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Vanishing viscosity selects the viscosity solution

Statement

Let $n \geq 1$ , $T > 0$ , $Z = R^{n} \times (0, T)$ , and $Z_{0} = R^{n} \times [0, T)$ . Let $H : R^{n} \times [0, T] \times R^{n} \to R$ satisfy the Lipschitz conditions of part (a) of Comparison for first-order Hamilton--Jacobi equations, let $u_{0} : R^{n} \to R$ be bounded and uniformly continuous, and let $u_{0}^{(ε)} \to u_{0}$ locally uniformly on $R^{n}$ . For each $ε \in (0, 1)$ let $u^{ε} : Z \to R$ be a viscosity solution of $u_{t}^{ε} + H (x, t, D u^{ε}) = ε Δ u^{ε} in Z,$ meaning that for every test function $ϕ \in C^{1, 2} (Z)$ the residual $ϕ_{t} + H (z, D ϕ) - ε Δ ϕ$ is nonpositive at each local maximum of $u^{ε} - ϕ$ and nonnegative at each local minimum. Assume that $u^{ε}$ has initial datum $u_{0}^{(ε)}$ in the relaxed Cauchy sense. Suppose the family is uniformly bounded on $Z$ and locally equicontinuous up to the initial face: there is $M < \infty$ with $∣ u^{ε} (z) ∣ \leq M$ for every $ε \in (0, 1)$ and $z \in Z$ , and for every compact $K \subseteq Z_{0}$ and every $η > 0$ there is $δ > 0$ such that $∣ u^{ε} (z) - u^{ε} (z^{'}) ∣ < η$ for all $ε \in (0, 1)$ and $z, z^{'} \in K \cap Z$ with $∣ z - z^{'} ∣ < δ$ . These estimates give each $u^{ε}$ a continuous trace on the initial face. Then $u^{ε} \to u$ locally uniformly on $Z$ , where $u$ is the unique bounded viscosity solution of $u_{t} + H (x, t, D u) = 0$ with datum $u_{0}$ . Neither existence of the approximants nor a compactness theorem is asserted: the boundedness and equicontinuity estimates are hypotheses. No choice principle is used.

Facts & Assumptions

Given: The Hamiltonian $H$ with the Lipschitz conditions of comparison case (a), bounded uniformly continuous $u_{0}$ , data $u_{0}^{(ε)} \to u_{0}$ locally uniformly, a uniformly bounded family $(u^{ε})$ of viscous solutions, locally equicontinuous up to the initial face, with data $u_{0}^{(ε)}$ in the relaxed sense, and the half-relaxed limits $\overline{u}, \underline{u}$ of the family (Half-relaxed limits of a locally bounded family).

[F1]

For every fixed $ϕ \in C^{1, 2} (Z)$ , at each local maximum of $u^{ε} - ϕ$ one has $ϕ_{t} + H (z, D ϕ) \leq ε Δ ϕ (z)$ , and at each local minimum $ϕ_{t} + H (z, D ϕ) \geq ε Δ ϕ (z)$ . Thus the errors are bounded in absolute value by $c_{ε} (z) : = ε ∣ Δ ϕ (z) ∣$ , which is locally bounded and tends to $0$ locally uniformly by the explicitly assumed second-order test inequalities in the statement; the limit equation is tested in the first-order sense of Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem.

[F2]

The case-(a) finite-cover argument of Half-relaxed limits of sub- and supersolutions with vanishing perturbations proves the subsolution inequality at a strict contact from the inequality for that fixed smooth test and a locally uniformly vanishing error; the dual argument proves the supersolution inequality. The same item proves passage of the relaxed initial datum under local equicontinuity up to that face.

[F3]

Comparison, case (a), applies to the bounded upper semicontinuous subsolution $\overline{u}$ and the bounded lower semicontinuous supersolution $\underline{u}$ when their relaxed initial data agree (Comparison for first-order Hamilton--Jacobi equations); uniqueness in the bounded class is Uniqueness and sup-norm contraction for the Cauchy problem.

[F4]

A compact subset of $Z$ has a finite subcover from every intrinsic open cover (Open cover, subcover, compact metric space, and compact subset of a metric space); every ambient indexed open-ball cover of it also has a finite subcover retaining the indices (A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, clauses 2--3). Upper semicontinuous real-valued functions attain maxima on nonempty compact Euclidean sets (Semicontinuous extreme value theorem on compact Euclidean sets).

[F5]

A nonnegative smooth compactly supported bump equal to $1$ on a smaller ball is supplied by A smooth bump between concentric Euclidean balls. Its integral is finite and positive, so normalization gives a unit-mass bump and the scaled family of The mollifier family generated by a unit-mass smooth bump. Here only compact Riemann integrals are needed: continuous integrands on compact boxes are integrable (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set), and monotonicity and linearity give $∣ \int_{Q} g ∣ \leq vol (Q) \sup_{Q} ∣ g ∣$ (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ). For a $C^{1}$ function $f$ near a compact ball, first multiply by such a smooth cutoff equal to $1$ on a slightly larger ball and extend by zero, obtaining a globally $C^{1}$ compactly supported function. Its convolution with the fixed bump is smooth: on a fixed integration box every kernel-derivative difference quotient converges uniformly, by the mean value theorem and uniform continuity of the next derivative, so the integral bound passes each derivative through the integral. For the affine changes $y = x - ρ z$ on a compact integration box, Change of variables for an injective $C^{1}$ map on a compact Jordan set applies: the derivative is the invertible matrix $- ρ I$ and the absolute determinant is $ρ^{n + 1}$ . Thus, using the fixed-kernel formula $\int f (x - ρ y) η (y) d y$ gives first derivatives by the same uniform difference-quotient argument. Unit mass then bounds the errors in $f$ and $D_{i} f$ by their moduli of continuity at distance $ρ R_{η}$ , which tend to zero by Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous. These compact-integral arguments use no choice (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

technique · half-relaxed limits, stability with a vanishing perturbation, comparison, and a compactness-free conversion to local uniform convergence

1.1F1F2F4F5

The relaxed limits are sub- and supersolutions with common initial data. First take a smooth strict upper test $ψ$ for $\overline{u}$ at an interior point. The case-(a) compact finite-cover proof in [F2] applies using the fixed smooth test $ψ + ∣ z - z_{0} ∣^{2}$ at the approximating contacts: its viscosity error is bounded by $ε ∣ Δ_{x} (ψ + ∣ z - z_{0} ∣^{2}) ∣$ , which tends uniformly to zero on the compact contact region, so the half-relaxed limit satisfies $ψ_{t} + H (z, D ψ) \leq 0$ . The dual argument gives the lower-limit supersolution inequality for smooth strict lower tests. To extend these inequalities to an arbitrary $C^{1}$ test $ϕ$ , strictify its contact by adding or subtracting a quartic (Strictification of a viscosity test function by a quartic perturbation). On a closed ball around the contact, convolve $ϕ$ locally with a fixed compactly supported smooth unit-mass bump at scales tending to zero; uniform continuity of $ϕ$ and $D ϕ$ on that ball gives smooth approximants converging in $C^{1}$ . Maximise $\overline{u} - ϕ_{j}$ on the ball for each approximant. The strict contact and uniform convergence imply that the sets of such maximisers are interior for large $j$ and their distance to the original contact tends uniformly to zero. Strictify each smooth test at its maximiser by a quartic and apply the fixed-test argument above. Passing to the limit using $C^{1}$ convergence and continuity of $H$ proves the required inequality for $ϕ$ ; the lower-test argument is dual. Thus $\overline{u}$ is a subsolution and $\underline{u}$ a supersolution for the full $C^{1}$ test definition. Finally, local equicontinuity gives each approximant a continuous initial trace. Its relaxed initial condition makes that trace equal to $u_{0}^{(ε)}$ ; local uniform convergence of these data and the shared boundary modulus then pass the initial trace to both half-relaxed limits.

2.1step 1.1F3

Comparison forces the two limits to agree. The subsolution $\overline{u}$ is bounded and upper semicontinuous and the supersolution $\underline{u}$ is bounded and lower semicontinuous, with the same relaxed initial data $u_{0}$ ; comparison [F3] gives $\overline{u} \leq \underline{u}$ on $Z$ . Since $\underline{u} \leq \overline{u}$ pointwise by the definition of the half-relaxed limits, the two coincide: $\overline{u} = \underline{u} = : u$ , which is therefore continuous; by [F3] it is the unique bounded viscosity solution with datum $u_{0}$ .

3.1step 2.1givenF4algebra∎

Locally uniform convergence. Let $K \subseteq Z$ be compact and $δ > 0$ . For each $z \in K$ , the equalities $\overline{u} (z) = \underline{u} (z) = u (z)$ and the definition of the joint half-relaxed limits in Given give a radius $r_{z} > 0$ such that $∣ u^{ε} (y) - u (z) ∣ < δ / 3$ whenever $0 < ε < r_{z}$ and $y \in Z$ satisfies $∣ y - z ∣ < r_{z}$ . Shrink the radius, if necessary, so that also $∣ u (y) - u (z) ∣ < δ / 3$ there. The family of all such admissible balls covers $K$ ; by [F4] take a finite subcover $B (z_{i}, r_{i})$ and put $ε_{0} : = \min_{i} r_{i} > 0$ . For every $0 < ε < ε_{0}$ and $y \in K$ , one of these balls contains $y$ , so $∣ u^{ε} (y) - u (y) ∣ < 2 δ / 3 < δ$ . This proves uniform convergence on $K$ without selecting a sequence of parameters or points.

Remarks

What is not asserted. No existence of the viscous family is proved and no subsequence is extracted from the family itself; the boundedness and local equicontinuity are hypotheses. The pointwise equality of the two relaxed limits is equivalent to local uniform convergence of the family, which is the content of step 3.1.
Choice. The half-relaxed limits are computed as infima and suprema over sets; the comparison and uniqueness steps are choice-free, and the final conversion uses a finite cover of each compact set.

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Sources

Hung Vinh Tran, Hamilton--Jacobi Equations: Theory and Applications, 2020 preliminary author manuscript of AMS Graduate Studies in Mathematics 213 (complete text) (standard reference, not scraped)
Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society 27 (1992), 1--67 (complete article) (standard reference, not scraped)
Alberto Bressan, Viscosity Solutions of Hamilton--Jacobi Equations and Optimal Control Problems, complete author lecture notes, Penn State University (PDF records Fall 2019 revision) (standard reference, not scraped)