“Broadening the knowledge base and supporting the long term professional sustainability of the Research University Centre of Excellence
at the University of Szeged by ensuring the rising generation of excellent scientists.””
Doctoral School of Mathematics and Computer Science
Stochastic Days in Szeged 26.07.2012.
Exploding solutions of hydrodynamic equations in computer simulations
Carlo Boldrighini
(Sapienza University of Rome)
TÁMOP‐4.2.2/B‐10/1‐2010‐0012 project
EXPLODING SOLUTIONS OF HYDRODYNAMIC EQUATIONS
IN COMPUTER SIMULATIONS
C. Boldrighini, Dipartimento di Matematica, Universit`a di Roma ”La Sapienza” Based on joint work with S. Frigio, and P. Maponi
Universit`a di Camerino
70th anniversary of Andras Kramli Szeged, July 26th, 2013
EXPLODING SOLUTIONS OF HYDRODYNAMIC EQUATIONS
IN COMPUTER SIMULATIONS C. Boldrighini, Dipartimento di Matematica,
Universit`a di Roma ”La Sapienza”
Based on joint work with S. Frigio, and P. Maponi Universit`a di Camerino
70th anniversary of Andras Kramli Szeged, July 26th, 2013
EXPLODING SOLUTIONS OF HYDRODYNAMIC EQUATIONS
IN COMPUTER SIMULATIONS C. Boldrighini, Dipartimento di Matematica,
Universit`a di Roma ”La Sapienza”
Based on joint work with S. Frigio, and P. Maponi
Universit`a di Camerino
70th anniversary of Andras Kramli Szeged, July 26th, 2013
EXPLODING SOLUTIONS OF HYDRODYNAMIC EQUATIONS
IN COMPUTER SIMULATIONS C. Boldrighini, Dipartimento di Matematica,
Universit`a di Roma ”La Sapienza”
Based on joint work with S. Frigio, and P. Maponi Universit`a di Camerino
70th anniversary of Andras Kramli Szeged, July 26th, 2013
EXPLODING SOLUTIONS OF HYDRODYNAMIC EQUATIONS
IN COMPUTER SIMULATIONS C. Boldrighini, Dipartimento di Matematica,
Universit`a di Roma ”La Sapienza”
Based on joint work with S. Frigio, and P. Maponi Universit`a di Camerino
70th anniversary of Andras Kramli Szeged, July 26th, 2013
1. INTRODUCTION
The question whether the 3-dimensional incompressible Navier-Stokes (NS) equations can become singular at finite times for regular initial data, is one of the main open problems of rigorous fluid mechanics.
Assuming that such singularities exist, how would they appear? Could they describe some kind of physical phenomena?
These are relevant questions if we want to address the problem of the NS singularities, either theoretically or by computer evidence.
1. INTRODUCTION
The question whether the 3-dimensional incompressible Navier-Stokes (NS) equations can become singular at finite times for regular initial data, is one of the main open problems of rigorous fluid mechanics.
Assuming that such singularities exist, how would they appear? Could they describe some kind of physical phenomena?
These are relevant questions if we want to address the problem of the NS singularities, either theoretically or by computer evidence.
1. INTRODUCTION
The question whether the 3-dimensional incompressible Navier-Stokes (NS) equations can become singular at finite times for regular initial data, is one of the main open problems of rigorous fluid mechanics.
Assuming that such singularities exist, how would they appear?
Could they describe some kind of physical phenomena?
These are relevant questions if we want to address the problem of the NS singularities, either theoretically or by computer evidence.
1. INTRODUCTION
The question whether the 3-dimensional incompressible Navier-Stokes (NS) equations can become singular at finite times for regular initial data, is one of the main open problems of rigorous fluid mechanics.
Assuming that such singularities exist, how would they appear?
Could they describe some kind of physical phenomena?
These are relevant questions if we want to address the problem of the NS singularities, either theoretically or by computer evidence.
1. INTRODUCTION
The question whether the 3-dimensional incompressible Navier-Stokes (NS) equations can become singular at finite times for regular initial data, is one of the main open problems of rigorous fluid mechanics.
Assuming that such singularities exist, how would they appear?
Could they describe some kind of physical phenomena?
These are relevant questions if we want to address the problem of the NS singularities, either theoretically or by computer evidence.
Leray supposed that such singularities do exist and that they are related to turbulence [Leray 1934].
Modern ideas on turbulence developed independently of the problem of singularities.
For a long time there was little improvement on the subject. A real breakthrough came only a few years ago, due to the work of Li and Sinai (2003 - ).
They gave explicit examples of singularities at finite time for a class of complex-valued solutions of the 3-d NS.
Their method also applies to other equations of fluid dynamics, such as the Burgers equations [Li, Sinai 2010].
Leray supposed that such singularities do exist and that they are related to turbulence [Leray 1934].
Modern ideas on turbulence developed independently of the problem of singularities.
For a long time there was little improvement on the subject. A real breakthrough came only a few years ago, due to the work of Li and Sinai (2003 - ).
They gave explicit examples of singularities at finite time for a class of complex-valued solutions of the 3-d NS.
Their method also applies to other equations of fluid dynamics, such as the Burgers equations [Li, Sinai 2010].
Leray supposed that such singularities do exist and that they are related to turbulence [Leray 1934].
Modern ideas on turbulence developed independently of the problem of singularities.
For a long time there was little improvement on the subject.
A real breakthrough came only a few years ago, due to the work of Li and Sinai (2003 - ).
They gave explicit examples of singularities at finite time for a class of complex-valued solutions of the 3-d NS.
Their method also applies to other equations of fluid dynamics, such as the Burgers equations [Li, Sinai 2010].
Leray supposed that such singularities do exist and that they are related to turbulence [Leray 1934].
Modern ideas on turbulence developed independently of the problem of singularities.
For a long time there was little improvement on the subject.
A real breakthrough came only a few years ago, due to the work of Li and Sinai (2003 - ).
They gave explicit examples of singularities at finite time for a class of complex-valued solutions of the 3-d NS.
Their method also applies to other equations of fluid dynamics, such as the Burgers equations [Li, Sinai 2010].
Leray supposed that such singularities do exist and that they are related to turbulence [Leray 1934].
Modern ideas on turbulence developed independently of the problem of singularities.
For a long time there was little improvement on the subject.
A real breakthrough came only a few years ago, due to the work of Li and Sinai (2003 - ).
They gave explicit examples of singularities at finite time for a class of complex-valued solutions of the 3-d NS.
Their method also applies to other equations of fluid dynamics, such as the Burgers equations [Li, Sinai 2010].
Leray supposed that such singularities do exist and that they are related to turbulence [Leray 1934].
Modern ideas on turbulence developed independently of the problem of singularities.
For a long time there was little improvement on the subject.
A real breakthrough came only a few years ago, due to the work of Li and Sinai (2003 - ).
They gave explicit examples of singularities at finite time for a class of complex-valued solutions of the 3-d NS.
Their method also applies to other equations of fluid dynamics, such as the Burgers equations [Li, Sinai 2010].
The singularities appear as a concentration of the energy in a small space region,
as it happens for some physical phenomena (tornadoes).
The total energy diverges for the complex-valued solutions. For real valued solution of the NS equations the energy is bounded by the energy inequality.
The singular real solutions would show instead a divergence of the enstrophy (the integral of the square of the vorticity). One can expect that they also involve a concentration of the energy in a finite region, and could provide a model of tornado-like phenomena.
The singularities appear as a concentration of the energy in a small space region, as it happens for some physical phenomena (tornadoes).
The total energy diverges for the complex-valued solutions. For real valued solution of the NS equations the energy is bounded by the energy inequality.
The singular real solutions would show instead a divergence of the enstrophy (the integral of the square of the vorticity). One can expect that they also involve a concentration of the energy in a finite region, and could provide a model of tornado-like phenomena.
The singularities appear as a concentration of the energy in a small space region, as it happens for some physical phenomena (tornadoes).
The total energy diverges for the complex-valued solutions.
For real valued solution of the NS equations the energy is bounded by the energy inequality.
The singular real solutions would show instead a divergence of the enstrophy (the integral of the square of the vorticity). One can expect that they also involve a concentration of the energy in a finite region, and could provide a model of tornado-like phenomena.
The singularities appear as a concentration of the energy in a small space region, as it happens for some physical phenomena (tornadoes).
The total energy diverges for the complex-valued solutions.
For real valued solution of the NS equations the energy is bounded by the energy inequality.
The singular real solutions would show instead a divergence of the enstrophy (the integral of the square of the vorticity). One can expect that they also involve a concentration of the energy in a finite region, and could provide a model of tornado-like phenomena.
The singularities appear as a concentration of the energy in a small space region, as it happens for some physical phenomena (tornadoes).
The total energy diverges for the complex-valued solutions.
For real valued solution of the NS equations the energy is bounded by the energy inequality.
The singular real solutions would show instead a divergence of the enstrophy (the integral of the square of the vorticity).
One can expect that they also involve a concentration of the energy in a finite region, and could provide a model of tornado-like phenomena.
The singularities appear as a concentration of the energy in a small space region, as it happens for some physical phenomena (tornadoes).
The total energy diverges for the complex-valued solutions.
For real valued solution of the NS equations the energy is bounded by the energy inequality.
The singular real solutions would show instead a divergence of the enstrophy (the integral of the square of the vorticity).
One can expect that they also involve a concentration of the energy in a finite region, and could provide a model of tornado-like phenomena.
A careful analysis of the exploding solutions found by Li and Sinai can provide guidelines for the search of real valued singularities, both by theoretical and computational methods.
I report the results of a study by computer simulations of the complex solutions of the 2-dimensional Burgers equations with no boundary conditions and no external forces.
I will also report some recent results on the complex-valued 3-d NS equations.
Why Burgers?
It is the simplest model of classical fluid equations for which the esistence of singularities is proved for suitable initial data [Li, Sinai 2010].
A careful analysis of the exploding solutions found by Li and Sinai can provide guidelines for the search of real valued singularities, both by theoretical and computational methods.
I report the results of a study by computer simulations of the complex solutions of the 2-dimensional Burgers equations with no boundary conditions and no external forces.
I will also report some recent results on the complex-valued 3-d NS equations.
Why Burgers?
It is the simplest model of classical fluid equations for which the esistence of singularities is proved for suitable initial data [Li, Sinai 2010].
A careful analysis of the exploding solutions found by Li and Sinai can provide guidelines for the search of real valued singularities, both by theoretical and computational methods.
I report the results of a study by computer simulations of the complex solutions of the 2-dimensional Burgers equations with no boundary conditions and no external forces.
I will also report some recent results on the complex-valued 3-d NS equations.
Why Burgers?
It is the simplest model of classical fluid equations for which the esistence of singularities is proved for suitable initial data [Li, Sinai 2010].
A careful analysis of the exploding solutions found by Li and Sinai can provide guidelines for the search of real valued singularities, both by theoretical and computational methods.
I report the results of a study by computer simulations of the complex solutions of the 2-dimensional Burgers equations with no boundary conditions and no external forces.
I will also report some recent results on the complex-valued 3-d NS equations.
Why Burgers?
It is the simplest model of classical fluid equations for which the esistence of singularities is proved for suitable initial data [Li, Sinai 2010].
A careful analysis of the exploding solutions found by Li and Sinai can provide guidelines for the search of real valued singularities, both by theoretical and computational methods.
I report the results of a study by computer simulations of the complex solutions of the 2-dimensional Burgers equations with no boundary conditions and no external forces.
I will also report some recent results on the complex-valued 3-d NS equations.
Why Burgers?
It is the simplest model of classical fluid equations for which the esistence of singularities is proved for suitable initial data [Li, Sinai 2010].
2. EXPLODING COMPLEX BURGERS SOLUTIONS
The Burgers equations for the velocity field u(x,t) = (u1(x,t),u2(x,t))are
∂u
∂t +
2
X
j=1
uj
∂
∂xj
u= ∆u, x= (x1,x2)∈R2.
We work with the Fourier transform (multiplied by i) v(k,t) = (v1(k,t),v2(k,t)) :
v(k,t) = i 2π
Z
R2
u(x,t)eihk,xidx, k=(k1,k2)∈R2 h·,·idenotes the scalar product inR2.
2. EXPLODING COMPLEX BURGERS SOLUTIONS
The Burgers equations for the velocity field u(x,t) = (u1(x,t),u2(x,t))are
∂u
∂t +
2
X
j=1
uj
∂
∂xj
u= ∆u, x= (x1,x2)∈R2.
We work with the Fourier transform (multiplied by i) v(k,t) = (v1(k,t),v2(k,t)) :
v(k,t) = i 2π
Z
R2
u(x,t)eihk,xidx, k=(k1,k2)∈R2 h·,·idenotes the scalar product inR2.
2. EXPLODING COMPLEX BURGERS SOLUTIONS
The Burgers equations for the velocity field u(x,t) = (u1(x,t),u2(x,t))are
∂u
∂t +
2
X
j=1
uj
∂
∂xj
u= ∆u, x= (x1,x2)∈R2.
We work with the Fourier transform (multiplied by i) v(k,t) = (v1(k,t),v2(k,t)) :
v(k,t) = i 2π
Z
R2
u(x,t)eihk,xidx, k=(k1,k2)∈R2 h·,·idenotes the scalar product inR2.
2. EXPLODING COMPLEX BURGERS SOLUTIONS
The Burgers equations for the velocity field u(x,t) = (u1(x,t),u2(x,t))are
∂u
∂t +
2
X
j=1
uj
∂
∂xj
u= ∆u, x= (x1,x2)∈R2.
We work with the Fourier transform (multiplied by i) v(k,t) = (v1(k,t),v2(k,t)) :
v(k,t) = i 2π
Z
R2
u(x,t)eihk,xidx, k=(k1,k2)∈R2 h·,·idenotes the scalar product inR2.
2. EXPLODING COMPLEX BURGERS SOLUTIONS
The Burgers equations for the velocity field u(x,t) = (u1(x,t),u2(x,t))are
∂u
∂t +
2
X
j=1
uj
∂
∂xj
u= ∆u, x= (x1,x2)∈R2.
We work with the Fourier transform (multiplied by i) v(k,t) = (v1(k,t),v2(k,t)) :
v(k,t) = i 2π
Z
R2
u(x,t)eihk,xidx, k=(k1,k2)∈R2 h·,·idenotes the scalar product inR2.
2. EXPLODING COMPLEX BURGERS SOLUTIONS
The Burgers equations for the velocity field u(x,t) = (u1(x,t),u2(x,t))are
∂u
∂t +
2
X
j=1
uj
∂
∂xj
u= ∆u, x= (x1,x2)∈R2.
We work with the Fourier transform (multiplied by i) v(k,t) = (v1(k,t),v2(k,t)) :
v(k,t) = i 2π
Z
R2
u(x,t)eihk,xidx, k=(k1,k2)∈R2 h·,·idenotes the scalar product inR2.
vsatisfies the integral equation
v(k,t) =e−tk2v(k,0) + +
Z t
0
e−(t−s)k2ds Z
R2
hv(k−k0,s),k0iv(k0,s)dk0, (1) with some initial conditionv(k,0).
(As usualk2=k12+k22.)
We consider real solutionsv(k,t), corresponding to complex solutions inx-space.
The choice of the initial data for the explosion is done according to the analysis of [Li, Sinai, 2010]: they are concentrated
around a pointk(0) = (a,a), witha>0 large enough:
vsatisfies the integral equation
v(k,t) =e−tk2v(k,0) + +
Z t
0
e−(t−s)k2ds Z
R2
hv(k−k0,s),k0iv(k0,s)dk0, (1)
with some initial conditionv(k,0). (As usualk2=k12+k22.)
We consider real solutionsv(k,t), corresponding to complex solutions inx-space.
The choice of the initial data for the explosion is done according to the analysis of [Li, Sinai, 2010]: they are concentrated
around a pointk(0) = (a,a), witha>0 large enough:
vsatisfies the integral equation
v(k,t) =e−tk2v(k,0) + +
Z t
0
e−(t−s)k2ds Z
R2
hv(k−k0,s),k0iv(k0,s)dk0, (1) with some initial conditionv(k,0).
(As usualk2=k12+k22.)
We consider real solutionsv(k,t), corresponding to complex solutions inx-space.
The choice of the initial data for the explosion is done according to the analysis of [Li, Sinai, 2010]: they are concentrated
around a pointk(0) = (a,a), witha>0 large enough:
vsatisfies the integral equation
v(k,t) =e−tk2v(k,0) + +
Z t
0
e−(t−s)k2ds Z
R2
hv(k−k0,s),k0iv(k0,s)dk0, (1) with some initial conditionv(k,0).
(As usualk2=k12+k22.)
We consider real solutionsv(k,t), corresponding to complex solutions inx-space.
The choice of the initial data for the explosion is done according to the analysis of [Li, Sinai, 2010]: they are concentrated
around a pointk(0) = (a,a), witha>0 large enough:
vsatisfies the integral equation
v(k,t) =e−tk2v(k,0) + +
Z t
0
e−(t−s)k2ds Z
R2
hv(k−k0,s),k0iv(k0,s)dk0, (1) with some initial conditionv(k,0).
(As usualk2=k12+k22.)
We consider real solutionsv(k,t), corresponding to complex solutions inx-space.
The choice of the initial data for the explosion is done according to the analysis of [Li, Sinai, 2010]:
they are concentrated around a pointk(0) = (a,a), witha>0 large enough:
vsatisfies the integral equation
v(k,t) =e−tk2v(k,0) + +
Z t
0
e−(t−s)k2ds Z
R2
hv(k−k0,s),k0iv(k0,s)dk0, (1) with some initial conditionv(k,0).
(As usualk2=k12+k22.)
We consider real solutionsv(k,t), corresponding to complex solutions inx-space.
The choice of the initial data for the explosion is done according to the analysis of [Li, Sinai, 2010]: they are concentrated
around a pointk(0) = (a,a), witha>0 large enough:
If|k−k(0)| ≤R for some0<R<|k(0)|, we set
v1(k,0) = B 2πσ2e−
(k−k(0))2 2σ2
B1(k) +φ(1)(k−k(0))
(2a) v2(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B2(k) +φ(2)(k−k(0))
(2b) andv(k,0) = 0if|k−k(0)|>R.
Here
B1(k) = 1 +b0+ (b(1),k−k(0)) B2(k) = 1−b0+ (b(2),k−k(0)). B∈R+,b0∈R,b(1),b(2) ∈R2, and
If|k−k(0)| ≤R for some0<R<|k(0)|, we set v1(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B1(k) +φ(1)(k−k(0))
(2a) v2(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B2(k) +φ(2)(k−k(0))
(2b)
andv(k,0) = 0if|k−k(0)|>R. Here
B1(k) = 1 +b0+ (b(1),k−k(0)) B2(k) = 1−b0+ (b(2),k−k(0)). B∈R+,b0∈R,b(1),b(2) ∈R2, and
If|k−k(0)| ≤R for some0<R<|k(0)|, we set v1(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B1(k) +φ(1)(k−k(0))
(2a) v2(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B2(k) +φ(2)(k−k(0))
(2b) andv(k,0) = 0if|k−k(0)|>R.
Here
B1(k) = 1 +b0+ (b(1),k−k(0)) B2(k) = 1−b0+ (b(2),k−k(0)). B∈R+,b0∈R,b(1),b(2) ∈R2, and
If|k−k(0)| ≤R for some0<R<|k(0)|, we set v1(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B1(k) +φ(1)(k−k(0))
(2a) v2(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B2(k) +φ(2)(k−k(0))
(2b) andv(k,0) = 0if|k−k(0)|>R.
Here
B1(k) = 1 +b0+ (b(1),k−k(0)) B2(k) = 1−b0+ (b(2),k−k(0)).
B∈R+,b0∈R,b(1),b(2) ∈R2, and
If|k−k(0)| ≤R for some0<R<|k(0)|, we set v1(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B1(k) +φ(1)(k−k(0))
(2a) v2(k,0) = B
2πσ2e−
(k−k(0))2 2σ2
B2(k) +φ(2)(k−k(0))
(2b) andv(k,0) = 0if|k−k(0)|>R.
Here
B1(k) = 1 +b0+ (b(1),k−k(0)) B2(k) = 1−b0+ (b(2),k−k(0)).
B∈R+,b0∈R,b(1),b(2) ∈R2, and
φ(1), φ(2) are elements of the Hilbert spaceHwith norm kφk2 = 1
2πσ2 Z
R2
|φ(k)|2e−k
2
2σ2d2k (3)
and are orthogonal to the constants and to the linear functions k1−a,k2−a.
Forσ,a and φ(1), φ(2) fixed, the equations (2a,b) define a 6-parameter family of initial conditions.
The parameters areB and b= (b0,b(1),b(2)): b(i)= (b1(i),b2(i)),i = 1,2.
Forb we use the normkbk:= max{|b0|,|bj(i)|,i,j = 1,2}.
φ(1), φ(2) are elements of the Hilbert spaceHwith norm kφk2 = 1
2πσ2 Z
R2
|φ(k)|2e−k
2
2σ2d2k (3)
and are orthogonal to the constants and to the linear functions k1−a,k2−a.
Forσ,a and φ(1), φ(2) fixed, the equations (2a,b) define a 6-parameter family of initial conditions.
The parameters areB and b= (b0,b(1),b(2)): b(i)= (b1(i),b2(i)),i = 1,2.
Forb we use the normkbk:= max{|b0|,|bj(i)|,i,j = 1,2}.
φ(1), φ(2) are elements of the Hilbert spaceHwith norm kφk2 = 1
2πσ2 Z
R2
|φ(k)|2e−k
2
2σ2d2k (3)
and are orthogonal to the constants and to the linear functions k1−a,k2−a.
Forσ,a andφ(1), φ(2) fixed, the equations (2a,b) define a 6-parameter family of initial conditions.
The parameters areB and b= (b0,b(1),b(2)): b(i)= (b1(i),b2(i)),i = 1,2.
Forb we use the normkbk:= max{|b0|,|bj(i)|,i,j = 1,2}.
φ(1), φ(2) are elements of the Hilbert spaceHwith norm kφk2 = 1
2πσ2 Z
R2
|φ(k)|2e−k
2
2σ2d2k (3)
and are orthogonal to the constants and to the linear functions k1−a,k2−a.
Forσ,a andφ(1), φ(2) fixed, the equations (2a,b) define a 6-parameter family of initial conditions.
The parameters areB and b= (b0,b(1),b(2)):
b(i)= (b1(i),b2(i)),i = 1,2.
Forb we use the normkbk:= max{|b0|,|bj(i)|,i,j = 1,2}.
φ(1), φ(2) are elements of the Hilbert spaceHwith norm kφk2 = 1
2πσ2 Z
R2
|φ(k)|2e−k
2
2σ2d2k (3)
and are orthogonal to the constants and to the linear functions k1−a,k2−a.
Forσ,a andφ(1), φ(2) fixed, the equations (2a,b) define a 6-parameter family of initial conditions.
The parameters areB and b= (b0,b(1),b(2)):
b(i)= (b1(i),b2(i)),i = 1,2.
Forb we use the normkbk:= max{|b0|,|bj(i)|,i,j = 1,2}.
φ(1), φ(2) are elements of the Hilbert spaceHwith norm kφk2 = 1
2πσ2 Z
R2
|φ(k)|2e−k
2
2σ2d2k (3)
and are orthogonal to the constants and to the linear functions k1−a,k2−a.
Forσ,a andφ(1), φ(2) fixed, the equations (2a,b) define a 6-parameter family of initial conditions.
The parameters areB and b= (b0,b(1),b(2)):
b(i)= (b1(i),b2(i)),i = 1,2.
Forb we use the normkbk:= max{|b0|,|bj(i)|,i,j = 1,2}.
The following theorem is proved in [Li, Sinai, 2010].
Theorem. For any family of initial data (2a,b), one can find ρ0 >0, a time interval J = [τ1, τ2],0< τ1 < τ2 and functions B(τ), andb(τ) onJ, with kb(τ)k ≤ρ0, such that the solution of the Burgers equations with initial data specified by B(τ),b(τ), τ ∈ J, develop a singularity of the energy at t=τ.
A sketch of the proof is necessary to understand the nature and the behavior of the singularities.
The proof is based on a variant of the renormalization group method.
Write the initial data asw(A)(k) =Aw(1)(k), where Ais a parameter, andw(1) is a function of the type (2a,b).
The following theorem is proved in [Li, Sinai, 2010].
Theorem. For any family of initial data (2a,b), one can find ρ0 >0, a time interval J = [τ1, τ2],0< τ1 < τ2 and functions B(τ), andb(τ) onJ, with kb(τ)k ≤ρ0, such that the solution of the Burgers equations with initial data specified by B(τ),b(τ), τ ∈ J, develop a singularity of the energy at t=τ.
A sketch of the proof is necessary to understand the nature and the behavior of the singularities.
The proof is based on a variant of the renormalization group method.
Write the initial data asw(A)(k) =Aw(1)(k), where Ais a parameter, andw(1) is a function of the type (2a,b).
The following theorem is proved in [Li, Sinai, 2010].
Theorem. For any family of initial data (2a,b), one can find ρ0 >0, a time interval J = [τ1, τ2],0< τ1 < τ2 and functions B(τ), andb(τ) onJ, with kb(τ)k ≤ρ0, such that the solution of the Burgers equations with initial data specified by B(τ),b(τ), τ ∈ J, develop a singularity of the energy at t=τ.
A sketch of the proof is necessary to understand the nature and the behavior of the singularities.
The proof is based on a variant of the renormalization group method.
Write the initial data asw(A)(k) =Aw(1)(k), where Ais a parameter, andw(1) is a function of the type (2a,b).
The following theorem is proved in [Li, Sinai, 2010].
Theorem. For any family of initial data (2a,b), one can find ρ0 >0, a time interval J = [τ1, τ2],0< τ1 < τ2 and functions B(τ), andb(τ) onJ, with kb(τ)k ≤ρ0, such that the solution of the Burgers equations with initial data specified by B(τ),b(τ), τ ∈ J, develop a singularity of the energy at t=τ.
A sketch of the proof is necessary to understand the nature and the behavior of the singularities.
The proof is based on a variant of the renormalization group method.
Write the initial data asw(A)(k) =Aw(1)(k), where Ais a parameter, andw(1) is a function of the type (2a,b).
The following theorem is proved in [Li, Sinai, 2010].
Theorem. For any family of initial data (2a,b), one can find ρ0 >0, a time interval J = [τ1, τ2],0< τ1 < τ2 and functions B(τ), andb(τ) onJ, with kb(τ)k ≤ρ0, such that the solution of the Burgers equations with initial data specified by B(τ),b(τ), τ ∈ J, develop a singularity of the energy at t=τ.
A sketch of the proof is necessary to understand the nature and the behavior of the singularities.
The proof is based on a variant of the renormalization group method.
Write the initial data asw(A)(k) =Aw(1)(k), where Ais a parameter, andw(1) is a function of the type (2a,b).
The solution of the Burgers equation is written as a series
w(A)(k,t) =A e−tk2w(1)(k)+ +
Z t
0
e−k2(t−s)
∞
X
p=2
Apg(p)(k,s)ds. (4) We find recursive relations inp forg(p)(k,t) which remind of the famous BBGKY hierarchy of statistical physics:
Settingg(1)(k,t) =e−tk2w(1)(k) and g(2)(k,t) =
Z
R2
hw(1)(k−k0,s),k0iw(1)(k0,s)e−t(k−k
0)2−t(k0)2dk0,
we find forp >2 the recursive relations
The solution of the Burgers equation is written as a series w(A)(k,t) =A e−tk2w(1)(k)+
+ Z t
0
e−k2(t−s)
∞
X
p=2
Apg(p)(k,s)ds. (4)
We find recursive relations inp forg(p)(k,t) which remind of the famous BBGKY hierarchy of statistical physics:
Settingg(1)(k,t) =e−tk2w(1)(k) and g(2)(k,t) =
Z
R2
hw(1)(k−k0,s),k0iw(1)(k0,s)e−t(k−k
0)2−t(k0)2dk0,
we find forp >2 the recursive relations
The solution of the Burgers equation is written as a series w(A)(k,t) =A e−tk2w(1)(k)+
+ Z t
0
e−k2(t−s)
∞
X
p=2
Apg(p)(k,s)ds. (4) We find recursive relations inp forg(p)(k,t) which remind of the famous BBGKY hierarchy of statistical physics:
Settingg(1)(k,t) =e−tk2w(1)(k) and g(2)(k,t) =
Z
R2
hw(1)(k−k0,s),k0iw(1)(k0,s)e−t(k−k
0)2−t(k0)2dk0,
we find forp >2 the recursive relations
The solution of the Burgers equation is written as a series w(A)(k,t) =A e−tk2w(1)(k)+
+ Z t
0
e−k2(t−s)
∞
X
p=2
Apg(p)(k,s)ds. (4) We find recursive relations inp forg(p)(k,t) which remind of the famous BBGKY hierarchy of statistical physics:
Settingg(1)(k,t) =e−tk2w(1)(k)
and g(2)(k,t) =
Z
R2
hw(1)(k−k0,s),k0iw(1)(k0,s)e−t(k−k
0)2−t(k0)2dk0,
we find forp >2 the recursive relations
The solution of the Burgers equation is written as a series w(A)(k,t) =A e−tk2w(1)(k)+
+ Z t
0
e−k2(t−s)
∞
X
p=2
Apg(p)(k,s)ds. (4) We find recursive relations inp forg(p)(k,t) which remind of the famous BBGKY hierarchy of statistical physics:
Settingg(1)(k,t) =e−tk2w(1)(k) and g(2)(k,t) =
Z
R2
hw(1)(k−k0,s),k0iw(1)(k0,s)e−t(k−k
0)2−t(k0)2dk0,
we find forp >2 the recursive relations
g(p)(k,t) = Z t
0
ds2·
· Z
R2
hw(1)(k−k0,s),k0ig(p−1)(k0,s)e−t(k−k
0)2−t(k0)2dk0,+
+ X
p1+p2=p p1,p2>1
Z t
0
ds1 Z t
0
ds2· (5)
· Z
R2
hg(p1)(k−k0,s1),k0ig(p2)(k0,s2)e−(t−s1)(k−k
0)2−(t−s2)(k0)2dk0+
+ Z t
0
ds1 Z
R2
hg(p−1)(k−k0,s1),k0iw1(k0)e−(t−s1)(k−k
0)2−t(k0)2dk0
g(p)(k,t) = Z t
0
ds2·
· Z
R2
hw(1)(k−k0,s),k0ig(p−1)(k0,s)e−t(k−k
0)2−t(k0)2dk0,+
+ X
p1+p2=p p1,p2>1
Z t
0
ds1 Z t
0
ds2· (5)
· Z
R2
hg(p1)(k−k0,s1),k0ig(p2)(k0,s2)e−(t−s1)(k−k
0)2−(t−s2)(k0)2dk0+
+ Z t
0
ds1 Z
R2
hg(p−1)(k−k0,s1),k0iw1(k0)e−(t−s1)(k−k
0)2−t(k0)2dk0
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations. We setk=pk(0)+√
paY and h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6). For largep the functionsh(p) are concentrated around|Y| ≈1 and we can neglect the first and third term of the recursive relation (5). We also neglect terms like Y
0
√p in the exponent, and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations. We setk=pk(0)+√
paY and h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6). For largep the functionsh(p) are concentrated around|Y| ≈1 and we can neglect the first and third term of the recursive relation (5). We also neglect terms like Y
0
√p in the exponent, and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations.
We setk=pk(0)+√
paY and h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6). For largep the functionsh(p) are concentrated around|Y| ≈1 and we can neglect the first and third term of the recursive relation (5). We also neglect terms like Y
0
√p in the exponent, and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations.
We setk=pk(0)+√
paY and
h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6). For largep the functionsh(p) are concentrated around|Y| ≈1 and we can neglect the first and third term of the recursive relation (5). We also neglect terms like Y
0
√p in the exponent, and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations.
We setk=pk(0)+√
paY and h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6).
For largep the functionsh(p) are concentrated around|Y| ≈1 and we can neglect the first and third term of the recursive relation (5). We also neglect terms like Y
0
√p in the exponent, and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations.
We setk=pk(0)+√
paY and h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6).
For largep the functions h(p) are concentrated around|Y| ≈1
and we can neglect the first and third term of the recursive relation (5). We also neglect terms like Y
0
√p in the exponent, and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations.
We setk=pk(0)+√
paY and h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6).
For largep the functions h(p) are concentrated around|Y| ≈1 and we can neglect the first and third term of the recursive relation (5).
We also neglect terms like Y
0
√p in the exponent, and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations.
We setk=pk(0)+√
paY and h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6).
For largep the functions h(p) are concentrated around|Y| ≈1 and we can neglect the first and third term of the recursive relation (5). We also neglect terms like Y
0
√p in the exponent,
and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
The main contribution to the integrals inds1,ds2 comes from a small neighborhood of the upper boundt.
This is due to viscosity and would not apply to the Euler system.
A scaling transformation ofkand the functiong(p) greatly simplifies the asymptotics for largep of the recursive equations.
We setk=pk(0)+√
paY and h(p)(Y,t) =g(p)(pk(0)+√
paY,s), Y∈R2. (6).
For largep the functions h(p) are concentrated around|Y| ≈1 and we can neglect the first and third term of the recursive relation (5). We also neglect terms like Y
0
√p in the exponent, and introduce the adapted variables
sj =s(1−θj/pj2), j = 1,2, γ =p1/p,p2/p = 1−γ.
Settingk0 =p2k(0)+√
paY0, we finally get the simple recursive relation
h(p)(Y,t) = p2
4a2· (7)
· X
p1+p2=p p1,p2>1
1 p12p22
Z
R2
p2 p
2
X
j=1
h(pj 1) Y−Y0
√γ ,t
!
h(p2) Y0
√1−γ,t
! dY0.
By induction, assume that there are nested time intervals J(p+1)⊆ J(p), such that for t∈ J(p),
h(r)(Y,t) =r Z(t)(Λp(t))re−Y
2 2σ2
2πσ2 (H(Y) +δr(Y,t)), (8) for some someσ >0 and all r<p, whereδr is small andZ,Λp are functions to be determined.
Settingk0 =p2k(0)+√
paY0, we finally get the simple recursive relation
h(p)(Y,t) = p2
4a2· (7)
· X
p1+p2=p p1,p2>1
1 p12p22
Z
R2
p2 p
2
X
j=1
h(pj 1) Y−Y0
√γ ,t
!
h(p2) Y0
√1−γ,t
! dY0.
By induction, assume that there are nested time intervals J(p+1)⊆ J(p), such that for t∈ J(p),
h(r)(Y,t) =r Z(t)(Λp(t))re−Y
2 2σ2
2πσ2 (H(Y) +δr(Y,t)), (8) for some someσ >0 and all r<p, whereδr is small andZ,Λp are functions to be determined.
Settingk0 =p2k(0)+√
paY0, we finally get the simple recursive relation
h(p)(Y,t) = p2
4a2· (7)
· X
p1+p2=p p1,p2>1
1 p12p22
Z
R2
p2 p
2
X
j=1
h(pj 1) Y−Y0
√γ ,t
!
h(p2) Y0
√1−γ,t
! dY0.
By induction, assume that there are nested time intervals J(p+1)⊆ J(p), such that for t∈ J(p),
h(r)(Y,t) =r Z(t)(Λp(t))re−Y
2 2σ2
2πσ2 (H(Y) +δr(Y,t)), (8) for some someσ >0 and all r<p,
whereδr is small andZ,Λp are functions to be determined.
