0
$\begingroup$

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?

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ Thank you for this! I understand the example of the rotated heat equation but is there a way to make precise the general principle you alluded to "Cauchy problem is usually ill-posed, because the order of space derivatives is less than that of the time derivative"? $\endgroup$ Jun 27, 2018 at 10:06
  • 1
    $\begingroup$ @SaalHardali . I edited my answer. Because the Cauchy problem for the rotated heat equation is actually ill-posed. $\endgroup$ Jun 27, 2018 at 11:32
  • $\begingroup$ Thanks, although I was aware of this fact I didn't think about this when forming the question. In any case is there an apriori reason why someone should expect the rotated heat equation to be ill posed from just looking at the derivatives? (without knowing beforehand the solution of the usual heat equation) $\endgroup$ Jun 27, 2018 at 12:39
  • $\begingroup$ Does this well-posedness hold for Lipschitz continuous weak solutions as well? Are such solutions still always given by the method of characteristics? $\endgroup$
    – Chris
    Mar 10, 2023 at 21:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.