AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 2. More info at http://summerschool.ssa.org.ua

1. Nonlinear transport phenomena:
models, method of solving and unusual
features
Vsevolod Vladimirov
AGH University of Science and technology, Faculty of Applied
Mathematics
´
Krakow, August 10, 2010
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 1 / 29

2. Burgers equation
Consider the second law of Newton for viscous incompressible
fluid:
∂ ui ∂ ui 1∂P
+ uj j
+ = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3,
∂t ∂x ρ ∂ xi
u(t, x) is the velocity field,
∂ j ∂
∂ t + u ∂ xj is the time ( substantial) derivative;
ρ is the constant density ;
P is the pressure ;
ν is the viscosity coefficient;
2
∆ = n ∂∂x2 is the Laplace operator.
i=1
i
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x . (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29

3. Burgers equation
Consider the second law of Newton for viscous incompressible
fluid:
∂ ui ∂ ui 1∂P
+ uj j
+ = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3,
∂t ∂x ρ ∂ xi
u(t, x) is the velocity field,
∂ j ∂
∂ t + u ∂ xj is the time ( substantial) derivative;
ρ is the constant density ;
P is the pressure ;
ν is the viscosity coefficient;
2
∆ = n ∂∂x2 is the Laplace operator.
i=1
i
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x . (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29

4. Burgers equation
Consider the second law of Newton for viscous incompressible
fluid:
∂ ui ∂ ui 1∂P
+ uj j
+ = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3,
∂t ∂x ρ ∂ xi
u(t, x) is the velocity field,
∂ j ∂
∂ t + u ∂ xj is the time ( substantial) derivative;
ρ is the constant density ;
P is the pressure ;
ν is the viscosity coefficient;
2
∆ = n ∂∂x2 is the Laplace operator.
i=1
i
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x . (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29

5. Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t + τ, x)
+ u(t, x) ux (t, x) = ν ux x (t, x).
∂t
Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up
∂t
to O(τ 2 ) the equation called the hyperbolic generalization of
the Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x . (2)
GBE appears when modeling transport phenomena in media
possessing internal structure: granular media,polymers, cellular
structures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29

6. Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t + τ, x)
+ u(t, x) ux (t, x) = ν ux x (t, x).
∂t
Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up
∂t
to O(τ 2 ) the equation called the hyperbolic generalization of
the Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x . (2)
GBE appears when modeling transport phenomena in media
possessing internal structure: granular media,polymers, cellular
structures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29

7. Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t + τ, x)
+ u(t, x) ux (t, x) = ν ux x (t, x).
∂t
Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up
∂t
to O(τ 2 ) the equation called the hyperbolic generalization of
the Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x . (2)
GBE appears when modeling transport phenomena in media
possessing internal structure: granular media,polymers, cellular
structures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29

8. Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t + τ, x)
+ u(t, x) ux (t, x) = ν ux x (t, x).
∂t
Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up
∂t
to O(τ 2 ) the equation called the hyperbolic generalization of
the Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x . (2)
GBE appears when modeling transport phenomena in media
possessing internal structure: granular media,polymers, cellular
structures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29

9. Various generalizations of Burgers equation
Convection-reaction diffusion equation
ut + u ux = ν [un ux ]x + f (u), (3)
and its hyperbolic generalization
τ ut t + ut + u ux = ν [un ux ]x + f (u) (4)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 4 / 29

10. Various generalizations of Burgers equation
Convection-reaction diffusion equation
ut + u ux = ν [un ux ]x + f (u), (3)
and its hyperbolic generalization
τ ut t + ut + u ux = ν [un ux ]x + f (u) (4)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 4 / 29

11. Solution to BE
Lemma 1. BE is connected with the equation
1 2
ψt + ψx = ν ψ x x (5)
2
by means of the transformation
u2
ψx = u, ψt = ν ux − . (6)
2
Lemma 2. The equation (5) is connected with the heat
transport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29

12. Solution to BE
Lemma 1. BE is connected with the equation
1 2
ψt + ψx = ν ψ x x (5)
2
by means of the transformation
u2
ψx = u, ψt = ν ux − . (6)
2
Lemma 2. The equation (5) is connected with the heat
transport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29

13. Solution to BE
Lemma 1. BE is connected with the equation
1 2
ψt + ψx = ν ψ x x (5)
2
by means of the transformation
u2
ψx = u, ψt = ν ux − . (6)
2
Lemma 2. The equation (5) is connected with the heat
transport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29

14. Corollary. Solution to the initial value problem
ut + u ux = ν u x x , (7)
u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φ x x , (8)
x
1
Φ(0, x) = exp − F (z) d z := θ(x)
2ν 0
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29

15. Corollary. Solution to the initial value problem
ut + u ux = ν u x x , (7)
u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φ x x , (8)
x
1
Φ(0, x) = exp − F (z) d z := θ(x)
2ν 0
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29

16. Corollary. Solution to the initial value problem
ut + u ux = ν u x x , (7)
u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φ x x , (8)
x
1
Φ(0, x) = exp − F (z) d z := θ(x)
2ν 0
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29

17. Let us remind, that solution to the initial value problem (8) can
be presented by the formula
∞ (x−ξ)2
1
Φ(t, x) = √ θ(ξ) e− 4ν t d ξ.
4πν t −∞
Corollary. Solution to the initial value problem (7) is given by
the formula
∞ x−ξ − f (ξ;t, x)
−∞ t e dξ
2ν
u(t, x) = f (ξ;t, x)
, (9)
∞ −
−∞ e 2ν d ξ
where
ξ
(x − ξ)2
f (ξ; t, x) = F (z) d z +
0 2t
.
So, the formula (9)completely defines the solution to Cauchy
problem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29

18. Let us remind, that solution to the initial value problem (8) can
be presented by the formula
∞ (x−ξ)2
1
Φ(t, x) = √ θ(ξ) e− 4ν t d ξ.
4πν t −∞
Corollary. Solution to the initial value problem (7) is given by
the formula
∞ x−ξ − f (ξ;t, x)
−∞ t e dξ
2ν
u(t, x) = f (ξ;t, x)
, (9)
∞ −
−∞ e 2ν d ξ
where
ξ
(x − ξ)2
f (ξ; t, x) = F (z) d z +
0 2t
.
So, the formula (9)completely defines the solution to Cauchy
problem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29

19. Let us remind, that solution to the initial value problem (8) can
be presented by the formula
∞ (x−ξ)2
1
Φ(t, x) = √ θ(ξ) e− 4ν t d ξ.
4πν t −∞
Corollary. Solution to the initial value problem (7) is given by
the formula
∞ x−ξ − f (ξ;t, x)
−∞ t e dξ
2ν
u(t, x) = f (ξ;t, x)
, (9)
∞ −
−∞ e 2ν d ξ
where
ξ
(x − ξ)2
f (ξ; t, x) = F (z) d z +
0 2t
.
So, the formula (9)completely defines the solution to Cauchy
problem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29

20. Let us remind, that solution to the initial value problem (8) can
be presented by the formula
∞ (x−ξ)2
1
Φ(t, x) = √ θ(ξ) e− 4ν t d ξ.
4πν t −∞
Corollary. Solution to the initial value problem (7) is given by
the formula
∞ x−ξ − f (ξ;t, x)
−∞ t e dξ
2ν
u(t, x) = f (ξ;t, x)
, (9)
∞ −
−∞ e 2ν d ξ
where
ξ
(x − ξ)2
f (ξ; t, x) = F (z) d z +
0 2t
.
So, the formula (9)completely defines the solution to Cauchy
problem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29

21. Example: solution of the ”point explosion” problem
Let
u(0, x) = F (x) = Aδ(x)H(x),
1 (x−ξ)2 1 if x ≥ 0,
δ(x) = lim √ e− 4 ν t , H(x) = .
t→ +0 4πν t 0 if x < 0
Figure:
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 8 / 29

22. Example: solution of the ”point explosion” problem
Let
u(0, x) = F (x) = Aδ(x)H(x),
1 (x−ξ)2 1 if x ≥ 0,
δ(x) = lim √ e− 4 ν t , H(x) = .
t→ +0 4πν t 0 if x < 0
Figure:
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 8 / 29

23. Performing simple but tedious calculations, we finally get the
following solution to the point explosion problem:
x2
ν eR − 1 e− 4 ν t
u(t, x) = √ ,
t π x
(eR + 1) + erf( √4 ν t ) (1 − eR )
2
where z
2 2
erf(z) = √ e−x d x,
π 0
A
R= 2ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29

24. Performing simple but tedious calculations, we finally get the
following solution to the point explosion problem:
x2
ν eR − 1 e− 4 ν t
u(t, x) = √ ,
t π x
(eR + 1) + erf( √4 ν t ) (1 − eR )
2
where z
2 2
erf(z) = √ e−x d x,
π 0
A
R= 2ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29

25. Performing simple but tedious calculations, we finally get the
following solution to the point explosion problem:
x2
ν eR − 1 e− 4 ν t
u(t, x) = √ ,
t π x
(eR + 1) + erf( √4 ν t ) (1 − eR )
2
where z
2 2
erf(z) = √ e−x d x,
π 0
A
R= 2ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29

26. Performing simple but tedious calculations, we finally get the
following solution to the point explosion problem:
x2
ν eR − 1 e− 4 ν t
u(t, x) = √ ,
t π x
(eR + 1) + erf( √4 ν t ) (1 − eR )
2
where z
2 2
erf(z) = √ e−x d x,
π 0
A
R= 2ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29

27. Suppose now, that ν becomes very large. Then
x
R→ 0 eR ≈ 1 + R, erf √ ≈ 0,
4ν t
and
x2
ν A
e− 4 ν t A x2
u(t, x) = 2ν
√ + O(R2 ) ≈ √ e− 4 ν t .
t π 4πν t
Corollary.Solution to the ”point explosion” problem for the BE
approaches solution to the ”heat explosion” problem for the
linear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29

28. Suppose now, that ν becomes very large. Then
x
R→ 0 eR ≈ 1 + R, erf √ ≈ 0,
4ν t
and
x2
ν A
e− 4 ν t A x2
u(t, x) = 2ν
√ + O(R2 ) ≈ √ e− 4 ν t .
t π 4πν t
Corollary.Solution to the ”point explosion” problem for the BE
approaches solution to the ”heat explosion” problem for the
linear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29

29. Suppose now, that ν becomes very large. Then
x
R→ 0 eR ≈ 1 + R, erf √ ≈ 0,
4ν t
and
x2
ν A
e− 4 ν t A x2
u(t, x) = 2ν
√ + O(R2 ) ≈ √ e− 4 ν t .
t π 4πν t
Corollary.Solution to the ”point explosion” problem for the BE
approaches solution to the ”heat explosion” problem for the
linear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29

30. Suppose now, that ν becomes very large. Then
x
R→ 0 eR ≈ 1 + R, erf √ ≈ 0,
4ν t
and
x2
ν A
e− 4 ν t A x2
u(t, x) = 2ν
√ + O(R2 ) ≈ √ e− 4 ν t .
t π 4πν t
Corollary.Solution to the ”point explosion” problem for the BE
approaches solution to the ”heat explosion” problem for the
linear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29

31. For large R the way of getting the approximating formula is less clear, so
we restore to the results of the numerical simulation. Below it is shown the
solution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
 x √
t
if t > 0, 0 < x < 2√ t,
A
u(t, x) = ,
0 if t > 0, x < 0 or x > 2 A t
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29

32. For large R the way of getting the approximating formula is less clear, so
we restore to the results of the numerical simulation. Below it is shown the
solution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
 x √
t
if t > 0, 0 < x < 2√ t,
A
u(t, x) = ,
0 if t > 0, x < 0 or x > 2 A t
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29

33. For large R the way of getting the approximating formula is less clear, so
we restore to the results of the numerical simulation. Below it is shown the
solution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
 x √
t
if t > 0, 0 < x < 2√ t,
A
u(t, x) = ,
0 if t > 0, x < 0 or x > 2 A t
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29

34. Figure:
A common solution
x
√
t if t > 0, 0 < x < 2√ t,
A
u(t, x) =
0 if t > 0, x < 0 or x > 2 A t,
to the Burgers and the Euler equations
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 12 / 29

35. So the solutions to the point explosion problem for BE are
completely different in the limiting cases: when
R = A/(2 ν) → 0 it coincides with the solution of the heat
explosion problem,
while for large R it reminds the shock wave solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 13 / 29

36. So the solutions to the point explosion problem for BE are
completely different in the limiting cases: when
R = A/(2 ν) → 0 it coincides with the solution of the heat
explosion problem,
while for large R it reminds the shock wave solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 13 / 29

37. The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolic
generalization of BE:
τ utt + ut + u ux = ν ux x , (10)
u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x ,
we can conclude, that the parameter C = ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supported
function, and D = max ϕ(x), then the number M = D/C (the
”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29

38. The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolic
generalization of BE:
τ utt + ut + u ux = ν ux x , (10)
u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x ,
we can conclude, that the parameter C = ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supported
function, and D = max ϕ(x), then the number M = D/C (the
”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29

39. The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolic
generalization of BE:
τ utt + ut + u ux = ν ux x , (10)
u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x ,
we can conclude, that the parameter C = ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supported
function, and D = max ϕ(x), then the number M = D/C (the
”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29

40. Results of the numerical simulation: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 15 / 29

41. Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 16 / 29

42. Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 17 / 29

43. Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 18 / 29

44. Figure: M = 0.3
The solution of the initial perturbation reminds the evolution of
the point explosion problem for BE in the case when
R = A/(2 ν) is large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 19 / 29

45. Results of the numerical simulation: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 20 / 29

46. Figure: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 21 / 29

47. Figure: M = 1.45
For M = 1 + ε a formation of the blow-up regime is observed at
the beginning of evolution,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 22 / 29

48. Figure: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 23 / 29

49. Figure: M = 1.45
but for larger t it is suppressed by viscosity and returns to the
shape of the BE solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 24 / 29

50. Results of the numerical simulation: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 25 / 29

51. Figure: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 26 / 29

52. Figure: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 27 / 29

53. Figure: M = 1.8
For M = 1.8 (and larger ones) a blow-up regime is formed at
the wave front in finite time!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 28 / 29

54. Appendix 1. Calculation of point explosion problem
for BE
Since,
ξ ∞
−A, if ξ < 0,
F (x) d x = −A lim δ(x) φB (x) H(x) d x =
0+ B→+0 −∞ 0, if ξ > 0,
∞
where φB (x) is any C0 function such that φ(x)|<B, ξ> ≡ 1, and
supp φ ⊂ < B/2, ξ + B/2 > then
(x−ξ)2
2t − A if ξ < 0,
f (ξ; t, x) = (x−ξ)2
2 t , if ξ > 0.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 29 / 29

55. Appendix 1. Calculation of point explosion problem
for BE
Since,
ξ ∞
−A, if ξ < 0,
F (x) d x = −A lim δ(x) φB (x) H(x) d x =
0+ B→+0 −∞ 0, if ξ > 0,
∞
where φB (x) is any C0 function such that φ(x)|<B, ξ> ≡ 1, and
supp φ ⊂ < B/2, ξ + B/2 > then
(x−ξ)2
2t − A if ξ < 0,
f (ξ; t, x) = (x−ξ)2
2 t , if ξ > 0.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 29 / 29