Dumb question: Usually in viscosity solution theory for Hamilton Jacobi equations (with convex, coercive Hamiltonians), solutions are said to be in the class $BUC(\mathbb{R}^n)$ or $C^{0,1}(\mathbb{R}^n)$. See for example, this paper or this paper on stochastic homogenization. I'm kind of confused because I can't see the viscosity solution being globally bounded even for trivial Hamilton Jacobi equations. Consider $$ u_t + |Du|_2 = 1, u(0) = 0,$$ which has viscosity solution $u(x,t) = t$ by the Hopf-Lax formula. It's certainly Lipschitz, but not globally bounded. So does $u(x,t) \in BUC(\mathbb{R}^n \times [0,\infty))$ mean bounded, uniformly-continuous on any compact? I'm guessing it does, but it'll be nice to hear from an expert.
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$\begingroup$ $BUC(X)$ means real-valued bounded and uniformly continuous on $X$, that's it. For the time-dependent problem, the correct way to state the existence result is that a viscosity solution exists in $BUC(\mathbb R^n \times [0,T])$ for every $T>0$, which happens to also be in $C^{0,1}(\mathbb R^n \times[0,T])$ for every $T>0$. $\endgroup$– Scott ArmstrongApr 17, 2014 at 13:13
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$\begingroup$ @user5678 Thanks, I looked it up a few days ago too, you're absolutely right. $\endgroup$– arjunApr 19, 2014 at 15:29
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I apologize, my question was rather lazy. The best reference I've found for viscosity solutions of HJB equations is Crandall and Lions' original paper from 1983.
They consider the model problem $$ u_t + H(Du) = 0 \quad u(x,0) = u_0(x). $$ Then (Theorem VI.2) says that if $H$ is continuous, there is a (nonlinear) solution semigroup $S(t):BUC(\mathbb{R}^n) \to BUC(\mathbb{R}^n)$ for this equation that's nonexpansive and strongly continuous with respect to the $L^{\infty}(\mathbb{R}^n)$ norm. This ensures that the solution $u(x,t)$ is in $BUC(\mathbb{R}^n \times [0,T])$ for any finite $T > 0$. The generalization for a Hamiltonian that varies in space $H(Du,x)$ ought not to be very different, and I suspect I only have to do some reference chasing.
Since it seems to appear everywhere, let's also consider also the so-called stationary HJB equation $$ v(x) + H(Dv,x) = m(x). $$ This also has a $BUC$ viscosity solution when $m(x)$ is in $BUC$ and $H$ is "nice enough".
Since the theory is fairly abstract as stated in the original papers, it's hard to build intuition. Something I've found very helpful is the optimal control interpretation for HJB equations with convex Hamiltonians (see Evans). For example, it's very easy to see why $v(x)$ ought to be bounded using its variational interpretation. The leap from convex Hamiltonians to general Hamiltonians appears to be a big one, and no wonder Lions won a Fields medal for it.
In any case, I've only started understanding the theory and would appreciate any corrections or comments.
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$\begingroup$ Regarding the stationary equation, do you have a bounded domain in mind? It's not clear to me the equation will have a bounded solution in the whole space. $\endgroup$ Mar 5, 2018 at 16:22