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Among the three equations
$$\Delta u\left( x \right) = 0,{\rm{ }}{u_t}\left( {x,t} \right) = \Delta u\left( {x,t} \right),{\rm{ }}{u_{tt}}\left( {x,t} \right) = \Delta u\left( {x,t} \right),{\rm{ }}x \in {R^n},{\rm{ }}t \in \left( { - \infty ,\infty } \right)$$
which one has the smoothing effect, namely a ${C^2}$ solution on ${R^n}$, or on ${R^n} \times \left( { - \infty ,\infty } \right)$ is automatically a ${C^\infty }$ solution? If an equation has no smoothing effect, give a solution to that equation whose regularity is in ${C^2}$ only (but not in ${C^k}$ for $k \ge 3$).

From Evans', I learned that both the Laplace and heat equations have smoothing effect. However the regularity of the wave equations are not discussed. Please help.

Patrick Windance
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  • In the case of parabolic equation: the solution can be written as a convolution of the initial condition with the (smooth) heat kernel (http://math.stackexchange.com/questions/89785/what-does-the-heat-kernel-in-the-heat-equation-represent-ux-t). The elliptic case is similar. – Martín-Blas Pérez Pinilla Mar 21 '16 at 07:51

1 Answers1

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Consider the one-dimensional wave equation $u_{tt}=u_{xx}$. The general solution is $u(x,t)=f(x+t)+g(x-t)$ where $f$ and $g$ are two arbitrary $C^2$ functions. Thus, if $f$ and $g$ are $C^2$ but not $C^3$, he corresponding solution is $C^2$ but not $C^3$.

In general, hyperbolic equations do not have smoothing effects. To the contrary, singularities in the initial or boundary data are transmitted through the characteristics.

Julián Aguirre
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  • Really to the contrary, hyperbolic equations quite frequently destroy regularity in the course of their evolution. A classic example where this makes intuitive sense is the nonlinear transport equation $u_t+(1-u)u_x=0,u(0,x)=u_0(x)$. This is a simple model of 1D traffic flow, where traffic is moving to the right and slows down where there is a higher density of cars. If $u_0$ is decreasing somewhere, then shocks will form as cars in the high density region to the left fail to catch up to cars in the low density region to the right. – Ian Mar 21 '16 at 13:54