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Singularity formation in incompressible fluids

Singularity formation in incompressible fluids

Tarek M. Elgindi (UC-San Diego) In-Jee Jeong (KIAS) Porquerolles August 2018

Singularity formation in incompressible fluids Basic Ideas

Introducing the Equations

Recall the incompressible Euler equation for the velocity field u : Ω × R → Rd and pressure p : Ω × R → R of an ideal fluid flowing through a suitable domain Ω ⊂ Rd:

Singularity formation in incompressible fluids Basic Ideas

Introducing the Equations

Recall the incompressible Euler equation for the velocity field u : Ω × R → Rd and pressure p : Ω × R → R of an ideal fluid flowing through a suitable domain Ω ⊂ Rd: ∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0,

Singularity formation in incompressible fluids Basic Ideas

Introducing the Equations

Recall the incompressible Euler equation for the velocity field u : Ω × R → Rd and pressure p : Ω × R → R of an ideal fluid flowing through a suitable domain Ω ⊂ Rd: ∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω.

Singularity formation in incompressible fluids Basic Ideas

Introducing the Equations

Recall the incompressible Euler equation for the velocity field u : Ω × R → Rd and pressure p : Ω × R → R of an ideal fluid flowing through a suitable domain Ω ⊂ Rd: ∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. The system was derived by Euler in 1755. It seems to be the second PDE ever written.

Singularity formation in incompressible fluids Basic Ideas

Introducing the Equations

Recall the incompressible Euler equation for the velocity field u : Ω × R → Rd and pressure p : Ω × R → R of an ideal fluid flowing through a suitable domain Ω ⊂ Rd: ∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. The system was derived by Euler in 1755. It seems to be the second PDE ever

• written. Many difficulties:

System

Singularity formation in incompressible fluids Basic Ideas

Introducing the Equations

Recall the incompressible Euler equation for the velocity field u : Ω × R → Rd and pressure p : Ω × R → R of an ideal fluid flowing through a suitable domain Ω ⊂ Rd: ∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. The system was derived by Euler in 1755. It seems to be the second PDE ever

• written. Many difficulties:

System Non-local

Singularity formation in incompressible fluids Basic Ideas

Introducing the Equations

Recall the incompressible Euler equation for the velocity field u : Ω × R → Rd and pressure p : Ω × R → R of an ideal fluid flowing through a suitable domain Ω ⊂ Rd: ∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. The system was derived by Euler in 1755. It seems to be the second PDE ever

• written. Many difficulties:

System Non-local Insufficient a-priori bounds

Singularity formation in incompressible fluids Basic Ideas

Introducing the Equations

Recall the incompressible Euler equation for the velocity field u : Ω × R → Rd and pressure p : Ω × R → R of an ideal fluid flowing through a suitable domain Ω ⊂ Rd: ∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. The system was derived by Euler in 1755. It seems to be the second PDE ever

• written. Many difficulties:

System Non-local Insufficient a-priori bounds Large class of stationary states

Singularity formation in incompressible fluids Basic Ideas

Two Important Problems in the Field

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. Global Regularity Question: Given a u0 which is smooth (and rapidly decaying at infinity) and an Ω which is smooth, does there always exist a solution u ∈ C ∞(Ω × [0, ∞))?

Singularity formation in incompressible fluids Basic Ideas

Two Important Problems in the Field

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. Global Regularity Question: Given a u0 which is smooth (and rapidly decaying at infinity) and an Ω which is smooth, does there always exist a solution u ∈ C ∞(Ω × [0, ∞))? Long-time Behavior of Global Solutions: Suppose we had a global solution u ∈ W 1,∞(Ω × [0, ∞)). Can we describe the behavior of u as t → ∞?

Singularity formation in incompressible fluids Basic Ideas

Two Important Problems in the Field

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. Global Regularity Question: Given a u0 which is smooth (and rapidly decaying at infinity) and an Ω which is smooth, does there always exist a solution u ∈ C ∞(Ω × [0, ∞))? Long-time Behavior of Global Solutions: Suppose we had a global solution u ∈ W 1,∞(Ω × [0, ∞)). Can we describe the behavior of u as t → ∞? Conjecture: Generic solutions are not pre-compact in L2 as t → ∞. In other words, generic global solutions will become rougher and rougher as t → ∞.

Singularity formation in incompressible fluids Basic Ideas

Two Basic Approaches to the Global Regularity Problem

Singularity formation in incompressible fluids Basic Ideas

Two Basic Approaches to the Global Regularity Problem

Approach A:

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.

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.)

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.

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, Cordoba2, Fontelos, Hou, Li, Lei, Luo, Kiselev, Sverak, Choi, Yao,...

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, Cordoba2, Fontelos, Hou, Li, Lei, Luo, Kiselev, Sverak, Choi, Yao,... Approach B:

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, Cordoba2, Fontelos, Hou, Li, Lei, Luo, Kiselev, Sverak, Choi, Yao,... Approach B: Search for special solutions to the 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. 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, Cordoba2, Fontelos, Hou, Li, Lei, Luo, Kiselev, Sverak, Choi, Yao,... Approach B: Search for special solutions to the equation. Some examples of this approach are:

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, Cordoba2, 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,...)

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, Cordoba2, 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, Cordoba2, 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, Cordoba2, 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

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω.

Singularity formation in incompressible fluids Basic Ideas

Scaling Invariant Solutions

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. We are going to follow ”Approach B,” which is to search for very special types

• f solutions by imposing a high degree of natural symmetries on the solution.

Singularity formation in incompressible fluids Basic Ideas

Scaling Invariant Solutions

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. We are going to follow ”Approach B,” which is to search for very special types

• f 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

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. We are going to follow ”Approach B,” which is to search for very special types

• f 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 Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 2)}. Then there is a space

X ⊂ W 1,∞ for which:

Singularity formation in incompressible fluids Basic Ideas

Scaling Invariant Solutions

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. We are going to follow ”Approach B,” which is to search for very special types

• f 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 Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 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

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. We are going to follow ”Approach B,” which is to search for very special types

• f 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 Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 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

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. We are going to follow ”Approach B,” which is to search for very special types

• f 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 Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 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 3D solutions) which blow-up in finite time.

Singularity formation in incompressible fluids Basic Ideas

Scaling Invariant Solutions

Theorem Let ǫ > 0 and Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 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 3D solutions) which blow-up in finite time. Remarks:

Singularity formation in incompressible fluids Basic Ideas

Scaling Invariant Solutions

Theorem Let ǫ > 0 and Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 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 3D 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 Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 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 3D 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 Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 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 3D 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 Ωǫ = {x ∈ R3 : (1 + ǫ|x3|)2 ≤ (x2

1 + x2 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 3D 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, 2D solutions are global which means that the blow-up is not solely coming from the setting but rather from the 3D Euler equation itself.

Singularity formation in incompressible fluids Basic Ideas

Scaling Invariant Solutions

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. Recall that whenever λ > 0 and Q is an orthogonal matrix (QQT = I) and u(x, t) is a solution to the Euler equation, 1 λu(λx, t) and QTu(Qx, t) are solutions as well.

Singularity formation in incompressible fluids Basic Ideas

Special Solutions to Fluid Equations

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. 1 λu(λx, t) and QTu(Qx, t) are solutions as well. Thus, formally if u0(x) = 1 λu0(λx) and u0(Qx) = Qu0(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

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. 1 λu(λx, t) and QTu(Qx, t) are solutions as well. Thus, formally if u0(x) = 1 λu0(λx) and u0(Qx) = Qu0(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

∂tu + u · ∇u + ∇p = 0, div(u) = 0, u|t=0 = u0, u · n = 0 on ∂Ω. 1 λu(λx, t) and QTu(Qx, t) are solutions as well. Thus, formally if u0(x) = 1 λu0(λx) and u0(Qx) = Qu0(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 u0(Qx) = Qu0(x) for all rotational matrices fixing the z−axis we just get the axi-symmetric 3D Euler equation.

Singularity formation in incompressible fluids Basic Ideas

Special Solutions to Fluid Equations

If u solves the incompressible Euler equation then 1 λu(λx, t) and QTu(Qx, t) are solutions as well. Formally if u0(x) = 1 λu0(λx) and u0(Qx) = Qu0(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 u0(Qx) = Qu0(x) for all rotational matrices fixing the z−axis we just get the axi-symmetric 3D Euler equation. Using the Scaling symmetry of the equation comes with many problems: If we assume

1 λu0(λx) = u0(x) for all x and λ then u0 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? λu0( x λ) ≡ u0(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: λu0( x λ) ≡ u0(x), for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λu( 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: λu0( x λ) ≡ u0(x), for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λu( 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: λu0( x λ) ≡ u0(x), for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λu( 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: λu0( x λ) ≡ u0(x), for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λu( 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: λu0( x λ) ≡ u0(x), for all x and λ > 0. We call such data scale-invariant. Formally, using scaling, we see that λu( 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 2D: ∂tω + u · ∇ω = 0, u = ∇⊥(∆)−1ω.

Singularity formation in incompressible fluids Scale-Invariant Solutions

The 2D Euler Equation with Non-decaying Vorticity

Let us see how things work in 2D: ∂tω + u · ∇ω = 0, u = ∇⊥(∆)−1ω. Formally, let’s believe that if ω0 is scale invariant (0−homogeneous in space), then ω(t) remains as such. Write: ω(r, θ, t) = g(θ, t). ∆−1ω = r 2G(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 2D: ∂tω + u · ∇ω = 0, u = ∇⊥(∆)−1ω. Formally, let’s believe that if ω0 is scale invariant (0−homogeneous in space), then ω(t) remains as such. Write: ω(r, θ, t) = g(θ, t). ∆−1ω = r 2G(θ, t). With this ansatz, the 2D Euler system collapses to an active scalar equation on S1: ∂tg + 2G∂θg = 0, 4G + ∂θθG = g.

Singularity formation in incompressible fluids Scale-Invariant Solutions

The 2D Euler Equation with Non-decaying Vorticity

Let us see how things work in 2D: ∂tω + u · ∇ω = 0, u = ∇⊥(∆)−1ω. Formally, let’s believe that if ω0 is scale invariant (0−homogeneous in space), then ω(t) remains as such. Write: ω(r, θ, t) = g(θ, t). ∆−1ω = r 2G(θ, t). ∂tg + 2G∂θg = 0, 4G + ∂θθG = 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 2D: ∂tω + u · ∇ω = 0, u = ∇⊥(∆)−1ω. Formally, let’s believe that if ω0 is scale invariant (0−homogeneous in space), then ω(t) remains as such. Write: ω(r, θ, t) = g(θ, t). ∆−1ω = r 2G(θ, t). ∂tg + 2G∂θg = 0, 4G + ∂θθG = 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 ω0 ∈ L∞(R2) be m−fold symmetric for some m ≥ 3. Then, there exists a unique global solution to 2D Euler in the class C(R : L∞

w∗(R2)) with ω m−fold symmetric and ω|t=0 = ω0.

Singularity formation in incompressible fluids Scale-Invariant Solutions

The 2D Euler Equation with Non-decaying Vorticity

Let us see how things work in 2D: ∂tω + u · ∇ω = 0, u = ∇⊥(∆)−1ω. Theorem (E., Jeong, 2016, to appear in CPAM) Let ω0 ∈ L∞(R2) be m−fold symmetric for some m ≥ 3. Then, there exists a unique global solution to 2D Euler in the class C(R : L∞

w∗(R2)) with ω m−fold symmetric and ω|t=0 = ω0.

Corollary Let ω0(r, θ) = g0(θ) ∈ L∞(S1) be 2π

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: ∂tg + 2G∂θg = 0, 4G + ∂θθG = g.

Singularity formation in incompressible fluids Scale-Invariant Solutions

The 2D Euler Equation with Non-decaying Vorticity

Recall the 2D Euler system: ∂tω + u · ∇ω = 0 u = ∇⊥(∆)−1ω Theorem (E., Jeong, 2016) Let ω0 ∈ L∞(R2) be m−fold symmetric for some m ≥ 3. Then, there exists a unique global solution to 2D Euler in the class C(R : L∞

w∗(R2)) with ω m−fold symmetric and ω|t=0 = ω0.

Corollary Let ω0(r, θ) = g0(θ) ∈ L∞(S1) be 2π

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: ∂tg + 2G∂θg = 0, 4G + ∂θθG = g. Remark: The elliptic problem 4G + ∂θθG = g can be solved since m ≥ 3.

Singularity formation in incompressible fluids Scale-Invariant Solutions

Some Remarks on Scale-Invariant Solutions to 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = 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

∂tg + 2G∂θg = 0 4G + ∂θθG = 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

∂tg + 2G∂θg = 0 4G + ∂θθG = 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

∂tg + 2G∂θg = 0 4G + ∂θθG = 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

∂tg + 2G∂θg = 0 4G + ∂θθG = 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

∂tg + 2G∂θg = 0 4G + ∂θθG = 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

∂tg + 2G∂θg = 0 4G + ∂θθG = 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

∂tg + 2G∂θg = 0 4G + ∂θθG = 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

∂tg + 2G∂θg = 0 4G + ∂θθG = 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: ∂tg + 2G∂θg = 0 4G + ∂θθG = g for which it is known that |∂θg|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: ∂tg + 2G∂θg = 0 4G + ∂θθG = g for which it is known that |∂θg|L∞ → ∞ as t → ∞. Question: Is it possible to construct compactly supported solutions to the 2D 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: ∂tg + 2G∂θg = 0 4G + ∂θθG = g for which it is known that |∂θg|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: |f | ˚

C0,α = |f |L∞ +

sup

|x−y|<1

||x|αf (x) − |y|αf (y)| |x − y|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: |f | ˚

C0,α = |f |L∞ +

sup

|x−y|<1

||x|αf (x) − |y|αf (y)| |x − y|α . Examples: If f (r, θ) = g(θ), f ∈ ˚ C 0,α(R2) if and only if g ∈ C α(S1)

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: |f | ˚

C0,α = |f |L∞ +

sup

|x−y|<1

||x|αf (x) − |y|αf (y)| |x − y|α . Examples: If f (r, θ) = g(θ), f ∈ ˚ C 0,α(R2) if and only if g ∈ C α(S1) If f (x) = sin(log(x)), then f ∈ ˚ C 0,1(R) while sin( 1

x ) ∈ ˚

C 0,α for any α > 0.

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: |f | ˚

C0,α = |f |L∞ +

sup

|x−y|<1

||x|αf (x) − |y|αf (y)| |x − y|α . Examples: If f (r, θ) = g(θ), f ∈ ˚ C 0,α(R2) if and only if g ∈ C α(S1) If f (x) = sin(log(x)), then f ∈ ˚ C 0,1(R) while sin( 1

x ) ∈ ˚

C 0,α for any α > 0. If f ∈ C α

c (R2) then f ∈ ˚

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: |f | ˚

C0,α = |f |L∞ +

sup

|x−y|<1

||x|αf (x) − |y|αf (y)| |x − y|α . Examples: If f (r, θ) = g(θ), f ∈ ˚ C 0,α(R2) if and only if g ∈ C α(S1) If f (x) = sin(log(x)), then f ∈ ˚ C 0,1(R) while sin( 1

x ) ∈ ˚

C 0,α for any α > 0. If f ∈ C α

c (R2) then f ∈ ˚

C 0,α. Denote by ˚ C 0,α

m

the space of ˚ C 0,α functions which are m−fold symmetric on R2.

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: |f | ˚

C0,α = |f |L∞ +

sup

|x−y|<1

||x|αf (x) − |y|αf (y)| |x − y|α . Examples: If f (r, θ) = g(θ), f ∈ ˚ C 0,α(R2) if and only if g ∈ C α(S1) If f (x) = sin(log(x)), then f ∈ ˚ C 0,1(R) while sin( 1

x ) ∈ ˚

C 0,α for any α > 0. If f ∈ C α

c (R2) then f ∈ ˚

C 0,α. Denote by ˚ C 0,α

m

the space of ˚ C 0,α functions which are m−fold symmetric on R2. Lemma D2∆−1 : ˚ C 0,α

m

→ ˚ C 0,α

m

for all 0 < α < 1 and m ≥ 3.

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: |f | ˚

C0,α = |f |L∞ +

sup

|x−y|<1

||x|αf (x) − |y|αf (y)| |x − y|α . Examples: If f (r, θ) = g(θ), f ∈ ˚ C 0,α(R2) if and only if g ∈ C α(S1) If f (x) = sin(log(x)), then f ∈ ˚ C 0,1(R) while sin( 1

x ) ∈ ˚

C 0,α for any α > 0. If f ∈ C α

c (R2) then f ∈ ˚

C 0,α. Denote by ˚ C 0,α

m

the space of ˚ C 0,α functions which are m−fold symmetric on R2. Lemma D2∆−1 : ˚ C 0,α

m

→ ˚ C 0,α

m

for all 0 < α < 1 and m ≥ 3. Theorem The 2D Euler equation (in vorticity form) is globally well-posed on ˚ C 0,α

m (R2) for

every 0 ≤ α ≤ 1 and m ≥ 3. Moreover, solutions satisfy: |ω| ˚

C0,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 (R2) for

every 0 ≤ α ≤ 1 and m ≥ 3. Moreover, solutions satisfy: |ω| ˚

C0,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 (R2) for

every 0 ≤ α ≤ 1 and m ≥ 3. Moreover, solutions satisfy: |ω| ˚

C0,α ≤ exp(C exp(Ct))

for all t > 0. Theorem Suppose g0 ∈ ˚ C 1,α(S1) is π

2 periodic and odd. Let φ ∈ C ∞(R) be bounded and

φ(0) = 1. Then, if ω0(r, θ) = φ(r)g0(θ), for all t > 0, |ω(·, t)| ˚

C0,1 ≥ |∂θg(·, 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 (R2) for

every 0 ≤ α ≤ 1 and m ≥ 3. Moreover, solutions satisfy: |ω| ˚

C0,α ≤ exp(C exp(Ct))

for all t > 0. Theorem Suppose g0 ∈ ˚ C 1,α(S1) is π

2 periodic and odd. Let φ ∈ C ∞(R) be bounded and

φ(0) = 1. Then, if ω0(r, θ) = φ(r)g0(θ), for all t > 0, |ω(·, t)| ˚

C0,1 ≥ |∂θg(·, t)|L∞.

Proof. Write ˜ ω = ω − g and prove that ˜ ω ∈ C α(R2) for all t > 0 and ˜ ω(0, t) = 0 for all t > 0. This uses the crucial observations that |˜ u(x)| |x|1+α as |x| → 0 as well as the fact that f ∈ C α, g ∈ ˚ C α and g(0) = 0 implies fg ∈ C α.

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: ∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω

Singularity formation in incompressible fluids Boussinesq System

Boussinesq System

Let us recall the 2D Boussinseq system: ∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω 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: ∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω 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: ∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = g

Singularity formation in incompressible fluids Boussinesq System

Boussinesq System

Let us recall the 2D Boussinseq system: ∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω 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: ∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = 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: ∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω 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: ∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = 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: ∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω 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: ∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = 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 1D Boussinesq System

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = g

Singularity formation in incompressible fluids Boussinesq System

Blow-up for the 1D Boussinesq System

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = 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 1D Boussinesq System

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = 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 1D Boussinesq System

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = 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 1D Boussinesq System

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = 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: ∂t L g ≥ c L g 2 .

Singularity formation in incompressible fluids Boussinesq System

Blow-up for finite-energy and bounded denisty solutions to the 2D Boussinesq system

∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω

Singularity formation in incompressible fluids Boussinesq System

Blow-up for finite-energy and bounded denisty solutions to the 2D Boussinesq system

∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω 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

∂tω + u · ∇ω = ∂yρ ∂tρ + u · ∇ρ = 0 u = ∇⊥(∆)−1ω A consequence of blow-up for the 1D Boussinesq system is the following theorem: Theorem For M > 0 let Ω = {x ∈ R2 : 0 ≤ x2 ≤ Mx1}. Then the 2D Boussinesq system (in vorticity form) is: LWP for (ω, ∇ρ) ∈ ˚ C 0,α. If ρ0 ≡ 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

∂tg + 2G∂θg = 0 4G + ∂θθG = g

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = g Problem 1: Growth of scale invariant solutions to 2D Euler on S1.

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = g Problem 1: Growth of scale invariant solutions to 2D Euler on S1. Determine whether the exponential bound on ∂θg is sharp.

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = g Problem 1: Growth of scale invariant solutions to 2D Euler on S1. Determine whether the exponential bound on ∂θg is sharp. Remark: It is known to be sharp when there are boundaries. It is also known that there are solutions which grow exponentially fast for some (arbitrarily long) time and then relax to quadratic growth.

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = g Problem 1: Growth of scale invariant solutions to 2D Euler on S1. Determine whether the exponential bound on ∂θg is sharp. Remark: It is known to be sharp when there are boundaries. It is also known that there are solutions which grow exponentially fast for some (arbitrarily long) time and then relax to quadratic growth. Problem 2: Long-time Behavior of scale invariant solutions to 2D Euler on S1.

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = g Problem 1: Growth of scale invariant solutions to 2D Euler on S1. Determine whether the exponential bound on ∂θg is sharp. Remark: It is known to be sharp when there are boundaries. It is also known that there are solutions which grow exponentially fast for some (arbitrarily long) time and then relax to quadratic growth. Problem 2: Long-time Behavior of scale invariant solutions to 2D Euler on S1. (A) Prove that all ”positive and odd” solutions on S1 of period π

2 converge

strongly to 0 as t → ∞. These are solutions that initially look like sin(4θ).

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = g Problem 1: Growth of scale invariant solutions to 2D Euler on S1. Determine whether the exponential bound on ∂θg is sharp. Remark: It is known to be sharp when there are boundaries. It is also known that there are solutions which grow exponentially fast for some (arbitrarily long) time and then relax to quadratic growth. Problem 2: Long-time Behavior of scale invariant solutions to 2D Euler on S1. (A) Prove that all ”positive and odd” solutions on S1 of period π

2 converge

strongly to 0 as t → ∞. These are solutions that initially look like sin(4θ). (B) Describe the behavior of solutions with non-zero mean (we already know there are periodic and quasi-periodic solutions).

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = g Problem 1: Growth of scale invariant solutions to 2D Euler on S1. Determine whether the exponential bound on ∂θg is sharp. Remark: It is known to be sharp when there are boundaries. It is also known that there are solutions which grow exponentially fast for some (arbitrarily long) time and then relax to quadratic growth. Problem 2: Long-time Behavior of scale invariant solutions to 2D Euler on S1. (A) Prove that all ”positive and odd” solutions on S1 of period π

2 converge

strongly to 0 as t → ∞. These are solutions that initially look like sin(4θ). (B) Describe the behavior of solutions with non-zero mean (we already know there are periodic and quasi-periodic solutions). Problem 3: Prove the existence of rotating (compactly supported) vortex patches with corners.

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Euler

∂tg + 2G∂θg = 0 4G + ∂θθG = g Problem 1: Growth of scale invariant solutions to 2D Euler on S1. Determine whether the exponential bound on ∂θg is sharp. Remark: It is known to be sharp when there are boundaries. It is also known that there are solutions which grow exponentially fast for some (arbitrarily long) time and then relax to quadratic growth. Problem 2: Long-time Behavior of scale invariant solutions to 2D Euler on S1. (A) Prove that all ”positive and odd” solutions on S1 of period π

2 converge

strongly to 0 as t → ∞. These are solutions that initially look like sin(4θ). (B) Describe the behavior of solutions with non-zero mean (we already know there are periodic and quasi-periodic solutions). Problem 3: Prove the existence of rotating (compactly supported) vortex patches with corners. Remark: These solutions have been observed numerically by Deem and Zabusky (1978) but never shown to exist analytically.

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Boussinesq

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = g

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Boussinesq

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = g Problem 1: Remove monotonicity conditions on vorticity for blow-up

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Boussinesq

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = g Problem 1: Remove monotonicity conditions on vorticity for blow-up Problem 2: Prove blow-up even when the density and vorticity vanish identically near the boundary.

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Boussinesq

∂tg + 2G∂θg = P sin(θ) + ∂θP cos(θ) ∂tP + 2G∂θP = P∂θG 4G + ∂θθG = g Problem 1: Remove monotonicity conditions on vorticity for blow-up Problem 2: Prove blow-up even when the density and vorticity vanish identically near the boundary.

Singularity formation in incompressible fluids Open Problems

Open Problems: 2D Boussinesq

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