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Conservation Laws for BeginnersIMA Workshop
Career Options for Women in Mathematical Sciences
Barbara Lee Keyfitz
The Ohio State [email protected]
March 4, 2013
Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 1 / 11
Outline
1 IntroductionHistoryHyperbolic PDE
2 What we know about conservation lawsOne space dimension, linear theoryOne space dimension: small data resultsOne space dimension: large dataBasics of multidimensional problemsSample results
3 What we don’t know (and would like to find out)Are multidimensional problems well-posed?Guderley Mach reflection
Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 2 / 11
Introduction History
‘History of Differential Equations’
A view from the distance . . .
Euler
Weierstrass
Lie
Poincare
Computers
When we study differential equations, what are we looking for?
Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 3 / 11
Introduction Hyperbolic PDE
Hyperbolic Partial Differential Equations: TRANSPORT
The idea:
ut + aux = 0 or∂u
∂t+ a
∂u
∂x= 0 vs ut + uux = 0
u = f (x − at)
characteristics dx/dt = a
weak solutions (for rough data)
solutions for all t
u=f(x-ut) (implicit)
characteristics dx/dt = u(speed = amplitude)
weak solutions always
require ut + f (u)x = 0
Discontinuousacross
characteristics
Locally smoothsolution
x
t
x
t
y
x−ut=y
Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 4 / 11
What we know about conservation laws One space dimension, linear theory
One Space Dimension: Linear vs Nonlinear
Linear
First-order system, 1-D:
ut + A(x , t)ux + b(x , t) = 0
Diagonalize:
vt + Λ(x , t)vx + b(x , t) = 0
Picard method: t
x
(x0,t
0)
Nonlinear (Quasilinear) System
ut + f (u)x ≡ ut + A(u)ux
b = b + P(∂tP
−1 + A∂xP−1)
If A = A(u) then P = P(u)
So b depends on ut and ux
Unless Λ is independent of u(which is the linear case)
Mechanism is still transport, butinteractions are now nonlinear
Nonlinear effects are important,but fundamental mechanism isstill transport
Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 5 / 11
What we know about conservation laws One space dimension: small data results
Approach to Nonlinear Problems
ut + A(u)ux = 0, e. v. A = {λ1(u) < λ2(u) . . . < λn(u)}
Calculate wave interactions ‘by hand’ (Glimm, Bressan)
Works but
only in total variation norm TVonly for small data: TV (u(·, t)) < ε
x
t
x
t
Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 6 / 11
What we know about conservation laws One space dimension: large data
Large Data: A Mystery – Several Mysteries
Phenomena include
Explosions (L∞ growth in u)
Singular Shocks (K. & Kranzer)
Example: Marco Mazzotti’s anti-Langmuir Chromatography
∂
∂t
(ui +
aiui1− u1 + u2
)+
∂
∂xui = 0 , i = 1, 2 a1 < a2
Figure 11b944
71
Author's personal copy
M. Mazzotti et al. / J. Chromatogr. A 1217 (2010) 2002–2012 2009
Fig. 7. Effect of feed concentration on the interaction between phenetole (comp. 1) and 4-tert-butylphenol (comp. 2) in frontal analysis experiments (Zurich laboratory). (a)5 cm column, high concentration range; (b) 25 cm column, low concentration range.
Two final remarks are worth making. The first remark refers tothe shape of the peak in the experiment at 100% concentration (seeFig. 7a, inset), which is in this case clearly different from that exhib-ited by the peaks obtained at higher concentration. Both beforeand after the main sharp peak, the UV profile reaches two plateaus,which are above the feed concentrations of the two species; theyelute for a time, namely between 0.2 and 0.3 min, which is compa-rable to the elution time of the main peak itself, i.e. about 0.2 min.
We do not have an explanation for this effect, which is commonto all three columns, but is not so evident or not at all exhibited athigher concentration.
The second remark refers to the two sharp fronts exhibited byall delta-shocks’ spikes. It is well known that sharp fronts in non-linear chromatography exhibit a constant pattern behavior, whichis called shock layer, when they separate two constant states andpropagate through long enough columns [11–13]. Although there
Components phenetole (C8H10O) and 4-tert-butylphenol (C10H14O)Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 7 / 11
What we know about conservation laws Basics of multidimensional problems
More Than One Space Dimension:Wave Propagation Is Different
First-order system for u = u(x , y , t) ∈ Rn
With n × n flux matrices A and B, A =∂f
∂x, B =
∂g
∂y
ut + A(x , y , t, u)ux + B(x , y , t, u)uy = 0
y
t
x
Characteristics are surfaces
Waves spread
Amplitudes decay
Wave interactions are complicated
Theorem (P. Brenner): Linear systems are well-posed only in L2
Theorem (J. Rauch): The same is true in quasilinear systems
Mismatch between known 1-D and multi-D constraints
Recent results (De Lellis &Szekelyhidi) on non-uniqueness of(standard definition) weak solutions
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What we know about conservation laws Sample results
Self-similar Problems: 2-D Riemann Problems (with Canic,Lieberman, Kim, Jegdic, Tesdall, Popivanov, Payne)
• Analogy with 1-D: focus on transport and wave interactions
• Occur in physically interesting problemsExample: Shock reflection by a wedge
X= tΞ
S= tΣFlowWedge
Incident Shock
ReflectedShock
t<0 t=0 t>0
• Expect to see well-posed problems (but some surprises)
• Interesting mathematics (hyperbolic + elliptic)
Approach
• Work completely in self-similar coordinates: ξ = xt , η = y
t
• Reduced eq’n (−ξ + A(U))Uξ + (−η + B(U))Uη = 0 changes type
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What we know about conservation laws Sample results
Local Picture for Regular Reflection
WEAK STRONG
Incident Shock Incident Shock
Reflected
Shock Reflected
Shock
Sonic LineELLIPTIC
REGION
ELLIPTIC
REGION
FREE BOUNDARY
DEGENERACY IN ELLIPTIC EQUATION
UTSD
ut + uux + vy = 0
vx − uy = 0
NLWS
ρt + mx + ny = 0
mt + p(ρ)x = 0
nt + p(ρ)y = 0
Incident Shock
Reflected Shock
Free Boundary
Cutoff Boundary
Incident Shock
Cutoff Boundary
Reflected Shock
Free Boundary
Sonic Line
"STRONG" "WEAK"
See also results of Chen, Feldman et al on potential flow
Barbara Keyfitz (Ohio State) Conservation Laws for Beginners March 4, 2013 10 / 11
What we don’t know (and would like to find out) Guderley MR
Guderley Mach Reflection (Hunter and Tesdall)
x/t
y/t
1.0746 1.0748 1.075 1.0752 1.0754 1.0756
0.41
0.4102
0.4104
0.4106
0.4108
Discovered in numerical simulations and verified experimentally byB. W. Skews & al. (JFM)
No theory as yet
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