Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?

Question

Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:

$$u_t = grad[V(u)]$$

For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-dimensional faces of the $n+1$-th cube $[0,1]^{n} \times [0,T)$. Where by "$grad$" I mean the gradient w.r.t. to the spatial coordinates (excluding the last coordinate).

Are there necessary and sufficient conditions one can put on $V$ for which the above problem will always be well-posed for some choice of $T$? (for initial conditions in reasonable function spaces).

Are there similar conditions (perhaps more complicated) for the second order equation:

$$u_{tt} = grad[V(u)]$$

With the same type of boundary conditions?

Answer 1

The first equation (first order) is a quasilinear transport equation. Developping, it writes $u_t+(Z\cdot\nabla_x)u=0$ where $Z:=-(\nabla_uV)\circ u$. The method of characteristics tells you that $u$ is constant along the integral curves of $Z$. Since $Z$ depends only on $u$, these curves are straight lines.

The boundary conditions that you mention are not appropriate her. Your equation is hyperbolic. A Cauchy problem (initial data prescribed at $t=0$) is well-suited. If your spatial domain is bounded, you must impose a Dirichlet condition only along the parts of the boundary where the characteristics are incoming. Mind however that because of nonlinearity, shock waves will usually develop in finite time (think that the characteristic lines are not parallel, and must meet after some time interval). Therefore you have to consider entropy solution. The Cauchy problem is uniquely solvable (Kruzhkov, 1970). An appropriate initial-boundary value problem is, too (F. Otto ?).

As for the second problem, the Cauchy problem is usually ill-posed, because the order of space derivatives is less than that of the time derivative. This is already true in the linear case if $n=1$, where your equation is a "rotated" heat equation $u_{tt}=u_x$. The pure Cauchy problem (domain ${\mathbb R}^n$) is ill-posed in every reasonable space ($C^\infty$, Sobolev spaces) because it displays growing modes $e^{\alpha t|\xi|}$ ($|\xi|$ the space frequency variable), where $\alpha=e^{i\pi/4}$ ; their modulus $\exp(t\frac{|\xi|\sqrt2}2)$ grows arbitrarily fast. This, with the Uniform Boundedness Principle, proves ill-posedness.