Cauchy problem

A Cauchy problem prescribes a solution and enough of its derivatives on a hypersurface to determine local evolution away from that hypersurface.

For $a{u}_{x_1}+b{u}_{x_2}=c$ with data along $(\xi (s),\eta (s))$, the initial curve is noncharacteristic when its tangent is linearly independent of the projected characteristic vector $(a,b)$ evaluated at the initial value of $u$. The determinant is $a{\eta }^{\prime }-b{\xi }^{\prime }$, up to sign. For $C^1$ coefficients and data this permits the inverse function theorem to invert the local characteristic parametrization and gives a unique local classical solution.

Continuous dependence means that convergence of the initial data in a specified topology implies convergence of the corresponding solutions in the specified solution topology. It is a stability requirement, not merely existence or uniqueness. Smooth-data instability of the Cauchy-Riemann Cauchy problem gives a counterexample even when the initial data converge uniformly with every fixed number of derivatives.

For a scalar equation of order $k$, Cauchy data consist of the first $k$ normal jets $u,{\partial }_{N}u,\dots ,{\partial }_N^{k-1}u$ on the initial hypersurface.

These prescribed values define the initial conditions of a Cauchy problem.

The boundary jet through order $m-1$ is the collection of traces ${\partial }^{\alpha }u{|}_{\partial \Omega }$ with $|\alpha |\le m-1$. Vanishing of the whole jet is stronger than vanishing of one derivative. On a smooth hypersurface, zero normal Cauchy derivatives through order $m-1$, together with their tangential derivatives, give a zero full boundary jet.

For an order-$k$ scalar quasilinear partial differential equation with real-analytic coefficients, real-analytic Cauchy data on a real-analytic non-characteristic hypersurface determine a unique real-analytic solution in a neighbourhood of each point of that hypersurface.

For a linear differential operator with real analytic coefficients, a classical solution with zero Cauchy data on a real analytic non-characteristic hypersurface vanishes locally near that hypersurface. The solution itself need not be real analytic. A proof solves the real analytic inhomogeneous formal adjoint problem against polynomial sources, annihilates all polynomial moments using boundary jets, and uses polynomial density to force the solution to vanish.

Near a non-characteristic plane $x=0$, use the small cap ${\omega }_{\epsilon }=\{x>0,x+y^2<\epsilon \}$. Its curved face is non-characteristic by continuity of the principal symbol. A polynomial coordinate change flattens that face. Uniform real analytic coefficient bounds and the uniform Cauchy radius for polynomial forcing solve $P^{\ast }v=g$ throughout a common cap, with zero boundary jet on the curved face. The complete boundary-jet condition for formal adjoints then gives ${\int }_{{\omega }_{\epsilon }}ug=0$ for every polynomial $g$. Uniform approximation of $\bar{u}$ gives $\int |u{|}^2=0$. Repeating on the other side of the plane proves two-sided local uniqueness.

For a linear real analytic non-characteristic Cauchy problem for a partial differential equation with zero data, the local real analytic radius may be chosen uniformly over polynomial forcing terms. Fix an analyticity exponent $R^{-1}$. Each polynomial, including a polynomial obtained by a polynomial coordinate change, satisfies the corresponding factorial derivative bound with some finite prefactor, uniformly over a compact parameter range. Divide the forcing by that prefactor to put all such data in one fixed majorant class. The Cauchy-Kovalevskaya theorem supplies a common radius, and linearity rescales the solution back without shrinking it. This argument does not bound the amplitudes of the resulting solutions uniformly.

A real analytic system solved for the highest normal derivatives, with real analytic Cauchy data, has a unique local real analytic solution. For common order $m$, its normal form is ${\partial }_t^{m}U=F(t,x,\{{\partial }_t^j{\partial }_x^{\alpha }U:j<m,j+|\alpha |\le m\})$. An invertible coefficient/Jacobian matrix for the highest normal derivatives is the system's non-characteristic hypersurface condition. A real analytic coordinate change flattens the initial hypersurface, and adjoining derivatives through order $m-1$ reduces the problem to a first-order real analytic system. Uniqueness here is in the real analytic class; it does not assert Hadamard well-posedness in smooth or Sobolev norms.

In null coordinates, the characteristic boundary problemis a Volterra integral equation. Compatible analytic boundary data determine a unique analytic solution.

On a compact characteristic rectangle, the Volterra iterates for the analytic Goursat problem are bounded by $C^m/(m!{)}^2$. Their series and differentiated series converge throughout the rectangle, extending the local analytic solution globally across it.