Cauchy data on a curve
/ Cauchy -> koh-SHEE /
To single out one solution of a PDE from the infinitely many it allows, you must feed it some data — and for a first-order equation the natural thing to prescribe is the solution's values along a curve. Cauchy data on a curve is exactly that: you draw a curve through the domain and announce what u equals at every point of it, then ask for the solution that grows out from that curve.
Picture it on the solution-surface side. The data curve, say (x(r), y(r)) parametrized by r, together with the prescribed values u = u0(r), is a thread already drawn in three-dimensional space — one curve of the surface, handed to you. The Cauchy problem is to build the rest of the surface so that it contains this thread and solves the PDE. The method of characteristics does it: through each point of the data curve, fire off a characteristic; the characteristics sweep out the surface, and where they meet the data curve they must match u0. For the transport equation, prescribing u along the line t = 0 is the everyday case — it is just the initial condition u(x,0) = f(x).
Cauchy data is the first-order analogue of an initial condition for an ODE, but with a crucial twist: it lives on a curve, and that curve cannot be just anywhere. If it runs along a characteristic, the data is carried by a single characteristic and cannot determine the neighbouring surface — the problem is either unsolvable or has infinitely many solutions. The data curve must cross the characteristics transversally; that requirement is the non-characteristic condition and it is what makes the Cauchy problem well-posed.
For u_x + u_y = 0, prescribing u = h(s) along the x-axis (the curve y = 0, x = s) is good Cauchy data: the axis crosses every characteristic x - y = const exactly once, so u(x,y) = h(x - y) is determined. But prescribing data along the line x = y — which IS a characteristic — fails: it cannot fix u off that line.
Good Cauchy data sits on a curve that meets each characteristic once; data along a characteristic determines nothing nearby.
Well-posedness is sensitive to the equation's type. Cauchy data is the right thing for first-order and hyperbolic equations, but prescribing Cauchy data for an elliptic equation like Laplace's is ill-posed (Hadamard's classic warning): tiny changes in the data can blow the solution up.