Nonlinear transport phenomena:models, method of solving and unusual
features
Vsevolod Vladimirov
AGH University of Science and technology, Faculty of AppliedMathematics
Krakow, August 10, 2010
Summer School: KPI, 2010 Nonlinear transport phenomena 1 / 38
THE AIM OF THIS LECTURE IS TO PRESENT THESIMPLEST PATTERNS SUPPORTED BY TRANSPORTEQUATIONS
WHAT ARE PATTERNS?
Summer School: KPI, 2010 Nonlinear transport phenomena 2 / 38
THE AIM OF THIS LECTURE IS TO PRESENT THESIMPLEST PATTERNS SUPPORTED BY TRANSPORTEQUATIONS
WHAT ARE PATTERNS?
Summer School: KPI, 2010 Nonlinear transport phenomena 2 / 38
Examples of patterns:I Solution to the system of reaction-diffusion equation
ut = Du ∆u + a(1− u)
vt = Dv ∆v + u v2 − k v
u(0, x) = f(x), v(0, x) = g(x),
∆ =∑2
i=1∂ 2
∂x2i
I periodically initiated cordial impulse; formation of thetrain of solitary waves;
I tsunami;I tornadoI etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 3 / 38
Examples of patterns:I Solution to the system of reaction-diffusion equation
ut = Du ∆u + a(1− u)
vt = Dv ∆v + u v2 − k v
u(0, x) = f(x), v(0, x) = g(x),
∆ =∑2
i=1∂ 2
∂x2i
I periodically initiated cordial impulse; formation of thetrain of solitary waves;
I tsunami;I tornadoI etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 3 / 38
Examples of patterns:I Solution to the system of reaction-diffusion equation
ut = Du ∆u + a(1− u)
vt = Dv ∆v + u v2 − k v
u(0, x) = f(x), v(0, x) = g(x),
∆ =∑2
i=1∂ 2
∂x2i
I periodically initiated cordial impulse; formation of thetrain of solitary waves;
I tsunami;I tornadoI etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 3 / 38
Examples of patterns:I Solution to the system of reaction-diffusion equation
ut = Du ∆u + a(1− u)
vt = Dv ∆v + u v2 − k v
u(0, x) = f(x), v(0, x) = g(x),
∆ =∑2
i=1∂ 2
∂x2i
I periodically initiated cordial impulse; formation of thetrain of solitary waves;
I tsunami;I tornadoI etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 3 / 38
Examples of patterns:I Solution to the system of reaction-diffusion equation
ut = Du ∆u + a(1− u)
vt = Dv ∆v + u v2 − k v
u(0, x) = f(x), v(0, x) = g(x),
∆ =∑2
i=1∂ 2
∂x2i
I periodically initiated cordial impulse; formation of thetrain of solitary waves;
I tsunami;I tornadoI etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 3 / 38
Examples of patterns:I Solution to the system of reaction-diffusion equation
ut = Du ∆u + a(1− u)
vt = Dv ∆v + u v2 − k v
u(0, x) = f(x), v(0, x) = g(x),
∆ =∑2
i=1∂ 2
∂x2i
I periodically initiated cordial impulse; formation of thetrain of solitary waves;
I tsunami;I tornadoI etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 3 / 38
Examples of patterns:I Solution to the system of reaction-diffusion equation
ut = Du ∆u + a(1− u)
vt = Dv ∆v + u v2 − k v
u(0, x) = f(x), v(0, x) = g(x),
∆ =∑2
i=1∂ 2
∂x2i
I periodically initiated cordial impulse; formation of thetrain of solitary waves;
I tsunami;I tornadoI etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 3 / 38
Our aim to discuss the patterns supported by the varioustransport equations in case of one spatial variable.
Advantages:I ability to present and analyze the analytical solutions to
non-linear modelsI the possibility to analyze the problem by means of
qualitative theory methods.
By patterns we temporarily mean:I non-monotonic solution to the modelling system,
maintaining their shape during the evolution;I or the non-monotonic solutions evolving in a self-similar
modeI or the non-monotonic solutions tending to infinity in finite
time (blowing-up patterns)Summer School: KPI, 2010 Nonlinear transport phenomena 4 / 38
Our aim to discuss the patterns supported by the varioustransport equations in case of one spatial variable.
Advantages:I ability to present and analyze the analytical solutions to
non-linear modelsI the possibility to analyze the problem by means of
qualitative theory methods.
By patterns we temporarily mean:I non-monotonic solution to the modelling system,
maintaining their shape during the evolution;I or the non-monotonic solutions evolving in a self-similar
modeI or the non-monotonic solutions tending to infinity in finite
time (blowing-up patterns)Summer School: KPI, 2010 Nonlinear transport phenomena 4 / 38
Our aim to discuss the patterns supported by the varioustransport equations in case of one spatial variable.
Advantages:I ability to present and analyze the analytical solutions to
non-linear modelsI the possibility to analyze the problem by means of
qualitative theory methods.
By patterns we temporarily mean:I non-monotonic solution to the modelling system,
maintaining their shape during the evolution;I or the non-monotonic solutions evolving in a self-similar
modeI or the non-monotonic solutions tending to infinity in finite
time (blowing-up patterns)Summer School: KPI, 2010 Nonlinear transport phenomena 4 / 38
Our aim to discuss the patterns supported by the varioustransport equations in case of one spatial variable.
Advantages:I ability to present and analyze the analytical solutions to
non-linear modelsI the possibility to analyze the problem by means of
qualitative theory methods.
By patterns we temporarily mean:I non-monotonic solution to the modelling system,
maintaining their shape during the evolution;I or the non-monotonic solutions evolving in a self-similar
modeI or the non-monotonic solutions tending to infinity in finite
time (blowing-up patterns)Summer School: KPI, 2010 Nonlinear transport phenomena 4 / 38
Our aim to discuss the patterns supported by the varioustransport equations in case of one spatial variable.
Advantages:I ability to present and analyze the analytical solutions to
non-linear modelsI the possibility to analyze the problem by means of
qualitative theory methods.
By patterns we temporarily mean:I non-monotonic solution to the modelling system,
maintaining their shape during the evolution;I or the non-monotonic solutions evolving in a self-similar
modeI or the non-monotonic solutions tending to infinity in finite
time (blowing-up patterns)Summer School: KPI, 2010 Nonlinear transport phenomena 4 / 38
Our aim to discuss the patterns supported by the varioustransport equations in case of one spatial variable.
Advantages:I ability to present and analyze the analytical solutions to
non-linear modelsI the possibility to analyze the problem by means of
qualitative theory methods.
By patterns we temporarily mean:I non-monotonic solution to the modelling system,
maintaining their shape during the evolution;I or the non-monotonic solutions evolving in a self-similar
modeI or the non-monotonic solutions tending to infinity in finite
time (blowing-up patterns)Summer School: KPI, 2010 Nonlinear transport phenomena 4 / 38
Heat transport: the balance equation
d
d t
∫Ω
ρ u(t, x)d x = −∫
∂ Ω~q d~σ +
∫Ω
f (u(t, x); t, x)d x, (1)
where ~q is the density of the heat flux on the boundary ∂ Ω.On virtue of the Fick law, is as follows:
~q = −κ∇u(t, x);
u(t, x) is the energy per unit mass (the temperature);ρ is the heat capacity per unit volume;
κ(u; t, x) is the heat transport coefficient;f (u(t, x); t, x) is the voluminal heat source.
Summer School: KPI, 2010 Nonlinear transport phenomena 5 / 38
Heat transport: the balance equation
d
d t
∫Ω
ρ u(t, x)d x = −∫
∂ Ω~q d~σ +
∫Ω
f (u(t, x); t, x)d x, (1)
where ~q is the density of the heat flux on the boundary ∂ Ω.On virtue of the Fick law, is as follows:
~q = −κ∇u(t, x);
u(t, x) is the energy per unit mass (the temperature);ρ is the heat capacity per unit volume;
κ(u; t, x) is the heat transport coefficient;f (u(t, x); t, x) is the voluminal heat source.
Summer School: KPI, 2010 Nonlinear transport phenomena 5 / 38
Heat transport: the balance equation
d
d t
∫Ω
ρ u(t, x)d x = −∫
∂ Ω~q d~σ +
∫Ω
f (u(t, x); t, x)d x, (1)
where ~q is the density of the heat flux on the boundary ∂ Ω.On virtue of the Fick law, is as follows:
~q = −κ∇u(t, x);
u(t, x) is the energy per unit mass (the temperature);ρ is the heat capacity per unit volume;
κ(u; t, x) is the heat transport coefficient;f (u(t, x); t, x) is the voluminal heat source.
Summer School: KPI, 2010 Nonlinear transport phenomena 5 / 38
Heat transport: the balance equation
d
d t
∫Ω
ρ u(t, x)d x = −∫
∂ Ω~q d~σ +
∫Ω
f (u(t, x); t, x)d x, (1)
where ~q is the density of the heat flux on the boundary ∂ Ω.On virtue of the Fick law, is as follows:
~q = −κ∇u(t, x);
u(t, x) is the energy per unit mass (the temperature);ρ is the heat capacity per unit volume;
κ(u; t, x) is the heat transport coefficient;f (u(t, x); t, x) is the voluminal heat source.
Summer School: KPI, 2010 Nonlinear transport phenomena 5 / 38
Using the Green-Gauss-Ostrogradsky theorem, we are able towrite down the following identities:
−∫
∂ Ω~q d~σ ≡
∫∂ Ω
κ∇u(t, x) d~σ =
∫Ω∇ [κ∇u(t, x)] d x.
So, the balance equation can be rewritten as∫Ω
ρ
∂ u(t, x)
∂ t−∇ [κ∇u(t, x)]− f [u(t, x); t, x]
d x = 0.
Since the volume Ω is arbitrary, fulfillment of the balanceequation is possible providing that the integrand is equal to zero
In the case of one spatial variable, we get this way the equation
∂ u(t, x)
∂ t=
∂
∂ x
[κ (u(t, x), t, x)
∂ u(t, x)
∂ x
]+f (u(t, x), t, x) ,
(2)
κ = κ/ρ, f = f/ρ.
Summer School: KPI, 2010 Nonlinear transport phenomena 6 / 38
Using the Green-Gauss-Ostrogradsky theorem, we are able towrite down the following identities:
−∫
∂ Ω~q d~σ ≡
∫∂ Ω
κ∇u(t, x) d~σ =
∫Ω∇ [κ∇u(t, x)] d x.
So, the balance equation can be rewritten as∫Ω
ρ
∂ u(t, x)
∂ t−∇ [κ∇u(t, x)]− f [u(t, x); t, x]
d x = 0.
Since the volume Ω is arbitrary, fulfillment of the balanceequation is possible providing that the integrand is equal to zero
In the case of one spatial variable, we get this way the equation
∂ u(t, x)
∂ t=
∂
∂ x
[κ (u(t, x), t, x)
∂ u(t, x)
∂ x
]+f (u(t, x), t, x) ,
(2)
κ = κ/ρ, f = f/ρ.
Summer School: KPI, 2010 Nonlinear transport phenomena 6 / 38
Using the Green-Gauss-Ostrogradsky theorem, we are able towrite down the following identities:
−∫
∂ Ω~q d~σ ≡
∫∂ Ω
κ∇u(t, x) d~σ =
∫Ω∇ [κ∇u(t, x)] d x.
So, the balance equation can be rewritten as∫Ω
ρ
∂ u(t, x)
∂ t−∇ [κ∇u(t, x)]− f [u(t, x); t, x]
d x = 0.
Since the volume Ω is arbitrary, fulfillment of the balanceequation is possible providing that the integrand is equal to zero
In the case of one spatial variable, we get this way the equation
∂ u(t, x)
∂ t=
∂
∂ x
[κ (u(t, x), t, x)
∂ u(t, x)
∂ x
]+f (u(t, x), t, x) ,
(2)
κ = κ/ρ, f = f/ρ.
Summer School: KPI, 2010 Nonlinear transport phenomena 6 / 38
Using the Green-Gauss-Ostrogradsky theorem, we are able towrite down the following identities:
−∫
∂ Ω~q d~σ ≡
∫∂ Ω
κ∇u(t, x) d~σ =
∫Ω∇ [κ∇u(t, x)] d x.
So, the balance equation can be rewritten as∫Ω
ρ
∂ u(t, x)
∂ t−∇ [κ∇u(t, x)]− f [u(t, x); t, x]
d x = 0.
Since the volume Ω is arbitrary, fulfillment of the balanceequation is possible providing that the integrand is equal to zero
In the case of one spatial variable, we get this way the equation
∂ u(t, x)
∂ t=
∂
∂ x
[κ (u(t, x), t, x)
∂ u(t, x)
∂ x
]+f (u(t, x), t, x) ,
(2)
κ = κ/ρ, f = f/ρ.
Summer School: KPI, 2010 Nonlinear transport phenomena 6 / 38
Self-similar solutions to the heat equation.
Statement 1. Any physical law can be written down in theform
a = ϕ (a1, a2, ..., an; an+1, ...an+m) , where
a1, a2, ..., an
are the physical quantities expressed in the basic units,
a, an+1, ...an+m
are the physical quantities expressed in derived units.Example of basic units:
the length[L], the time [T ], the mass [M ]
Examples of physical quantities expressed in derivedunits:
V =d x
d t
[L
T
], F = m
d2 x
d t2
[M · LT 2
], etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 7 / 38
Self-similar solutions to the heat equation.
Statement 1. Any physical law can be written down in theform
a = ϕ (a1, a2, ..., an; an+1, ...an+m) , where
a1, a2, ..., an
are the physical quantities expressed in the basic units,
a, an+1, ...an+m
are the physical quantities expressed in derived units.Example of basic units:
the length[L], the time [T ], the mass [M ]
Examples of physical quantities expressed in derivedunits:
V =d x
d t
[L
T
], F = m
d2 x
d t2
[M · LT 2
], etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 7 / 38
Self-similar solutions to the heat equation.
Statement 1. Any physical law can be written down in theform
a = ϕ (a1, a2, ..., an; an+1, ...an+m) , where
a1, a2, ..., an
are the physical quantities expressed in the basic units,
a, an+1, ...an+m
are the physical quantities expressed in derived units.Example of basic units:
the length[L], the time [T ], the mass [M ]
Examples of physical quantities expressed in derivedunits:
V =d x
d t
[L
T
], F = m
d2 x
d t2
[M · LT 2
], etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 7 / 38
Self-similar solutions to the heat equation.
Statement 1. Any physical law can be written down in theform
a = ϕ (a1, a2, ..., an; an+1, ...an+m) , where
a1, a2, ..., an
are the physical quantities expressed in the basic units,
a, an+1, ...an+m
are the physical quantities expressed in derived units.Example of basic units:
the length[L], the time [T ], the mass [M ]
Examples of physical quantities expressed in derivedunits:
V =d x
d t
[L
T
], F = m
d2 x
d t2
[M · LT 2
], etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 7 / 38
Self-similar solutions to the heat equation.
Statement 1. Any physical law can be written down in theform
a = ϕ (a1, a2, ..., an; an+1, ...an+m) , where
a1, a2, ..., an
are the physical quantities expressed in the basic units,
a, an+1, ...an+m
are the physical quantities expressed in derived units.Example of basic units:
the length[L], the time [T ], the mass [M ]
Examples of physical quantities expressed in derivedunits:
V =d x
d t
[L
T
], F = m
d2 x
d t2
[M · LT 2
], etc., etc.
Summer School: KPI, 2010 Nonlinear transport phenomena 7 / 38
Statement 2. Derived physical quantities a, an+1, ...an+m canbe expressed through the basic ones in the following form:
a = ar11 ar2
2 ...arnn Π,
an+j = arj1
1 arj2
2 ...arjn
n Πj , j = 1, 2, ....,m,
where Π, Π1, ...., Πm are dimensionless parameters.So, the physical law
a = F (a1, ...an; ...an+j , ....)
is equivalent to
Π =1
ar11 ar2
2 ...arnn
F
[a1, a2, ..., an;...a
rj1
1 arj2
2 ...arjn
n Πj ...
]. (3)
Summer School: KPI, 2010 Nonlinear transport phenomena 8 / 38
Statement 2. Derived physical quantities a, an+1, ...an+m canbe expressed through the basic ones in the following form:
a = ar11 ar2
2 ...arnn Π,
an+j = arj1
1 arj2
2 ...arjn
n Πj , j = 1, 2, ....,m,
where Π, Π1, ...., Πm are dimensionless parameters.So, the physical law
a = F (a1, ...an; ...an+j , ....)
is equivalent to
Π =1
ar11 ar2
2 ...arnn
F
[a1, a2, ..., an;...a
rj1
1 arj2
2 ...arjn
n Πj ...
]. (3)
Summer School: KPI, 2010 Nonlinear transport phenomena 8 / 38
The essence of Π theorem
Theorem. The RHS of the formula (3) depends, at most,on the m dimensionless parameters Π1, ...., Πm, and doesnot depend on the dimension quantities a1, a2, ..., an.
1
So, the physical law
a = F (a1, ...an; ...an+j , ....)
in dimensionless variables takes the form to
Π = Ψ(Π1, Π2, ...Πm ) . (4)
1A. Samarskij, A. Mikhailov, Mathematical Modelling: Ideas, Methods,Examples, Mosclw, 1997, Ch. V.
Summer School: KPI, 2010 Nonlinear transport phenomena 9 / 38
The essence of Π theorem
Theorem. The RHS of the formula (3) depends, at most,on the m dimensionless parameters Π1, ...., Πm, and doesnot depend on the dimension quantities a1, a2, ..., an.
1
So, the physical law
a = F (a1, ...an; ...an+j , ....)
in dimensionless variables takes the form to
Π = Ψ(Π1, Π2, ...Πm ) . (4)
1A. Samarskij, A. Mikhailov, Mathematical Modelling: Ideas, Methods,Examples, Mosclw, 1997, Ch. V.
Summer School: KPI, 2010 Nonlinear transport phenomena 9 / 38
The essence of Π theorem
Theorem. The RHS of the formula (3) depends, at most,on the m dimensionless parameters Π1, ...., Πm, and doesnot depend on the dimension quantities a1, a2, ..., an.
1
So, the physical law
a = F (a1, ...an; ...an+j , ....)
in dimensionless variables takes the form to
Π = Ψ(Π1, Π2, ...Πm ) . (4)
1A. Samarskij, A. Mikhailov, Mathematical Modelling: Ideas, Methods,Examples, Mosclw, 1997, Ch. V.
Summer School: KPI, 2010 Nonlinear transport phenomena 9 / 38
Applications. Point explosion (linear case)
ut = κ ux x, (5)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (5) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[E · L2
T
], Q[E · L].
Derived units:
u =Q√κ t
Π, x =√
κ t Π1.
Thus, passing to dimensionless variables we get
Π =
√κ t
Qf(t, κ, Q;
√κ t Π1) = ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 10 / 38
Applications. Point explosion (linear case)
ut = κ ux x, (5)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (5) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[E · L2
T
], Q[E · L].
Derived units:
u =Q√κ t
Π, x =√
κ t Π1.
Thus, passing to dimensionless variables we get
Π =
√κ t
Qf(t, κ, Q;
√κ t Π1) = ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 10 / 38
Applications. Point explosion (linear case)
ut = κ ux x, (5)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (5) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[E · L2
T
], Q[E · L].
Derived units:
u =Q√κ t
Π, x =√
κ t Π1.
Thus, passing to dimensionless variables we get
Π =
√κ t
Qf(t, κ, Q;
√κ t Π1) = ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 10 / 38
Applications. Point explosion (linear case)
ut = κ ux x, (5)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (5) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[E · L2
T
], Q[E · L].
Derived units:
u =Q√κ t
Π, x =√
κ t Π1.
Thus, passing to dimensionless variables we get
Π =
√κ t
Qf(t, κ, Q;
√κ t Π1) = ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 10 / 38
Applications. Point explosion (linear case)
ut = κ ux x, (5)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (5) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[E · L2
T
], Q[E · L].
Derived units:
u =Q√κ t
Π, x =√
κ t Π1.
Thus, passing to dimensionless variables we get
Π =
√κ t
Qf(t, κ, Q;
√κ t Π1) = ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 10 / 38
Corollary.Ansatz
u =Q√κ t
ϕ(ξ), ξ = Π1
should strongly simplify the point explosion problem (5).In fact, inserting this ansatz into the heat transport equation,we get
2 ϕ[ξ] + ξ ϕ[ξ] + ϕ[ξ] ≡ d
d ξ(ξϕ[ξ] + 2 ϕ[ξ]) = 0, (6)
with the additional condition (as well expressed in thedimensionless variables):∫ ∞
−∞ϕ[ξ] dξ = 1. (7)
Summer School: KPI, 2010 Nonlinear transport phenomena 11 / 38
Corollary.Ansatz
u =Q√κ t
ϕ(ξ), ξ = Π1
should strongly simplify the point explosion problem (5).In fact, inserting this ansatz into the heat transport equation,we get
2 ϕ[ξ] + ξ ϕ[ξ] + ϕ[ξ] ≡ d
d ξ(ξϕ[ξ] + 2 ϕ[ξ]) = 0, (6)
with the additional condition (as well expressed in thedimensionless variables):∫ ∞
−∞ϕ[ξ] dξ = 1. (7)
Summer School: KPI, 2010 Nonlinear transport phenomena 11 / 38
Corollary.Ansatz
u =Q√κ t
ϕ(ξ), ξ = Π1
should strongly simplify the point explosion problem (5).In fact, inserting this ansatz into the heat transport equation,we get
2 ϕ[ξ] + ξ ϕ[ξ] + ϕ[ξ] ≡ d
d ξ(ξϕ[ξ] + 2 ϕ[ξ]) = 0, (6)
with the additional condition (as well expressed in thedimensionless variables):∫ ∞
−∞ϕ[ξ] dξ = 1. (7)
Summer School: KPI, 2010 Nonlinear transport phenomena 11 / 38
Statement 3. Solution to the problem
d
d ξ(ξϕ[ξ] + 2 ϕ[ξ]) = 0,
∫ ∞
−∞ϕ[ξ] dξ = 1
is given by the formula
ϕ[ξ] =1√4 π
e−ξ2
4 .
Corollary. The only solution to the linear point explosionproblem (5) is given by the formula
u =Q√
4 κπ te−
x2
4 κ t . (8)
Summer School: KPI, 2010 Nonlinear transport phenomena 12 / 38
Statement 3. Solution to the problem
d
d ξ(ξϕ[ξ] + 2 ϕ[ξ]) = 0,
∫ ∞
−∞ϕ[ξ] dξ = 1
is given by the formula
ϕ[ξ] =1√4 π
e−ξ2
4 .
Corollary. The only solution to the linear point explosionproblem (5) is given by the formula
u =Q√
4 κπ te−
x2
4 κ t . (8)
Summer School: KPI, 2010 Nonlinear transport phenomena 12 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 13 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 14 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 15 / 38
Remark 1. For Q = 1 the previous formula written as
E =1√
4 π κ te−
x2
4 κ t
defines so called fundamental solution of the operator
L =∂
∂ t− κ
∂2
∂ x2.
Knowledge of this solution enables to solve the initial valueproblem.
Theorem Solution to the initial value problem
ut − κux x, u(0, x) = F (x)
is given by the formula
u(t, x) = E ∗ F (t, x) ≡∫ +∞
−∞E (t, x− y) · F (y) d y
provided that the integral in the RHS does exist.
Summer School: KPI, 2010 Nonlinear transport phenomena 16 / 38
Remark 1. For Q = 1 the previous formula written as
E =1√
4 π κ te−
x2
4 κ t
defines so called fundamental solution of the operator
L =∂
∂ t− κ
∂2
∂ x2.
Knowledge of this solution enables to solve the initial valueproblem.
Theorem Solution to the initial value problem
ut − κux x, u(0, x) = F (x)
is given by the formula
u(t, x) = E ∗ F (t, x) ≡∫ +∞
−∞E (t, x− y) · F (y) d y
provided that the integral in the RHS does exist.
Summer School: KPI, 2010 Nonlinear transport phenomena 16 / 38
Peculiarity of the solution
u =Q√
4 κ π te−
x2
4 κ t
to the linear point explosion:
At any moment of time all the space ”knows ” that the pointexplosion took place. This is physically incorrect!!!
Summer School: KPI, 2010 Nonlinear transport phenomena 17 / 38
Peculiarity of the solution
u =Q√
4 κ π te−
x2
4 κ t
to the linear point explosion:
At any moment of time all the space ”knows ” that the pointexplosion took place. This is physically incorrect!!!
Summer School: KPI, 2010 Nonlinear transport phenomena 17 / 38
Point explosion (nonlinear case)
ut = κ [u ux]x , (9)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (9) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[L2
E · T
], Q[E · L].
Derived units:
u =Q2/3
[κ t]1/3Π, x = [κ Q t]1/3 Π1.
Thus, passing to dimensionless variables we get
Π =[κ t]1/3
Q2/3f
(t, κ, Q; Π1 [κ Q t]1/3
)= ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 18 / 38
Point explosion (nonlinear case)
ut = κ [u ux]x , (9)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (9) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[L2
E · T
], Q[E · L].
Derived units:
u =Q2/3
[κ t]1/3Π, x = [κ Q t]1/3 Π1.
Thus, passing to dimensionless variables we get
Π =[κ t]1/3
Q2/3f
(t, κ, Q; Π1 [κ Q t]1/3
)= ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 18 / 38
Point explosion (nonlinear case)
ut = κ [u ux]x , (9)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (9) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[L2
E · T
], Q[E · L].
Derived units:
u =Q2/3
[κ t]1/3Π, x = [κ Q t]1/3 Π1.
Thus, passing to dimensionless variables we get
Π =[κ t]1/3
Q2/3f
(t, κ, Q; Π1 [κ Q t]1/3
)= ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 18 / 38
Point explosion (nonlinear case)
ut = κ [u ux]x , (9)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (9) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[L2
E · T
], Q[E · L].
Derived units:
u =Q2/3
[κ t]1/3Π, x = [κ Q t]1/3 Π1.
Thus, passing to dimensionless variables we get
Π =[κ t]1/3
Q2/3f
(t, κ, Q; Π1 [κ Q t]1/3
)= ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 18 / 38
Point explosion (nonlinear case)
ut = κ [u ux]x , (9)
u(0, x) = 0, x ∈ R, x 6= 0,
∫ ∞
−∞u(t, x) d x = Q = const.
The solution to (9) can be expressed as u = f(t, κ, Q; x). Basicunits:
t[T ], κ
[L2
E · T
], Q[E · L].
Derived units:
u =Q2/3
[κ t]1/3Π, x = [κ Q t]1/3 Π1.
Thus, passing to dimensionless variables we get
Π =[κ t]1/3
Q2/3f
(t, κ, Q; Π1 [κ Q t]1/3
)= ϕ (Π1) .
Summer School: KPI, 2010 Nonlinear transport phenomena 18 / 38
Corollary. Ansatz
u =Q
[κ Q t]1/3ϕ(ξ), ξ = Π1 =
x
[κ Q t]1/3
should strongly simplify the point explosion problem (9).In fact, inserting this ansatz into the heat transport equation,we get
d
d ξ(ξϕ[ξ] + 3 ϕ[ξ] ϕ[ξ]) = 0, (10)
with the additional condition (as well expressed in thedimensionless variables):∫ ∞
−∞ϕ[ξ] dξ = 1. (11)
Summer School: KPI, 2010 Nonlinear transport phenomena 19 / 38
Corollary. Ansatz
u =Q
[κ Q t]1/3ϕ(ξ), ξ = Π1 =
x
[κ Q t]1/3
should strongly simplify the point explosion problem (9).In fact, inserting this ansatz into the heat transport equation,we get
d
d ξ(ξϕ[ξ] + 3 ϕ[ξ] ϕ[ξ]) = 0, (10)
with the additional condition (as well expressed in thedimensionless variables):∫ ∞
−∞ϕ[ξ] dξ = 1. (11)
Summer School: KPI, 2010 Nonlinear transport phenomena 19 / 38
Corollary. Ansatz
u =Q
[κ Q t]1/3ϕ(ξ), ξ = Π1 =
x
[κ Q t]1/3
should strongly simplify the point explosion problem (9).In fact, inserting this ansatz into the heat transport equation,we get
d
d ξ(ξϕ[ξ] + 3 ϕ[ξ] ϕ[ξ]) = 0, (10)
with the additional condition (as well expressed in thedimensionless variables):∫ ∞
−∞ϕ[ξ] dξ = 1. (11)
Summer School: KPI, 2010 Nonlinear transport phenomena 19 / 38
Statement 4. Solution to the problemd
d ξ(ξϕ[ξ] + 3 ϕ[ξ] ϕ[ξ]) = 0,∫ ∞
−∞ϕ[ξ] dξ = 1
is given by the formula
ϕ[ξ] =
16
(ξ2Φ − ξ2
)if |ξ| < ξΦ,
0 otherwise,
where ξΦ = (9/2)1/3.
Corollary. The only solution to the nonlinear pointexplosion problem (9) is given by the formula
u =Q
6 [κQ t]1/3
(9/2)1/3 − x2
[κ Q t]2/3 if |x| <[
9 (κ t Q)2
2
]1/6
,
0 otherwise(12)
Summer School: KPI, 2010 Nonlinear transport phenomena 20 / 38
Statement 4. Solution to the problemd
d ξ(ξϕ[ξ] + 3 ϕ[ξ] ϕ[ξ]) = 0,∫ ∞
−∞ϕ[ξ] dξ = 1
is given by the formula
ϕ[ξ] =
16
(ξ2Φ − ξ2
)if |ξ| < ξΦ,
0 otherwise,
where ξΦ = (9/2)1/3.
Corollary. The only solution to the nonlinear pointexplosion problem (9) is given by the formula
u =Q
6 [κQ t]1/3
(9/2)1/3 − x2
[κ Q t]2/3 if |x| <[
9 (κ t Q)2
2
]1/6
,
0 otherwise(12)
Summer School: KPI, 2010 Nonlinear transport phenomena 20 / 38
Statement 4. Solution to the problemd
d ξ(ξϕ[ξ] + 3 ϕ[ξ] ϕ[ξ]) = 0,∫ ∞
−∞ϕ[ξ] dξ = 1
is given by the formula
ϕ[ξ] =
16
(ξ2Φ − ξ2
)if |ξ| < ξΦ,
0 otherwise,
where ξΦ = (9/2)1/3.
Corollary. The only solution to the nonlinear pointexplosion problem (9) is given by the formula
u =Q
6 [κQ t]1/3
(9/2)1/3 − x2
[κ Q t]2/3 if |x| <[
9 (κ t Q)2
2
]1/6
,
0 otherwise(12)
Summer School: KPI, 2010 Nonlinear transport phenomena 20 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 21 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 22 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 23 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 24 / 38
Conclusion. Nonlinearity of the heat transport coefficientcauses the localization of the heat wave within the finitedomain
< −L(t), L(t) >,
where
L(t) =
[9 (κ t Q)2
2
]1/6
= C t1/3.
Summer School: KPI, 2010 Nonlinear transport phenomena 25 / 38
Unlimited growth of temperature and the completelocalization of the heat energy
ut = κ [uσ ux]x , (13)u(t, 0) = A (−t)−n , t < 0, n > 0, x ∈ R+, lim
t→−∞u(t, x) = 0(14)
In general, the problem is self-similar and the ansatz
u =A
(−t)nϕ(ξ), ξ =
x√Aσ κ t1−n σ
reduces the initial problem to corresponding ODE.
If we assume that n = σ = 1, than ξ will not depend on t:
u =A
(−t)ϕ(ξ), ξ =
x√A κ
.
Summer School: KPI, 2010 Nonlinear transport phenomena 26 / 38
Unlimited growth of temperature and the completelocalization of the heat energy
ut = κ [uσ ux]x , (13)u(t, 0) = A (−t)−n , t < 0, n > 0, x ∈ R+, lim
t→−∞u(t, x) = 0(14)
In general, the problem is self-similar and the ansatz
u =A
(−t)nϕ(ξ), ξ =
x√Aσ κ t1−n σ
reduces the initial problem to corresponding ODE.
If we assume that n = σ = 1, than ξ will not depend on t:
u =A
(−t)ϕ(ξ), ξ =
x√A κ
.
Summer School: KPI, 2010 Nonlinear transport phenomena 26 / 38
Unlimited growth of temperature and the completelocalization of the heat energy
ut = κ [uσ ux]x , (13)u(t, 0) = A (−t)−n , t < 0, n > 0, x ∈ R+, lim
t→−∞u(t, x) = 0(14)
In general, the problem is self-similar and the ansatz
u =A
(−t)nϕ(ξ), ξ =
x√Aσ κ t1−n σ
reduces the initial problem to corresponding ODE.
If we assume that n = σ = 1, than ξ will not depend on t:
u =A
(−t)ϕ(ξ), ξ =
x√A κ
.
Summer School: KPI, 2010 Nonlinear transport phenomena 26 / 38
Unlimited growth of temperature and the completelocalization of the heat energy
ut = κ [uσ ux]x , (13)u(t, 0) = A (−t)−n , t < 0, n > 0, x ∈ R+, lim
t→−∞u(t, x) = 0(14)
In general, the problem is self-similar and the ansatz
u =A
(−t)nϕ(ξ), ξ =
x√Aσ κ t1−n σ
reduces the initial problem to corresponding ODE.
If we assume that n = σ = 1, than ξ will not depend on t:
u =A
(−t)ϕ(ξ), ξ =
x√A κ
.
Summer School: KPI, 2010 Nonlinear transport phenomena 26 / 38
The reduced equation
ϕϕ + ϕ2 − ϕ = 0,
together with the initial condition ϕ(0) = 1 (arising from thecondition u(t, 0) = A
(−t)), has the solution
ϕ(ξ) =
(1− ξ/
√6)2, if 0 < ξ <
√6,
0 if ξ ≥√
6,
The corresponding solution to the BV problem (13) is asfollows:
u(t, x) =
A−t
(1− x√
6 κ A
)2, if 0 < x <
√6 κ A,
0 if x ≥√
6 κ A.
Moral of the tale: [1]. Heat energy is completely localized inthe segment 0 < x <
√6 κ A0;
[2]. The temperature u(t, x) tends to +∞ in every pointx ∈ [0,
√6 κ A0) as t → 0−.
Summer School: KPI, 2010 Nonlinear transport phenomena 27 / 38
The reduced equation
ϕϕ + ϕ2 − ϕ = 0,
together with the initial condition ϕ(0) = 1 (arising from thecondition u(t, 0) = A
(−t)), has the solution
ϕ(ξ) =
(1− ξ/
√6)2, if 0 < ξ <
√6,
0 if ξ ≥√
6,
The corresponding solution to the BV problem (13) is asfollows:
u(t, x) =
A−t
(1− x√
6 κ A
)2, if 0 < x <
√6 κ A,
0 if x ≥√
6 κ A.
Moral of the tale: [1]. Heat energy is completely localized inthe segment 0 < x <
√6 κ A0;
[2]. The temperature u(t, x) tends to +∞ in every pointx ∈ [0,
√6 κ A0) as t → 0−.
Summer School: KPI, 2010 Nonlinear transport phenomena 27 / 38
The reduced equation
ϕϕ + ϕ2 − ϕ = 0,
together with the initial condition ϕ(0) = 1 (arising from thecondition u(t, 0) = A
(−t)), has the solution
ϕ(ξ) =
(1− ξ/
√6)2, if 0 < ξ <
√6,
0 if ξ ≥√
6,
The corresponding solution to the BV problem (13) is asfollows:
u(t, x) =
A−t
(1− x√
6 κ A
)2, if 0 < x <
√6 κ A,
0 if x ≥√
6 κ A.
Moral of the tale: [1]. Heat energy is completely localized inthe segment 0 < x <
√6 κ A0;
[2]. The temperature u(t, x) tends to +∞ in every pointx ∈ [0,
√6 κ A0) as t → 0−.
Summer School: KPI, 2010 Nonlinear transport phenomena 27 / 38
The reduced equation
ϕϕ + ϕ2 − ϕ = 0,
together with the initial condition ϕ(0) = 1 (arising from thecondition u(t, 0) = A
(−t)), has the solution
ϕ(ξ) =
(1− ξ/
√6)2, if 0 < ξ <
√6,
0 if ξ ≥√
6,
The corresponding solution to the BV problem (13) is asfollows:
u(t, x) =
A−t
(1− x√
6 κ A
)2, if 0 < x <
√6 κ A,
0 if x ≥√
6 κ A.
Moral of the tale: [1]. Heat energy is completely localized inthe segment 0 < x <
√6 κ A0;
[2]. The temperature u(t, x) tends to +∞ in every pointx ∈ [0,
√6 κ A0) as t → 0−.
Summer School: KPI, 2010 Nonlinear transport phenomena 27 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 28 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 29 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 30 / 38
Blow-up regime caused by the source term incorporation
Model equation:ut = κ [u ux]x + q0 u2. (15)
Ansatz u(t, x) = P (t) Q(x) leads, after some algebraicmanipulation, to the equations
P−2(t)P (t) =[κ(Q(x) Q(x)′)′ + q0 Q(x)2
]Q−1(x) = C1. (16)
Theorem. The equation (15) has the following solution:
u(t, x) =4
3 q0 (tf − t)
cos2
[√q0
8 κx], if |x| <
√2 π2 κ
q0,
0 otherwise.(17)
The above solution
I is localized within the interval [−L, L], L =√
2 π2 κq0
;
I tends to +∞ ∀ x ∈ (−L, L), as t → tf − 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 31 / 38
Blow-up regime caused by the source term incorporation
Model equation:ut = κ [u ux]x + q0 u2. (15)
Ansatz u(t, x) = P (t) Q(x) leads, after some algebraicmanipulation, to the equations
P−2(t)P (t) =[κ(Q(x) Q(x)′)′ + q0 Q(x)2
]Q−1(x) = C1. (16)
Theorem. The equation (15) has the following solution:
u(t, x) =4
3 q0 (tf − t)
cos2
[√q0
8 κx], if |x| <
√2 π2 κ
q0,
0 otherwise.(17)
The above solution
I is localized within the interval [−L, L], L =√
2 π2 κq0
;
I tends to +∞ ∀ x ∈ (−L, L), as t → tf − 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 31 / 38
Blow-up regime caused by the source term incorporation
Model equation:ut = κ [u ux]x + q0 u2. (15)
Ansatz u(t, x) = P (t) Q(x) leads, after some algebraicmanipulation, to the equations
P−2(t)P (t) =[κ(Q(x) Q(x)′)′ + q0 Q(x)2
]Q−1(x) = C1. (16)
Theorem. The equation (15) has the following solution:
u(t, x) =4
3 q0 (tf − t)
cos2
[√q0
8 κx], if |x| <
√2 π2 κ
q0,
0 otherwise.(17)
The above solution
I is localized within the interval [−L, L], L =√
2 π2 κq0
;
I tends to +∞ ∀ x ∈ (−L, L), as t → tf − 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 31 / 38
Blow-up regime caused by the source term incorporation
Model equation:ut = κ [u ux]x + q0 u2. (15)
Ansatz u(t, x) = P (t) Q(x) leads, after some algebraicmanipulation, to the equations
P−2(t)P (t) =[κ(Q(x) Q(x)′)′ + q0 Q(x)2
]Q−1(x) = C1. (16)
Theorem. The equation (15) has the following solution:
u(t, x) =4
3 q0 (tf − t)
cos2
[√q0
8 κx], if |x| <
√2 π2 κ
q0,
0 otherwise.(17)
The above solution
I is localized within the interval [−L, L], L =√
2 π2 κq0
;
I tends to +∞ ∀ x ∈ (−L, L), as t → tf − 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 31 / 38
Blow-up regime caused by the source term incorporation
Model equation:ut = κ [u ux]x + q0 u2. (15)
Ansatz u(t, x) = P (t) Q(x) leads, after some algebraicmanipulation, to the equations
P−2(t)P (t) =[κ(Q(x) Q(x)′)′ + q0 Q(x)2
]Q−1(x) = C1. (16)
Theorem. The equation (15) has the following solution:
u(t, x) =4
3 q0 (tf − t)
cos2
[√q0
8 κx], if |x| <
√2 π2 κ
q0,
0 otherwise.(17)
The above solution
I is localized within the interval [−L, L], L =√
2 π2 κq0
;
I tends to +∞ ∀ x ∈ (−L, L), as t → tf − 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 31 / 38
Blow-up regime caused by the source term incorporation
Model equation:ut = κ [u ux]x + q0 u2. (15)
Ansatz u(t, x) = P (t) Q(x) leads, after some algebraicmanipulation, to the equations
P−2(t)P (t) =[κ(Q(x) Q(x)′)′ + q0 Q(x)2
]Q−1(x) = C1. (16)
Theorem. The equation (15) has the following solution:
u(t, x) =4
3 q0 (tf − t)
cos2
[√q0
8 κx], if |x| <
√2 π2 κ
q0,
0 otherwise.(17)
The above solution
I is localized within the interval [−L, L], L =√
2 π2 κq0
;
I tends to +∞ ∀ x ∈ (−L, L), as t → tf − 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 31 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 32 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 33 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 34 / 38
Figure:
Summer School: KPI, 2010 Nonlinear transport phenomena 35 / 38
THE ROLE OF SELF-SIMILAR SOLUTIONS
Summer School: KPI, 2010 Nonlinear transport phenomena 36 / 38
Maximum principle. Comparison theorems ( initialvalue problem).
Let us consider the Cauchy problem
ut = [κ(u) ux]x , (18)u(0, x) = u0(x) ≥ 0, x ∈ R, κ(u) > 0.
Theorem 1. Maximum of the solution u(t, x) at any timet > 0 does not exceed the maximum of the initial data:
maxt>0, −∞<x<∞u(t, x) ≤ max−∞<x<∞u0(x).
2
2A. Samarskij, A. Mikhailov, Mathematical Modelling: Ideas, Methods,Examples, Moscow, 1997, A. Samarskij, V. Galaktionov et al, Blow-upregimes in quasilinear parabolic equations, Berlin: Walter de Gruyter, 1995.
Summer School: KPI, 2010 Nonlinear transport phenomena 37 / 38
Maximum principle. Comparison theorems ( initialvalue problem).
Let us consider the Cauchy problem
ut = [κ(u) ux]x , (18)u(0, x) = u0(x) ≥ 0, x ∈ R, κ(u) > 0.
Theorem 1. Maximum of the solution u(t, x) at any timet > 0 does not exceed the maximum of the initial data:
maxt>0, −∞<x<∞u(t, x) ≤ max−∞<x<∞u0(x).
2
2A. Samarskij, A. Mikhailov, Mathematical Modelling: Ideas, Methods,Examples, Moscow, 1997, A. Samarskij, V. Galaktionov et al, Blow-upregimes in quasilinear parabolic equations, Berlin: Walter de Gruyter, 1995.
Summer School: KPI, 2010 Nonlinear transport phenomena 37 / 38
Maximum principle. Comparison theorems ( initialvalue problem).
Let us consider the Cauchy problem
ut = [κ(u) ux]x , (18)u(0, x) = u0(x) ≥ 0, x ∈ R, κ(u) > 0.
Theorem 1. Maximum of the solution u(t, x) at any timet > 0 does not exceed the maximum of the initial data:
maxt>0, −∞<x<∞u(t, x) ≤ max−∞<x<∞u0(x).
2
2A. Samarskij, A. Mikhailov, Mathematical Modelling: Ideas, Methods,Examples, Moscow, 1997, A. Samarskij, V. Galaktionov et al, Blow-upregimes in quasilinear parabolic equations, Berlin: Walter de Gruyter, 1995.
Summer School: KPI, 2010 Nonlinear transport phenomena 37 / 38
Maximum principle. Comparison theorems ( initialvalue problem).
Let us consider the Cauchy problem
ut = [κ(u) ux]x , (18)u(0, x) = u0(x) ≥ 0, x ∈ R, κ(u) > 0.
Theorem 1. Maximum of the solution u(t, x) at any timet > 0 does not exceed the maximum of the initial data:
maxt>0, −∞<x<∞u(t, x) ≤ max−∞<x<∞u0(x).
2
2A. Samarskij, A. Mikhailov, Mathematical Modelling: Ideas, Methods,Examples, Moscow, 1997, A. Samarskij, V. Galaktionov et al, Blow-upregimes in quasilinear parabolic equations, Berlin: Walter de Gruyter, 1995.
Summer School: KPI, 2010 Nonlinear transport phenomena 37 / 38
Corollary (comparison theorem). Let u1(t, x),u2(t, x), u3(t, x), are the solutions to the equation
ut = [κ(u) ux]x ,
u(0, x) = u0(x) ≥ 0, x ∈ R, κ(u) > 0,
corresponding to the initial data u10(x), u2
0(x), and u30(x)
such thatu1
0(x) ≤ u20(x) ≤ u3
0(x)
for −∞ < x < ∞. Then
u1(t, x) ≤ u2(t, x) ≤ u3(t, x)
for all −∞ < x < ∞, and t > 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 38 / 38
Corollary (comparison theorem). Let u1(t, x),u2(t, x), u3(t, x), are the solutions to the equation
ut = [κ(u) ux]x ,
u(0, x) = u0(x) ≥ 0, x ∈ R, κ(u) > 0,
corresponding to the initial data u10(x), u2
0(x), and u30(x)
such thatu1
0(x) ≤ u20(x) ≤ u3
0(x)
for −∞ < x < ∞. Then
u1(t, x) ≤ u2(t, x) ≤ u3(t, x)
for all −∞ < x < ∞, and t > 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 38 / 38
Corollary (comparison theorem). Let u1(t, x),u2(t, x), u3(t, x), are the solutions to the equation
ut = [κ(u) ux]x ,
u(0, x) = u0(x) ≥ 0, x ∈ R, κ(u) > 0,
corresponding to the initial data u10(x), u2
0(x), and u30(x)
such thatu1
0(x) ≤ u20(x) ≤ u3
0(x)
for −∞ < x < ∞. Then
u1(t, x) ≤ u2(t, x) ≤ u3(t, x)
for all −∞ < x < ∞, and t > 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 38 / 38
Corollary (comparison theorem). Let u1(t, x),u2(t, x), u3(t, x), are the solutions to the equation
ut = [κ(u) ux]x ,
u(0, x) = u0(x) ≥ 0, x ∈ R, κ(u) > 0,
corresponding to the initial data u10(x), u2
0(x), and u30(x)
such thatu1
0(x) ≤ u20(x) ≤ u3
0(x)
for −∞ < x < ∞. Then
u1(t, x) ≤ u2(t, x) ≤ u3(t, x)
for all −∞ < x < ∞, and t > 0.
Summer School: KPI, 2010 Nonlinear transport phenomena 38 / 38