Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Take the Euler equation and remove terms which we deem unimportant. Take the important terms and try to reduce their complexity (remove geometry, dimensionality, parity, etc.) Prove either global regularity or blow-up for the reduced equation. Many examples: Constantin, Lax, Majda, De Gregorio, Cordoba 2 , Fontelos, Hou, Li, Lei, Luo, Kiselev, Sverak, Choi, Yao,... Approach B: Search for special solutions to the equation. Some examples of this approach are: Self-similar ansatz (Sverak, Chae, Tsai, Shvydkoy, Xue, Hou,...) Stagnation point ansatz (Stuart, Childress, Gibbon, Constantin, Saxton, Wu, Sarria,...)
Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Take the Euler equation and remove terms which we deem unimportant. Take the important terms and try to reduce their complexity (remove geometry, dimensionality, parity, etc.) Prove either global regularity or blow-up for the reduced equation. Many examples: Constantin, Lax, Majda, De Gregorio, Cordoba 2 , Fontelos, Hou, Li, Lei, Luo, Kiselev, Sverak, Choi, Yao,... Approach B: Search for special solutions to the equation. Some examples of this approach are: Self-similar ansatz (Sverak, Chae, Tsai, Shvydkoy, Xue, Hou,...) Stagnation point ansatz (Stuart, Childress, Gibbon, Constantin, Saxton, Wu, Sarria,...) Other infinite energy solutions (Childress, Gibbon,...)
Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Take the Euler equation and remove terms which we deem unimportant. Take the important terms and try to reduce their complexity (remove geometry, dimensionality, parity, etc.) Prove either global regularity or blow-up for the reduced equation. Many examples: Constantin, Lax, Majda, De Gregorio, Cordoba 2 , Fontelos, Hou, Li, Lei, Luo, Kiselev, Sverak, Choi, Yao,... Approach B: Search for special solutions to the equation. Some examples of this approach are: Self-similar ansatz (Sverak, Chae, Tsai, Shvydkoy, Xue, Hou,...) Stagnation point ansatz (Stuart, Childress, Gibbon, Constantin, Saxton, Wu, Sarria,...) Other infinite energy solutions (Childress, Gibbon,...)
Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Approach B: Search for special solutions to the equation. Remarks:
Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Approach B: Search for special solutions to the equation. Remarks: It is not clear how to go back from the models found in Approach A to the Euler equation
Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Approach B: Search for special solutions to the equation. Remarks: It is not clear how to go back from the models found in Approach A to the Euler equation Infinite energy solutions (of Approach B) are very unstable and predict blow-up even in 2d!
Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Approach B: Search for special solutions to the equation. Remarks: It is not clear how to go back from the models found in Approach A to the Euler equation Infinite energy solutions (of Approach B) are very unstable and predict blow-up even in 2d! It seems to be very difficult to find self-similar solutions due to a lack of compactness in the Euler equation. Most results on self-similar solutions are towards ruling them out (except recent works of Elling and then Vishik).
Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Approach B: Search for special solutions to the equation. Remarks: It is not clear how to go back from the models found in Approach A to the Euler equation Infinite energy solutions (of Approach B) are very unstable and predict blow-up even in 2d! It seems to be very difficult to find self-similar solutions due to a lack of compactness in the Euler equation. Most results on self-similar solutions are towards ruling them out (except recent works of Elling and then Vishik). Remark*:
Singularity formation in incompressible fluids Basic Ideas Two Basic Approaches to the Global Regularity Problem Approach A: Find reduced models in a (seemingly) ”ad-hoc” way. Approach B: Search for special solutions to the equation. Remarks: It is not clear how to go back from the models found in Approach A to the Euler equation Infinite energy solutions (of Approach B) are very unstable and predict blow-up even in 2d! It seems to be very difficult to find self-similar solutions due to a lack of compactness in the Euler equation. Most results on self-similar solutions are towards ruling them out (except recent works of Elling and then Vishik). Remark*: One way to salvage Approach A is to try to prove stability of the blow-ups found after the ”ad-hoc” reductions.
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω .
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . We are going to follow ”Approach B,” which is to search for very special types of solutions by imposing a high degree of natural symmetries on the solution.
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . We are going to follow ”Approach B,” which is to search for very special types of solutions by imposing a high degree of natural symmetries on the solution. Using ”scale-invariant” solutions, which we will discuss soon, we will show the following theorem:
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . We are going to follow ”Approach B,” which is to search for very special types of solutions by imposing a high degree of natural symmetries on the solution. Using ”scale-invariant” solutions, which we will discuss soon, we will show the following theorem: Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which:
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . We are going to follow ”Approach B,” which is to search for very special types of solutions by imposing a high degree of natural symmetries on the solution. Using ”scale-invariant” solutions, which we will discuss soon, we will show the following theorem: Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which: The 3D Euler equation is locally well-posed (and the various well-known blow-up criteria hold).
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . We are going to follow ”Approach B,” which is to search for very special types of solutions by imposing a high degree of natural symmetries on the solution. Using ”scale-invariant” solutions, which we will discuss soon, we will show the following theorem: Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which: The 3D Euler equation is locally well-posed (and the various well-known blow-up criteria hold). 2D solutions are global.
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . We are going to follow ”Approach B,” which is to search for very special types of solutions by imposing a high degree of natural symmetries on the solution. Using ”scale-invariant” solutions, which we will discuss soon, we will show the following theorem: Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which: The 3D Euler equation is locally well-posed (and the various well-known blow-up criteria hold). 2D solutions are global. There are finite-energy solutions (truly 3 D solutions) which blow-up in finite time.
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which: The 3D Euler equation is locally well-posed (and the various well-known blow-up criteria hold). 2D solutions are global. There are finite-energy solutions (truly 3 D solutions) which blow-up in finite time. Remarks :
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which: The 3D Euler equation is locally well-posed (and the various well-known blow-up criteria hold). 2D solutions are global. There are finite-energy solutions (truly 3 D solutions) which blow-up in finite time. Remarks : Compact domains with similar behavior are OK too.
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which: The 3D Euler equation is locally well-posed (and the various well-known blow-up criteria hold). 2D solutions are global. There are finite-energy solutions (truly 3 D solutions) which blow-up in finite time. Remarks : Compact domains with similar behavior are OK too. The construction is based heavily on properly introducing and understanding the dynamics of ”scale-invariant solutions”
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which: The 3D Euler equation is locally well-posed (and the various well-known blow-up criteria hold). 2D solutions are global. There are finite-energy solutions (truly 3 D solutions) which blow-up in finite time. Remarks : Compact domains with similar behavior are OK too. The construction is based heavily on properly introducing and understanding the dynamics of ”scale-invariant solutions” The solutions and domain are not ”smooth” but they are ”strong.”
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions Theorem Let ǫ > 0 and Ω ǫ = { x ∈ R 3 : (1 + ǫ | x 3 | ) 2 ≤ ( x 2 1 + x 2 2 ) } . Then there is a space X ⊂ W 1 , ∞ for which: The 3D Euler equation is locally well-posed (and the various well-known blow-up criteria hold). 2D solutions are global. There are finite-energy solutions (truly 3 D solutions) which blow-up in finite time. Remarks : Compact domains with similar behavior are OK too. The construction is based heavily on properly introducing and understanding the dynamics of ”scale-invariant solutions” The solutions and domain are not ”smooth” but they are ”strong.” Despite this drawback, 2 D solutions are global which means that the blow-up is not solely coming from the setting but rather from the 3 D Euler equation itself.
Singularity formation in incompressible fluids Basic Ideas Scaling Invariant Solutions ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . Recall that whenever λ > 0 and Q is an orthogonal matrix ( QQ T = I ) and u ( x , t ) is a solution to the Euler equation, 1 Q T u ( Qx , t ) λ u ( λ x , t ) and are solutions as well.
Singularity formation in incompressible fluids Basic Ideas Special Solutions to Fluid Equations ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . 1 Q T u ( Qx , t ) λ u ( λ x , t ) and are solutions as well. Thus, formally if u 0 ( x ) = 1 λ u 0 ( λ x ) and u 0 ( Qx ) = Qu 0 ( x ) for some class of orthogonal matrices Q and λ > 0, then u will obey the same symmetries so long as the solution exists. Maybe then we will have enough control on solutions to say something.
Singularity formation in incompressible fluids Basic Ideas Special Solutions to Fluid Equations ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . 1 Q T u ( Qx , t ) λ u ( λ x , t ) and are solutions as well. Thus, formally if u 0 ( x ) = 1 λ u 0 ( λ x ) and u 0 ( Qx ) = Qu 0 ( x ) for some class of orthogonal matrices Q and λ > 0, then u will obey the same symmetries so long as the solution exists. Maybe then we will have enough control on solutions to say something. These claims require a versitile uniqueness theorem.
Singularity formation in incompressible fluids Basic Ideas Special Solutions to Fluid Equations ∂ t u + u · ∇ u + ∇ p = 0 , div( u ) = 0 , u | t =0 = u 0 , u · n = 0 on ∂ Ω . 1 Q T u ( Qx , t ) λ u ( λ x , t ) and are solutions as well. Thus, formally if u 0 ( x ) = 1 λ u 0 ( λ x ) and u 0 ( Qx ) = Qu 0 ( x ) for some class of orthogonal matrices Q and λ > 0, then u will obey the same symmetries so long as the solution exists. Maybe then we will have enough control on solutions to say something. These claims require a versitile uniqueness theorem. Using the rotational symmetry of the equation is classical: if we assume u 0 ( Qx ) = Qu 0 ( x ) for all rotational matrices fixing the z − axis we just get the axi-symmetric 3 D Euler equation.
Singularity formation in incompressible fluids Basic Ideas Special Solutions to Fluid Equations If u solves the incompressible Euler equation then 1 Q T u ( Qx , t ) λ u ( λ x , t ) and are solutions as well. Formally if u 0 ( x ) = 1 λ u 0 ( λ x ) and u 0 ( Qx ) = Qu 0 ( x ) for some class of orthogonal matrices Q and λ > 0, then u will obey the same symmetries so long as the solution exists. Maybe then we will have enough control on solutions to say something. Using the rotational symmetry of the equation is classical: if we assume u 0 ( Qx ) = Qu 0 ( x ) for all rotational matrices fixing the z − axis we just get the axi-symmetric 3 D Euler equation. Using the Scaling symmetry of the equation comes with many problems: 1 If we assume λ u 0 ( λ x ) = u 0 ( x ) for all x and λ then u 0 is automatically growing at infinity and, at best, Lipschitz continuous in space.
Singularity formation in incompressible fluids Basic Ideas Scale-Invariant Data What happens if one tries to take data of the following form? λ u 0 ( x λ ) ≡ u 0 ( x ) , for all x and λ > 0. We call such data scale-invariant .
Singularity formation in incompressible fluids Basic Ideas Scale-Invariant Data One can try take data of the following form: λ u 0 ( x λ ) ≡ u 0 ( x ) , for all x and λ > 0. We call such data scale-invariant . Formally, using scaling, we see that λ u ( x λ, t ) = u ( x , t ) for all x and λ . Problems:
Singularity formation in incompressible fluids Basic Ideas Scale-Invariant Data Idea: One can try take data of the following form: λ u 0 ( x λ ) ≡ u 0 ( x ) , for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λ u ( x λ, t ) = u ( x , t ) for all x and λ . Problems: (A) Such data is necessarily only Lipschitz continuous (i.e. outside of known well-posedness classes).
Singularity formation in incompressible fluids Basic Ideas Scale-Invariant Data One can try take data of the following form: λ u 0 ( x λ ) ≡ u 0 ( x ) , for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λ u ( x λ, t ) = u ( x , t ) for all x and λ . Problems: (A) Such data is necessarily only Lipschitz continuous (i.e. outside of known well-posedness classes). (B) Such data has linearly growing velocity field (even if C ∞ , we don’t have a uniqueness theory for solutions with linearly growing velocity).
Singularity formation in incompressible fluids Basic Ideas Scale-Invariant Data Idea: One can try take data of the following form: λ u 0 ( x λ ) ≡ u 0 ( x ) , for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λ u ( x λ, t ) = u ( x , t ) for all x and λ . Problems: (A) Such data is necessarily only Lipschitz continuous (i.e. outside of known well-posedness classes). (B) Such data has linearly growing velocity field (even if C ∞ , we don’t have a uniqueness theory for solutions with linearly growing velocity). Remark 1: For the problem of existence/uniqueness for growing velocity, the the works of Benedetto, Marchioro, Pulvirenti, Serfati, Kelliher, Cozzi-Kelliher,... Remark 2: Both of these problems are due to the non-local pressure.
Singularity formation in incompressible fluids Basic Ideas Scale-Invariant Data Idea: One can try take data of the following form: λ u 0 ( x λ ) ≡ u 0 ( x ) , for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λ u ( x λ, t ) = u ( x , t ) for all x and λ . Problems: (A) Such data is necessarily only Lipschitz continuous (i.e. outside of known well-posedness classes). (B) Such data has linearly growing velocity field (even if C ∞ , we don’t have a uniqueness theory for solutions with linearly growing velocity). (C) What would such solutions say about finite energy solutions?
Singularity formation in incompressible fluids Scale-Invariant Solutions The 2D Euler Equation with Non-decaying Vorticity Let us see how things work in 2 D : ∂ t ω + u · ∇ ω = 0 , u = ∇ ⊥ (∆) − 1 ω.
Singularity formation in incompressible fluids Scale-Invariant Solutions The 2D Euler Equation with Non-decaying Vorticity Let us see how things work in 2 D : ∂ t ω + u · ∇ ω = 0 , u = ∇ ⊥ (∆) − 1 ω. Formally, let’s believe that if ω 0 is scale invariant (0 − homogeneous in space), then ω ( t ) remains as such. Write: ω ( r , θ, t ) = g ( θ, t ). ∆ − 1 ω = r 2 G ( θ, t ).
Singularity formation in incompressible fluids Scale-Invariant Solutions The 2D Euler Equation with Non-decaying Vorticity Let us see how things work in 2 D : ∂ t ω + u · ∇ ω = 0 , u = ∇ ⊥ (∆) − 1 ω. Formally, let’s believe that if ω 0 is scale invariant (0 − homogeneous in space), then ω ( t ) remains as such. Write: ω ( r , θ, t ) = g ( θ, t ). ∆ − 1 ω = r 2 G ( θ, t ). With this ansatz, the 2 D Euler system collapses to an active scalar equation on S 1 : ∂ t g + 2 G ∂ θ g = 0 , 4 G + ∂ θθ G = g .
Singularity formation in incompressible fluids Scale-Invariant Solutions The 2D Euler Equation with Non-decaying Vorticity Let us see how things work in 2 D : ∂ t ω + u · ∇ ω = 0 , u = ∇ ⊥ (∆) − 1 ω. Formally, let’s believe that if ω 0 is scale invariant (0 − homogeneous in space), then ω ( t ) remains as such. Write: ω ( r , θ, t ) = g ( θ, t ). ∆ − 1 ω = r 2 G ( θ, t ). ∂ t g + 2 G ∂ θ g = 0 , 4 G + ∂ θθ G = g . Let us note that to solve the second equation, we need either to be on a thin domain or to look for solutions with high periodicity and both of these are OK assumptions to make. All of this is formal. Now let’s write the theorem that makes this rigorous:
Singularity formation in incompressible fluids Scale-Invariant Solutions The 2D Euler Equation with Non-decaying Vorticity Let us see how things work in 2 D : ∂ t ω + u · ∇ ω = 0 , u = ∇ ⊥ (∆) − 1 ω. Formally, let’s believe that if ω 0 is scale invariant (0 − homogeneous in space), then ω ( t ) remains as such. Write: ω ( r , θ, t ) = g ( θ, t ). ∆ − 1 ω = r 2 G ( θ, t ). ∂ t g + 2 G ∂ θ g = 0 , 4 G + ∂ θθ G = g . Let us note that to solve the second equation, we need either to be on a thin domain or to look for solutions with high periodicity and both of these are OK assumptions to make. All of this is formal. Now let’s write the theorem that makes this rigorous: Theorem (E., Jeong, 2016, to appear in CPAM) Let ω 0 ∈ L ∞ ( R 2 ) be m − fold symmetric for some m ≥ 3 . Then, there exists a unique global solution to 2 D Euler in the class C ( R : L ∞ w ∗ ( R 2 )) with ω m − fold symmetric and ω | t =0 = ω 0 .
Singularity formation in incompressible fluids Scale-Invariant Solutions The 2D Euler Equation with Non-decaying Vorticity Let us see how things work in 2 D : ∂ t ω + u · ∇ ω = 0 , u = ∇ ⊥ (∆) − 1 ω. Theorem (E., Jeong, 2016, to appear in CPAM) Let ω 0 ∈ L ∞ ( R 2 ) be m − fold symmetric for some m ≥ 3 . Then, there exists a unique global solution to 2 D Euler in the class C ( R : L ∞ w ∗ ( R 2 )) with ω m − fold symmetric and ω | t =0 = ω 0 . Corollary Let ω 0 ( r , θ ) = g 0 ( θ ) ∈ L ∞ ( S 1 ) be 2 π m periodic for some m ≥ 3 . Then, the unique global solution to 2D Euler ω ( t , r , θ ) = g ( θ, t ) and g satisfies the following PDE system: ∂ t g + 2 G ∂ θ g = 0 , 4 G + ∂ θθ G = g .
Singularity formation in incompressible fluids Scale-Invariant Solutions The 2D Euler Equation with Non-decaying Vorticity Recall the 2D Euler system: ∂ t ω + u · ∇ ω = 0 u = ∇ ⊥ (∆) − 1 ω Theorem (E., Jeong, 2016) Let ω 0 ∈ L ∞ ( R 2 ) be m − fold symmetric for some m ≥ 3 . Then, there exists a unique global solution to 2 D Euler in the class C ( R : L ∞ w ∗ ( R 2 )) with ω m − fold symmetric and ω | t =0 = ω 0 . Corollary Let ω 0 ( r , θ ) = g 0 ( θ ) ∈ L ∞ ( S 1 ) be 2 π m periodic for some m ≥ 3 . Then, the unique global solution to 2D Euler must satisfy ω ( t , r , θ ) = g ( θ, t ) and g satisfies the following PDE system: ∂ t g + 2 G ∂ θ g = 0 , 4 G + ∂ θθ G = g . Remark: The elliptic problem 4 G + ∂ θθ G = g can be solved since m ≥ 3.
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) :
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) : Scale-invariant solutions to 2D Euler are necessarily globally regular.
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) : Scale-invariant solutions to 2D Euler are necessarily globally regular. Spatial derivatives of g may grow at most exponentially as t → ∞ .
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) : Scale-invariant solutions to 2D Euler are necessarily globally regular. Spatial derivatives of g may grow at most exponentially as t → ∞ . As of now, we have that a large class of solutions have derivatives which grow quadratically-in-time (and not faster).
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) : Scale-invariant solutions to 2D Euler are necessarily globally regular. Spatial derivatives of g may grow at most exponentially as t → ∞ . As of now, we have that a large class of solutions have derivatives which grow quadratically-in-time (and not faster). The exponential bound can be shown to be sharp if there are solid boundaries.
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) : Scale-invariant solutions to 2D Euler are necessarily globally regular. Spatial derivatives of g may grow at most exponentially as t → ∞ . As of now, we have that a large class of solutions have derivatives which grow quadratically-in-time (and not faster). The exponential bound can be shown to be sharp if there are solid boundaries. We conjecture that without boundaries, exponential growth is impossible.
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) : Scale-invariant solutions to 2D Euler are necessarily globally regular. Spatial derivatives of g may grow at most exponentially as t → ∞ . As of now, we have that a large class of solutions have derivatives which grow quadratically-in-time (and not faster). The exponential bound can be shown to be sharp if there are solid boundaries. We conjecture that without boundaries, exponential growth is impossible.
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) : Scale-invariant solutions to 2D Euler are necessarily globally regular. Spatial derivatives of g may grow at most exponentially as t → ∞ . As of now, we have that a large class of solutions have derivatives which grow quadratically-in-time (and not faster). The exponential bound can be shown to be sharp if there are solid boundaries. We conjecture that without boundaries, exponential growth is impossible. Remark 2 (Regarding Some Special Solutions) :
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Remark 1 (Regarding Norm Growth as t → ∞ ) : Scale-invariant solutions to 2D Euler are necessarily globally regular. Spatial derivatives of g may grow at most exponentially as t → ∞ . As of now, we have that a large class of solutions have derivatives which grow quadratically-in-time (and not faster). The exponential bound can be shown to be sharp if there are solid boundaries. We conjecture that without boundaries, exponential growth is impossible. Remark 2 (Regarding Some Special Solutions) : It is possible to construct periodic and quasi-periodic solutions to the system.
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler Next, let us suppose that we had a solution to this system: ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g for which it is known that | ∂ θ g | L ∞ → ∞ as t → ∞ .
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler Next, let us suppose that we had a solution to this system: ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g for which it is known that | ∂ θ g | L ∞ → ∞ as t → ∞ . Question: Is it possible to construct compactly supported solutions to the 2 D Euler equation exhibiting this same behavior?
Singularity formation in incompressible fluids Scale-Invariant Solutions Some Remarks on Scale-Invariant Solutions to 2D Euler Next, let us suppose that we had a solution to this system: ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g for which it is known that | ∂ θ g | L ∞ → ∞ as t → ∞ . General Principle: Whenever it is known that a scale-invariant solution experiences ”growth”, it can be shown that there are compactly supported solutions which grow at least as fast.
Singularity formation in incompressible fluids Cut-off Argument Propagation of Angular Regularity The first step towards cutting off scaling invariant solutions is to define the following scale of spaces for 0 ≤ α ≤ 1: || x | α f ( x ) − | y | α f ( y ) | C 0 ,α = | f | L ∞ + | f | ˚ sup . | x − y | α | x − y | < 1
Singularity formation in incompressible fluids Cut-off Argument Propagation of Angular Regularity The first step towards cutting off scaling invariant solutions is to define the following scale of spaces for 0 ≤ α ≤ 1: || x | α f ( x ) − | y | α f ( y ) | | f | ˚ C 0 ,α = | f | L ∞ + sup . | x − y | α | x − y | < 1 Examples: If f ( r , θ ) = g ( θ ), f ∈ ˚ C 0 ,α ( R 2 ) if and only if g ∈ C α ( S 1 )
Singularity formation in incompressible fluids Cut-off Argument Propagation of Angular Regularity The first step towards cutting off scaling invariant solutions is to define the following scale of spaces for 0 ≤ α ≤ 1: || x | α f ( x ) − | y | α f ( y ) | | f | ˚ C 0 ,α = | f | L ∞ + sup . | x − y | α | x − y | < 1 Examples: If f ( r , θ ) = g ( θ ), f ∈ ˚ C 0 ,α ( R 2 ) if and only if g ∈ C α ( S 1 ) C 0 ,α for any α > 0. If f ( x ) = sin(log( x )) , then f ∈ ˚ x ) �∈ ˚ C 0 , 1 ( R ) while sin( 1
Singularity formation in incompressible fluids Cut-off Argument Propagation of Angular Regularity The first step towards cutting off scaling invariant solutions is to define the following scale of spaces for 0 ≤ α ≤ 1: || x | α f ( x ) − | y | α f ( y ) | | f | ˚ C 0 ,α = | f | L ∞ + sup . | x − y | α | x − y | < 1 Examples: If f ( r , θ ) = g ( θ ), f ∈ ˚ C 0 ,α ( R 2 ) if and only if g ∈ C α ( S 1 ) C 0 ,α for any α > 0. If f ( x ) = sin(log( x )) , then f ∈ ˚ x ) �∈ ˚ C 0 , 1 ( R ) while sin( 1 c ( R 2 ) then f ∈ ˚ If f ∈ C α C 0 ,α .
Singularity formation in incompressible fluids Cut-off Argument Propagation of Angular Regularity The first step towards cutting off scaling invariant solutions is to define the following scale of spaces for 0 ≤ α ≤ 1: || x | α f ( x ) − | y | α f ( y ) | | f | ˚ C 0 ,α = | f | L ∞ + sup . | x − y | α | x − y | < 1 Examples: If f ( r , θ ) = g ( θ ), f ∈ ˚ C 0 ,α ( R 2 ) if and only if g ∈ C α ( S 1 ) C 0 ,α for any α > 0. If f ( x ) = sin(log( x )) , then f ∈ ˚ x ) �∈ ˚ C 0 , 1 ( R ) while sin( 1 c ( R 2 ) then f ∈ ˚ If f ∈ C α C 0 ,α . C 0 ,α functions which are m − fold symmetric on Denote by ˚ the space of ˚ C 0 ,α m R 2 .
Singularity formation in incompressible fluids Cut-off Argument Propagation of Angular Regularity The first step towards cutting off scaling invariant solutions is to define the following scale of spaces for 0 ≤ α ≤ 1: || x | α f ( x ) − | y | α f ( y ) | | f | ˚ C 0 ,α = | f | L ∞ + sup . | x − y | α | x − y | < 1 Examples: If f ( r , θ ) = g ( θ ), f ∈ ˚ C 0 ,α ( R 2 ) if and only if g ∈ C α ( S 1 ) C 0 ,α for any α > 0. If f ( x ) = sin(log( x )) , then f ∈ ˚ x ) �∈ ˚ C 0 , 1 ( R ) while sin( 1 c ( R 2 ) then f ∈ ˚ If f ∈ C α C 0 ,α . C 0 ,α functions which are m − fold symmetric on Denote by ˚ the space of ˚ C 0 ,α m R 2 . Lemma D 2 ∆ − 1 : ˚ → ˚ C 0 ,α C 0 ,α for all 0 < α < 1 and m ≥ 3 . m m
Singularity formation in incompressible fluids Cut-off Argument Propagation of Angular Regularity The first step towards cutting off scaling invariant solutions is to define the following scale of spaces for 0 ≤ α ≤ 1: || x | α f ( x ) − | y | α f ( y ) | | f | ˚ C 0 ,α = | f | L ∞ + sup . | x − y | α | x − y | < 1 Examples: If f ( r , θ ) = g ( θ ), f ∈ ˚ C 0 ,α ( R 2 ) if and only if g ∈ C α ( S 1 ) C 0 ,α for any α > 0. If f ( x ) = sin(log( x )) , then f ∈ ˚ x ) �∈ ˚ C 0 , 1 ( R ) while sin( 1 c ( R 2 ) then f ∈ ˚ If f ∈ C α C 0 ,α . C 0 ,α functions which are m − fold symmetric on Denote by ˚ the space of ˚ C 0 ,α m R 2 . Lemma D 2 ∆ − 1 : ˚ → ˚ C 0 ,α C 0 ,α for all 0 < α < 1 and m ≥ 3 . m m Theorem The 2D Euler equation (in vorticity form) is globally well-posed on ˚ C 0 ,α m ( R 2 ) for every 0 ≤ α ≤ 1 and m ≥ 3 . Moreover, solutions satisfy: | ω | ˚ C 0 ,α ≤ exp( C exp( Ct ))
Singularity formation in incompressible fluids Cut-off Argument Cut-off argument in ˚ C 0 ,α Theorem The 2D Euler equation (in vorticity form) is globally well-posed on ˚ C 0 ,α m ( R 2 ) for every 0 ≤ α ≤ 1 and m ≥ 3 . Moreover, solutions satisfy: | ω | ˚ C 0 ,α ≤ exp( C exp( Ct )) for all t > 0 .
Singularity formation in incompressible fluids Cut-off Argument Cut-off argument in ˚ C 0 ,α Theorem The 2D Euler equation (in vorticity form) is globally well-posed on ˚ C 0 ,α m ( R 2 ) for every 0 ≤ α ≤ 1 and m ≥ 3 . Moreover, solutions satisfy: | ω | ˚ C 0 ,α ≤ exp( C exp( Ct )) for all t > 0 . Theorem Suppose g 0 ∈ ˚ C 1 ,α ( S 1 ) is π 2 periodic and odd. Let φ ∈ C ∞ ( R ) be bounded and φ (0) = 1 . Then, if ω 0 ( r , θ ) = φ ( r ) g 0 ( θ ) , for all t > 0 , | ω ( · , t ) | ˚ C 0 , 1 ≥ | ∂ θ g ( · , t ) | L ∞ .
Singularity formation in incompressible fluids Cut-off Argument Cut-off argument in ˚ C 0 ,α Theorem The 2D Euler equation (in vorticity form) is globally well-posed on ˚ C 0 ,α m ( R 2 ) for every 0 ≤ α ≤ 1 and m ≥ 3 . Moreover, solutions satisfy: | ω | ˚ C 0 ,α ≤ exp( C exp( Ct )) for all t > 0 . Theorem Suppose g 0 ∈ ˚ C 1 ,α ( S 1 ) is π 2 periodic and odd. Let φ ∈ C ∞ ( R ) be bounded and φ (0) = 1 . Then, if ω 0 ( r , θ ) = φ ( r ) g 0 ( θ ) , for all t > 0 , | ω ( · , t ) | ˚ C 0 , 1 ≥ | ∂ θ g ( · , t ) | L ∞ . Proof. ω ∈ C α ( R 2 ) for all t > 0 and ˜ Write ˜ ω = ω − g and prove that ˜ ω (0 , t ) = 0 for u ( x ) | � | x | 1+ α as | x | → 0 as all t > 0. This uses the crucial observations that | ˜ C α and g (0) = 0 implies fg ∈ C α . well as the fact that f ∈ C α , g ∈ ˚
Singularity formation in incompressible fluids Boussinesq System Boussinesq System
Singularity formation in incompressible fluids Boussinesq System Boussinesq System Let us recall the 2D Boussinseq system: ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω
Singularity formation in incompressible fluids Boussinesq System Boussinesq System Let us recall the 2D Boussinseq system: ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω Idea: Let’s study the behavior of solutions which are of the form ω ( r , θ, t ) = g ( θ, t ) and ρ ( r , θ, t ) = r P ( θ, t ) .
Singularity formation in incompressible fluids Boussinesq System Boussinesq System Let us recall the 2D Boussinseq system: ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω Idea: Let’s study the behavior of solutions which are of the form ω ( r , θ, t ) = g ( θ, t ) and ρ ( r , θ, t ) = r P ( θ, t ) . Then we get the 1D system: ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g
Singularity formation in incompressible fluids Boussinesq System Boussinesq System Let us recall the 2D Boussinseq system: ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω Idea: Let’s study the behavior of solutions which are of the form ω ( r , θ, t ) = g ( θ, t ) and ρ ( r , θ, t ) = r P ( θ, t ) . Then we get the 1D system: ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g Small problem: it isn’t possible to impose that g , P have high periodicity (another way to say this: Boussinesq doesn’t have a rotational symmetry).
Singularity formation in incompressible fluids Boussinesq System Boussinesq System Let us recall the 2D Boussinseq system: ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω Idea: Let’s study the behavior of solutions which are of the form ω ( r , θ, t ) = g ( θ, t ) and ρ ( r , θ, t ) = r P ( θ, t ) . Then we get the 1D system: ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g Small problem: it isn’t possible to impose that g , P have high periodicity (another way to say this: Boussinesq doesn’t have a rotational symmetry). Solution: another way is to just impose a solid boundary and look at the problem on [ − L , L ] with L small enough. We are able to manage with L < π 2 and in 2D this means the fluid domain will be a corner of angle θ < π .
Singularity formation in incompressible fluids Boussinesq System Boussinesq System Let us recall the 2D Boussinseq system: ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω Idea: Let’s study the behavior of solutions which are of the form ω ( r , θ, t ) = g ( θ, t ) and ρ ( r , θ, t ) = r P ( θ, t ) . Then we get the 1D system: ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g Small problem: it isn’t possible to impose that g , P have high periodicity (another way to say this: Boussinesq doesn’t have a rotational symmetry). Solution: another way is to just impose a solid boundary and look at the problem on [ − L , L ] with L small enough. We are able to manage with L < π 2 and in 2D this means the fluid domain will be a corner of angle θ < π . Restricted to such domains, the 1D system becomes well-posed locally in time.
Singularity formation in incompressible fluids Boussinesq System Blow-up for the 1 D Boussinesq System ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g
Singularity formation in incompressible fluids Boussinesq System Blow-up for the 1 D Boussinesq System ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g Our goal is to use P to grow g and then use g to grow P , etc. For g to grow, we need P , P ′ ≥ 0. It turns out that it is easy to propagate the following information: g is odd on [ − L , L ] and P is even on [ − L , L ] .
Singularity formation in incompressible fluids Boussinesq System Blow-up for the 1 D Boussinesq System ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g Our goal is to use P to grow g and then use g to grow P , etc. For g to grow, we need P , P ′ ≥ 0. It turns out that it is easy to propagate the following information: g is odd on [ − L , L ] and P is even on [ − L , L ] . g ≥ 0 on [0 , L ] and P , P ′ ≥ 0 on [0 , L ]. This already implies that g is increasing but we were unable to show blow-up just using this information (though it may well be true).
Singularity formation in incompressible fluids Boussinesq System Blow-up for the 1 D Boussinesq System ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g Our goal is to use P to grow g and then use g to grow P , etc. For g to grow, we need P , P ′ ≥ 0. It turns out that it is easy to propagate the following information: g is odd on [ − L , L ] and P is even on [ − L , L ] . g ≥ 0 on [0 , L ] and P , P ′ ≥ 0 on [0 , L ]. This already implies that g is increasing but we were unable to show blow-up just using this information (though it may well be true). After some thinking, we can propagate the following information: g ′ ≥ 0 and P ′′ + P ≥ 0.
Singularity formation in incompressible fluids Boussinesq System Blow-up for the 1 D Boussinesq System ∂ t g + 2 G ∂ θ g = P sin( θ ) + ∂ θ P cos( θ ) ∂ t P + 2 G ∂ θ P = P ∂ θ G 4 G + ∂ θθ G = g Our goal is to use P to grow g and then use g to grow P , etc. For g to grow, we need P , P ′ ≥ 0. It turns out that it is easy to propagate the following information: g is odd on [ − L , L ] and P is even on [ − L , L ] . g ≥ 0 on [0 , L ] and P , P ′ ≥ 0 on [0 , L ]. This already implies that g is increasing but we were unable to show blow-up just using this information (though it may well be true). After some thinking, we can propagate the following information: g ′ ≥ 0 and P ′′ + P ≥ 0. Next, one just integrates the g equation and it is relatively simple to get: � L � � L � 2 ∂ t g ≥ c g . 0 0
Singularity formation in incompressible fluids Boussinesq System Blow-up for finite-energy and bounded denisty solutions to the 2D Boussinesq system ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω
Singularity formation in incompressible fluids Boussinesq System Blow-up for finite-energy and bounded denisty solutions to the 2D Boussinesq system ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω A consequence of blow-up for the 1D Boussinesq system is the following theorem:
Singularity formation in incompressible fluids Boussinesq System Blow-up for finite-energy and bounded denisty solutions to the 2D Boussinesq system ∂ t ω + u · ∇ ω = ∂ y ρ ∂ t ρ + u · ∇ ρ = 0 u = ∇ ⊥ (∆) − 1 ω A consequence of blow-up for the 1D Boussinesq system is the following theorem: Theorem For M > 0 let Ω = { x ∈ R 2 : 0 ≤ x 2 ≤ Mx 1 } . Then the 2 D Boussinesq system (in vorticity form) is: LWP for ( ω, ∇ ρ ) ∈ ˚ C 0 ,α . If ρ 0 ≡ 0 , the solution is global and grows at most double exponentially. There are compactly supported solutions which blow-up in finite time.
Singularity formation in incompressible fluids Open Problems Open Problems: 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g
Singularity formation in incompressible fluids Open Problems Open Problems: 2D Euler ∂ t g + 2 G ∂ θ g = 0 4 G + ∂ θθ G = g Problem 1: Growth of scale invariant solutions to 2 D Euler on S 1 .
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