will develop singularities afer some time A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Classical solutions How to understand the condition ∣ ∇ u ∣ = on the free boundary ∂ { u > } ? ▸ Supersolutions : ∣ ∇ u ( x , t )∣ ≤ for any ( x , t ) ∈ ∂ { u > } limsup { u > }∋( x , t )→( x , t ) ▸ Subsolutions : { u > }∋( x , t )→( x , t ) ∣ ∇ u ( x , t )∣ ≥ for any ( x , t ) ∈ ∂ { u > } . liminf ▸ Short time existence and uniqueness, provided ∂ { u > } and u are smooth [ B aconneau -L unardi 04 ] ▸ Generally classical solutions will develop singularities afer some time ▸ If u is radially symmetric, the solutions will stay classical until its extinction, i.e when u becomes identically [ G alaktionov -H ulshof -V azquez 97 ] A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Limit solutions Most natural notion of solution of ( P ) reflecting its origin. A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Limit solutions Most natural notion of solution of ( P ) reflecting its origin. → u in the sense ▸ Let u ε − u ∥ L ∞ ( � n ) → , → supp u , ∥ u ε supp u ε A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Limit solutions Most natural notion of solution of ( P ) reflecting its origin. → u in the sense ▸ Let u ε − u ∥ L ∞ ( � n ) → , → supp u , ∥ u ε supp u ε ▸ Solve the approximating problem ( P ε ) ∆ u ε − ∂ t u ε = β ε ( u ε ) in � n × ( , ∞) , u ε (⋅ , ) = u ε . A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Limit solutions Most natural notion of solution of ( P ) reflecting its origin. → u in the sense ▸ Let u ε − u ∥ L ∞ ( � n ) → , → supp u , ∥ u ε supp u ε ▸ Solve the approximating problem ( P ε ) ∆ u ε − ∂ t u ε = β ε ( u ε ) in � n × ( , ∞) , u ε (⋅ , ) = u ε . ▸ Te family { u ε } ε > will be uniformly bounded in C , x ∩ C , / norm on any K ⋐ � n × ( , ∞) [ C affarelli -V t azquez 95 ] A. Petrosyan Nonuniqeness in a free boundary problem from combustion
Limit solutions Most natural notion of solution of ( P ) reflecting its origin. → u in the sense ▸ Let u ε − u ∥ L ∞ ( � n ) → , → supp u , ∥ u ε supp u ε ▸ Solve the approximating problem ( P ε ) ∆ u ε − ∂ t u ε = β ε ( u ε ) in � n × ( , ∞) , u ε (⋅ , ) = u ε . ▸ Te family { u ε } ε > will be uniformly bounded in C , x ∩ C , / norm on any K ⋐ � n × ( , ∞) [ C affarelli -V t azquez 95 ] ▸ Hence, for a subsequence ε = ε j → + , u ε j → u locally uniformly in � n × ( , ∞) A. Petrosyan Nonuniqeness in a free boundary problem from combustion
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