The Hamilton--Jacobi correspondence in one dimension

Conjugate calculus and localization. Strict convexity makes $f^{\mathrm{\prime }}$ strictly increasing: convex secant inequalities give monotonicity, and equality at two distinct points would make $f$ affine between them. Superlinearity makes its limits $\pm \mathrm{\infty }$ (a finite derivative bound at either end would bound $f$ linearly there). Thus $p(q):=(f^{\mathrm{\prime }}{)}^{-1}(q)$ is continuous, and is the unique maximiser of $pq-f(p)$. The inequalities $p(q)h\le L(q+h)-L(q)\le p(q+h)h$ for $h>0$, and their reversed versions for $h<0$, show that $L^{\mathrm{\prime }}(q)=p(q)$. If $W$ is $M$-Lipschitz and $y$ minimises $W(y)+tL((x-y)\mathrm{/}t)$, perturb $y$ in each direction and use the Lipschitz bound to obtain $\mathrm{\mid }{L}^{\mathrm{\prime }}((x-y)\mathrm{/}t)\mathrm{\mid }\le M$. Hence $\mathrm{\mid }x-y\mathrm{\mid }\le Ct$, where $C={\max }_{\mathrm{\mid }p\mathrm{\mid }\le M}\mathrm{\mid }{f}^{\mathrm{\prime }}(p)\mathrm{\mid }$. Translating competitors shows that $Q_{t}W$ is $M$-Lipschitz in $x$. Moreover $L(v)-M\mathrm{\mid }v\mathrm{\mid }\ge -C_0$, where $C_0=\max \{f(M),f(-M)\}$ by the conjugate definition, and the competitor $y=x$ gives $Q_{h}W-W\le hL(0)$. These bounds and the semigroup law give a uniform time Lipschitz bound on finite horizons. Also $L(q)\ge -f(0)$ for all $q$, with equality at $q=f^{\mathrm{\prime }}(0)$, so $Q_{t}0=-tf(0)$. Sup contraction therefore gives $\mathrm{\parallel }{Q}_t{U}_0+tf(0){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$, proving boundedness on each finite slab, with no time-uniform bound asserted.

Viscous primitives for a smooth compact datum. Let $a\in C_c^{\mathrm{\infty }}$, $P_0(x)={\int }_{-\mathrm{\infty }}^{x}a$, and let $u^{\epsilon }$ be [F2]'s viscous solution for $g=f-f(0)$. Define $W^{\epsilon }(t)=H_{\epsilon t}{P}_0-{\int }_0^t{H}_{\epsilon (t-s)}f(u^{\epsilon }(s))ds\mathrm{.}$ Differentiating in $x$ gives exactly the mild identity for $u^{\epsilon }$, so $W_x^{\epsilon }=u^{\epsilon }$. Heat-potential cancellation as in the viscous construction makes $W^{\epsilon }$ classical at positive times, and its equation is $W_t^{\epsilon }+f(W_x^{\epsilon })=\epsilon W_{xx}^{\epsilon }$. At $x\to -\mathrm{\infty }$ its initial heat term tends to zero, while the integral tends to $tf(0)$: $g(u^{\epsilon }(s))\in L^1$, and convolution of an $L^1$ function with a bounded Gaussian tends to zero at spatial infinity; boundedness dominates the finite time integral. Thus $W^{\epsilon }(t,x)={\int }_{-\mathrm{\infty }}^x{u}^{\epsilon }(t,y)dy-tf(0)$. Testing the smoothed $\mathrm{\mid }{u}^{\epsilon }\mathrm{\mid }$ balance with an exterior cutoff, then removing a second outer cutoff, gives $\int \mathrm{\mid }{u}^{\epsilon }(t)\mathrm{\mid }(1-{\chi }_R)\le \int \mathrm{\mid }a\mathrm{\mid }(1-{\chi }_R)+CT(R^{-1}+R^{-2})\mathrm{\parallel }a{\mathrm{\parallel }}_1(0\le t\le T,0<\epsilon \le 1)\mathrm{.}$ This is the cutoff calculation of the $L^1$ bound in [F2], with $\mathrm{\mid }g(u)\mathrm{\mid }\le C\mathrm{\mid }u\mathrm{\mid }$ and [F3]'s derivative bounds. It supplies uniform tails.

A time modulus for the viscous primitives. The Gaussian convolution identity $H_r{H}_s=H_{r+s}$ follows by completing the square in the kernel product and using unit mass and Fubini. Applying it to the definition in step 1.2 gives $W^{\epsilon }(t+h)=H_{\epsilon h}{W}^{\epsilon }(t)-{\int }_t^{t+h}{H}_{\epsilon (t+h-s)}f(u^{\epsilon }(s))ds$. Put $M=\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $B={\max }_{\mathrm{\mid }z\mathrm{\mid }\le M}\mathrm{\mid }f(z)\mathrm{\mid }$. Since $W_x^{\epsilon }=u^{\epsilon }$, these primitives are $M$-Lipschitz in $x$, including at $t=0$. Gaussian scaling gives $\int \mathrm{\mid }z\mathrm{\mid }\mathrm{\Gamma }(z,r)dz=C\sqrt{r}$, with $C<\mathrm{\infty }$ by the Gaussian bound. Hence unit mass gives $\mathrm{\parallel }{H}_r{W}^{\epsilon }(t)-W^{\epsilon }(t){\mathrm{\parallel }}_{\mathrm{\infty }}\le CM\sqrt{r}$, and heat contraction bounds the time integral by $Bh$. Thus $\mathrm{\parallel }{W}^{\epsilon }(t+h)-W^{\epsilon }(t){\mathrm{\parallel }}_{\mathrm{\infty }}\le CM\sqrt{h}+Bh$ for $0\le t<t+h\le T$, uniformly in $0<\epsilon \le 1$.

Uniform convergence of primitives. By [F2], choose a subsequence $u^{\epsilon }\to u$ locally in space--time $L^1$, where $u$ is the entropy solution of datum $a$, continuous into local $L^1$. A further subsequence converges on almost every time slice locally in $L^1$. The uniform tails of step 1.2 pass to these slices by monotone exhaustion, and to every time by local continuity on bounded annuli followed by exhaustion. They imply $u(t)\in L^1$ with uniformly small tails; local continuity then gives global $L^1$ continuity on $[0,T]$. Moreover ${\int }_0^T\mathrm{\parallel }{u}^{\epsilon }(t)-u(t){\mathrm{\parallel }}_{1}dt\to 0$: the tails are uniformly small outside large intervals, the compact space--time convergence handles times away from $0,T$, and the bound $\mathrm{\mid }{u}^{\epsilon }\mathrm{\mid },\mathrm{\mid }u\mathrm{\mid }\le M$ controls the remaining small time intervals on the fixed spatial interval. For $P(t,x)={\int }_{-\mathrm{\infty }}^{x}u(t,y)dy-tf(0)$, the primitive formula of step 1.2 gives $\mathrm{\parallel }{W}^{\epsilon }(t)-P(t){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{u}^{\epsilon }(t)-u(t){\mathrm{\parallel }}_1$, so the integral in time of the left side tends to zero. The function $P$ is continuous in time in the supremum norm by global $L^1$ continuity. Together with the common modulus of step 2.1, this implies uniform convergence on $[0,T]\times \mathbb{R}$: a discrepancy of size $d>0$ at any time would persist with size at least $d\mathrm{/}2$ on a one-sided interval of length bounded below independently of $\epsilon$, contradicting that vanishing time integral.

The viscosity limit. At a strict local maximum of $P-\varphi$, with smooth $\varphi$, step 3.1 gives nearby local maxima of $W^{\epsilon }-\varphi$. The classical equation in step 1.2 gives ${\varphi }_t+f({\varphi }_x)\le \epsilon {\varphi }_{xx}$ there; passing to the limit proves the subsolution inequality. Local minima give the supersolution inequality. Adding a fourth-power distance term makes a contact strict without changing its first derivatives; approximation in $C^1$ on a compact contact neighbourhood reduces $C^1$ tests to smooth tests. Thus $P$ is a viscosity solution with initial datum $P_0$. It is bounded by $\mathrm{\parallel }a{\mathrm{\parallel }}_1+T\mathrm{\mid }f(0)\mathrm{\mid }$, spatially $M$-Lipschitz, and uniformly continuous in time on $[0,T]$ by step 3.1. Uniqueness in [F1] gives $P=Q_t{P}_0$.

Compactly supported bounded data. For compactly supported $a\in L^1\cap L^{\mathrm{\infty }}$, choose smooth compactly supported $a_j\to a$ in $L^1$ with $\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}$, using [F5]. Their primitives converge uniformly since ${\sup }_x\mathrm{\mid }{\int }_{-\mathrm{\infty }}^x(a_j-a)\mathrm{\mid }\le \mathrm{\parallel }{a}_j-a{\mathrm{\parallel }}_1$. By [F4], their entropy solutions converge uniformly in time in $L^1$ to the solution of datum $a$; their normalized primitives therefore converge uniformly as well. The Hopf--Lax sup contraction in [F1] passes the identity of step 4.1 to $P(t)=Q_t{P}_0$. In particular $P_x=u$ a.e. by [F5]. This proves (ii), including the normalization $-tf(0)$.

A bounded Lipschitz primitive with nonintegrable derivative. Let $U_0$ be as in (i), and set $U_0^R(x)=U_0(\max \{-R,\min \{x,R\}\})$. It has the same sup and Lipschitz bounds, and derivative $u_0^R=u_0{\mathbf{1}}_{(-R,R)}$ a.e. The difference between $U_0^R$ and the normalized primitive of $u_0^R$ is its constant value $U_0(-R)$; adding this constant commutes with Hopf--Lax. Thus step 5.1 shows that $(Q_t{U}_0^R{)}_x$ is an entropy solution with datum $u_0^R$. Step 1.1 places every minimiser for both $U_0$ and $U_0^R$ within $Ct$ of $x$. Consequently $Q_t{U}_0^R(x)=Q_t{U}_0(x)$ whenever $\mathrm{\mid }x\mathrm{\mid }+Ct<R$, since all those competitors see identical data. Every compact positive-time cylinder is contained in such a region for large $R$, so the a.e. derivative $v=(Q_t{U}_0{)}_x$ is bounded by $M$ and obeys the weak equation and every entropy inequality locally, hence globally. For a compact spatial set, fix $R$ large enough that this equality holds throughout $0\le t\le T$ on that set; the strong local trace of the compact-data solution supplies the trace of $v$ equal to $u_0$. This proves (i), with entropy uniqueness from [F4].

Conclusion. Step 6.1 proves the derivative correspondence for the entire bounded Lipschitz primitive class, including nonintegrable derivatives, while step 5.1 proves the normalized primitive correspondence for compactly supported integrable data. In that latter class, a.e. differentiation returns $u$, and integration from $-\mathrm{\infty }$ with the time shift $-tf(0)$ returns the prescribed viscosity potential. All arguments hold on an arbitrary finite horizon; [F1] gives viscosity uniqueness on each such slab, while [F4] gives compatibility of the entropy solutions on overlapping horizons. These are the global solutions and inverse correspondences asserted.