Some new results of global existence for reaction-diffusion-advection systems Michel Pierre Ecole Normale Sup´ erieure de Rennes and Institut de Recherche Math´ ematique de Rennes, France Workshop ”New Trends in Modeling, Control and Inverse Problems” Toulouse, June 16th-19th, 2014.
Introduction: a family of systems ◮ � ∂ t u i + A i u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , u i (0 , · ) = u 0 i ≥ 0 ,
Introduction: a family of systems ◮ � ∂ t u i + A i u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ A i are various ”diffusion-advection” operators, possibly A i = A i ( t )
Introduction: a family of systems ◮ � ∂ t u i + A i u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ A i are various ”diffusion-advection” operators, possibly A i = A i ( t ) ◮ f i : [0 , ∞ ) m �→ I R are regular nonlinearities such that : - (P) : the positivity of the solutions is preserved for all time: f = ( f 1 , ..., f m ) is quasi-positive - (M) : some mass dissipativity conditions holds like � i f i ( u ) ≤ 0
Introduction: a family of systems ◮ � ∂ t u i + A i u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ A i are various ”diffusion-advection” operators, possibly A i = A i ( t ) ◮ f i : [0 , ∞ ) m �→ I R are regular nonlinearities such that : - (P) : the positivity of the solutions is preserved for all time: f = ( f 1 , ..., f m ) is quasi-positive - (M) : some mass dissipativity conditions holds like � i f i ( u ) ≤ 0 ◮ or more general mass control property � i f i ( u ) ≤ C [1 + � i u i ].
A simple choice for the A i ◮ ∂ t u i − d i ∆ u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , where d i ∈ (0 , ∞ ). Local existence of nonnegative solutions on some maximal interval (0 , T ∗ ) always holds for u 0 i ∈ L ∞ (Ω) + .
A simple choice for the A i ◮ ∂ t u i − d i ∆ u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , where d i ∈ (0 , ∞ ). Local existence of nonnegative solutions on some maximal interval (0 , T ∗ ) always holds for u 0 i ∈ L ∞ (Ω) + . ◮ If the d i = d are all equal and � i f i ( u ) ≤ 0, then �� � �� � ∂ t u i − d ∆ u i ≤ 0 , i i so that, ∀ t ∈ (0 , T ∗ )
A simple choice for the A i ◮ ∂ t u i − d i ∆ u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , where d i ∈ (0 , ∞ ). Local existence of nonnegative solutions on some maximal interval (0 , T ∗ ) always holds for u 0 i ∈ L ∞ (Ω) + . ◮ If the d i = d are all equal and � i f i ( u ) ≤ 0, then �� � �� � ∂ t u i − d ∆ u i ≤ 0 , i i so that, ∀ t ∈ (0 , T ∗ ) ◮ � � u 0 � u i ( t ) � L ∞ (Ω) ≤ � i � L ∞ (Ω) i i which implies T ∗ = + ∞ and global existence on [0 , ∞ ).
The known results ∂ t u i − d i ∆ u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , ( S ) ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ In all cases, we keep L 1 (Ω)-estimates uniform in time, namely � � � � ∂ t u i − 0 = f i ( u ) ≤ 0 , Ω Ω i i � � u 0 ⇒ � u i ( t ) � L 1 (Ω) ≤ � i � L 1 (Ω) i i ⇒ ∀ t ∈ [0 , T ∗ ) , max � u 0 � u i ( t ) � L 1 (Ω) ≤ � i � L 1 (Ω) . i i
The known results ∂ t u i − d i ∆ u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , ( S ) ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ In all cases, we keep L 1 (Ω)-estimates uniform in time, namely � � � � ∂ t u i − 0 = f i ( u ) ≤ 0 , Ω Ω i i � � u 0 ⇒ � u i ( t ) � L 1 (Ω) ≤ � i � L 1 (Ω) i i ⇒ ∀ t ∈ [0 , T ∗ ) , max � u 0 � u i ( t ) � L 1 (Ω) ≤ � i � L 1 (Ω) . i i ◮ Negative result: if the d i are not equal, then L ∞ (Ω)-blow up may occur in finite time (in any dimension)+(for any superquadratic growth and high dimension) .
The known results ∂ t u i − d i ∆ u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , ( S ) ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ In all cases, we keep L 1 (Ω)-estimates uniform in time, namely � � � � ∂ t u i − 0 = f i ( u ) ≤ 0 , Ω Ω i i � � u 0 ⇒ � u i ( t ) � L 1 (Ω) ≤ � i � L 1 (Ω) i i ⇒ ∀ t ∈ [0 , T ∗ ) , max � u 0 � u i ( t ) � L 1 (Ω) ≤ � i � L 1 (Ω) . i i ◮ Negative result: if the d i are not equal, then L ∞ (Ω)-blow up may occur in finite time (in any dimension)+(for any superquadratic growth and high dimension) . ◮ Positive results of global existence: two main families.
The known results: 1) strong solutions ∂ t u i − d i ∆ u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , ( S ) ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ Theorem. Assume f = ( f 1 , ..., f m ) satisfies (P),(M) and has a triangular structure which means: � ∀ u ∈ [0 , ∞ ) m , Qf ( u ) ≤ 0 R m ] , [ or Qf ( u ) ≤ b (1+ u i ) , b ∈ I i for some (lower) triangular matrix Q , with nonnegative entries and invertible, and if the growth of the f i is at most polynomial, then the system ( S ) has a global classical solution.
The known results: 1) strong solutions ∂ t u i − d i ∆ u i = f i ( u 1 , ..., u m ) on (0 , ∞ ) × Ω , ( S ) ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ Theorem. Assume f = ( f 1 , ..., f m ) satisfies (P),(M) and has a triangular structure which means: � ∀ u ∈ [0 , ∞ ) m , Qf ( u ) ≤ 0 R m ] , [ or Qf ( u ) ≤ b (1+ u i ) , b ∈ I i for some (lower) triangular matrix Q , with nonnegative entries and invertible, and if the growth of the f i is at most polynomial, then the system ( S ) has a global classical solution. ◮ A typical example with m = 2 where α, β ≥ 1: � 1 � 1 u β f 1 ( u ) = − u α � 2 , f 1 ( u ) ≤ 0 0 f 1 ( u ) + f 2 ( u ) = 0 . Q = 1 u β f 2 ( u ) = u α 1 1 2
The known results: 1) strong solutions 1 u β ∂ t u 1 − d 1 ∆ u 1 = − u α 2 , 1 u β ∂ t u 2 − d 2 ∆ u 2 = u α 2 , ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ We obviously have on the maximum interval (0 , T ∗ ) � u 1 � L ∞ ( Q T ∗ ) ≤ � u 0 1 � L ∞ (Ω) .
The known results: 1) strong solutions 1 u β ∂ t u 1 − d 1 ∆ u 1 = − u α 2 , 1 u β ∂ t u 2 − d 2 ∆ u 2 = u α 2 , ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ We obviously have on the maximum interval (0 , T ∗ ) � u 1 � L ∞ ( Q T ∗ ) ≤ � u 0 1 � L ∞ (Ω) . ◮ Next, a main estimate is that ∂ t u 2 − d 2 ∆ u 2 = − [ ∂ t u 1 − d 1 ∆ u 1 ] implies the existence of C = C ( p , T , Ω) such that: ∀ p ∈ (1 , ∞ ) , � u 2 � L p ( Q T ) ≤ C � u 1 � L p ( Q T ) [ Q T = (0 , T ) × Ω] . Follows from the L p -regularity theory for the heat operator.
The known results: 1) strong solutions 1 u β ∂ t u 1 − d 1 ∆ u 1 = − u α 2 , 1 u β ∂ t u 2 − d 2 ∆ u 2 = u α 2 , ∂ ν u i = 0 on (0 , ∞ ) × ∂ Ω , u i (0 , · ) = u 0 i ≥ 0 , ◮ We obviously have on the maximum interval (0 , T ∗ ) � u 1 � L ∞ ( Q T ∗ ) ≤ � u 0 1 � L ∞ (Ω) . ◮ Next, a main estimate is that ∂ t u 2 − d 2 ∆ u 2 = − [ ∂ t u 1 − d 1 ∆ u 1 ] implies the existence of C = C ( p , T , Ω) such that: ∀ p ∈ (1 , ∞ ) , � u 2 � L p ( Q T ) ≤ C � u 1 � L p ( Q T ) [ Q T = (0 , T ) × Ω] . Follows from the L p -regularity theory for the heat operator. ◮ This implies that u 2 is bounded in L p ( Q T ∗ ) for all p < ∞ ...and also in L ∞ ( Q T ∗ ) thanks to the polynomial 2 . Whence T ∗ = + ∞ . 1 u β growth of u α
The known results: 1)The L p -estimate by duality ◮ ∂ t u 2 − d 2 ∆ u 2 ≤ − [ ∂ t u 1 − d 1 ∆ u 1 ] , u 2 ≥ 0 , implies the existence of C = C ( p , T , Ω) such that: ∀ p ∈ (1 , ∞ ) , � u 2 � L p ( Q T ) ≤ C � u 1 � L p ( Q T ) .
The known results: 1)The L p -estimate by duality ◮ ∂ t u 2 − d 2 ∆ u 2 ≤ − [ ∂ t u 1 − d 1 ∆ u 1 ] , u 2 ≥ 0 , implies the existence of C = C ( p , T , Ω) such that: ∀ p ∈ (1 , ∞ ) , � u 2 � L p ( Q T ) ≤ C � u 1 � L p ( Q T ) . ◮ Solve the dual problem � − ( ∂ t ψ + d 2 ∆ ψ ) = Θ ∈ C ∞ 0 ( Q T ) , Θ ≥ 0 , ψ ( T ) = 0 , ∂ ν ψ = 0 on Σ T .
The known results: 1)The L p -estimate by duality ◮ ∂ t u 2 − d 2 ∆ u 2 ≤ − [ ∂ t u 1 − d 1 ∆ u 1 ] , u 2 ≥ 0 , implies the existence of C = C ( p , T , Ω) such that: ∀ p ∈ (1 , ∞ ) , � u 2 � L p ( Q T ) ≤ C � u 1 � L p ( Q T ) . ◮ Solve the dual problem � − ( ∂ t ψ + d 2 ∆ ψ ) = Θ ∈ C ∞ 0 ( Q T ) , Θ ≥ 0 , ψ ( T ) = 0 , ∂ ν ψ = 0 on Σ T . ◮ � � � ( u 0 1 + u 0 u 2 Θ ≤ 2 ) ψ (0) + ( d 1 − d 2 ) u 1 ∆ ψ. Q T Ω Q T
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