MTH2032 Chap.8 Cauchy Problems and Domains of Definition
Cauchy Problems and Domains of Definition
Week 8 applies characteristics to Cauchy problems, where the solution is prescribed along a curve rather than at a point. The data can be fitted to a general solution or used as an initial curve from which characteristics are swept out.
The Cauchy-Kovalevskaya theorem guarantees a unique solution near a non-characteristic point of the curve, and the remainder of the week is about where the solution stops: the domain of definition, which must always be stated. The chapter ends with practice on fitting data and judging uniqueness.
What this chapter covers
- 01
Cauchy surface and Cauchy data
- 02
Method 1: fitting the arbitrary function
- 03
Method 2: sweeping out the initial curve
- 04
Cauchy-Kovalevskaya theorem
- 05
Non-characteristic condition and the Jacobian
- 06
Bounded data, blow-up and crossing characteristics
Worked example · free
Data on the x-axis carried along slanted lines
- 1dy/dx = 1/2 gives characteristics x − 2y = s, and u = F(x − 2y).
- 1On y = 0 the data give F(x) = x², so u = (x − 2y)².
- 1Check: 2u_x + u_y = 4(x − 2y) − 4(x − 2y) = 0.
- 1The x-axis meets every characteristic exactly once, so the domain of definition is the whole plane.
Key terms
- Cauchy Problem
- The task of finding a PDE solution that takes prescribed values along a given curve or surface.
- Initial Curve
- The curve (x0(s), y0(s), u0(s)) in three dimensions from which characteristics are launched at τ = 0.
- Non-Characteristic Curve
- A data curve that crosses the characteristics transversally, so that dy/dx differs from B/A along it.
- Transformation Jacobian
- The determinant of the map from (τ, s) to (x, y), whose vanishing marks where the solution cannot be inverted.
- Blow-Up
- The failure of a solution that becomes unbounded at a finite point along a characteristic, which ends the domain of definition.
Cauchy Problems and Domains of Definition FAQ
Which method should I use for a Cauchy problem?
If the general solution is easy to find, fit the data to its arbitrary function, as with constants in Part I. Otherwise parametrise the data as an initial curve at τ = 0 and solve the characteristic equations from each point. The second method also produces the Jacobian, which helps locate the edge of the domain of definition.
What happens if the data lie along a characteristic?
The solution is constant along that characteristic, or follows its ODE, so the data either conflict with the equation and no solution exists, or they fix the arbitrary function at only one value and infinitely many solutions exist. The Cauchy-Kovalevskaya theorem excludes this by requiring the curve to be non-characteristic.
How do I find the domain of definition?
Every point in the domain must lie on exactly one characteristic that crosses the data. Look at the ends of the data curve, at places where the data run along a characteristic, at points where the ODE along a characteristic blows up, and at points where characteristic projections cross. The region left after all four checks is the domain.
What should a sketch of a domain of definition include?
Draw the data curve, a handful of characteristics leaving it, and the boundary of the region they cover, labelling any curve where the solution blows up or where characteristics meet. Mark the region clearly and write its description in symbols underneath. The sketch is evidence for the domain you state, not a substitute for stating it.
Exam move
Treat every Cauchy problem as two tasks: the formula and the region. For the formula, practise both methods on the same example so that you can check one against the other, and keep the parameter s attached to the data and τ running along characteristics. For the region, build a habit of four checks: the ends of the data, data that run along a characteristic, blow-up along a characteristic, and crossing projections.
Draw a rough sketch of a few characteristics through the data for every problem, even when the question does not ask for one; it shows quickly which points are reached. Compute the Jacobian of the map from (τ, s) to (x, y) and find where it vanishes, because that is the systematic way to locate crossings. Finally, write the domain as a set in symbols with a sentence naming what bounds it.
The notes state that a solution is not complete without it, and in a midterm or final this sentence is easy to forget under time pressure. Pay particular attention to quasilinear problems, where the characteristics depend on the data and can meet after a finite time.
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