Well-posedness of scalar conservation laws with semilinear multiplicative rough-path forcing (Lions–Perthame–Souganidis problem (9))

Lions, Perthame and Souganidis (Stoch. PDE: Anal. Comput. 1 (2013), 664–686) built a pathwise well-posedness theory — pathwise stochastic entropy solutions with existence, uniqueness and a pathwise $L^1$-contraction — for scalar conservation laws whose flux is driven by a continuous (rough) signal in Stratonovich form, $du + \sum_{i=1}^N (A_i(u))_{x_i}\circ dW^i = 0$. The same paper left, as an explicit open question, the well-posedness of the semilinear version $du + \sum_{i=1}^N (A_i(u))_{x_i}\circ dW^i = \Phi(u)\circ d\tilde{\mathbf{W}}$ with genuinely nonlinear $\Phi$: the pathwise quasilinear approach and the Doss–Sussman transformation used for semilinear stochastic dependence are incompatible, because that transformation does not preserve shocks unless the forcing is linear. Subsequent literature resolved adjacent questions: the announced spatially dependent extension was resolved by the authors themselves (2014) and by Gess–Souganidis (2015) and Gess–Perthame–Souganidis (2016); Hofmanová (2016) proved well-posedness for rough flux combined with Itô forcing; Gess–Souganidis (2017) settled regularizing effects in the periodic, spatially homogeneous case. None of these treats the pathwise semilinear problem as stated, and Gess–Souganidis (2015) explicitly leave the semilinear extension 'to the interested reader'. The open core therefore survives: construct, or rigorously refute, a well-posedness theory for scalar conservation laws with semilinear multiplicative rough-path forcing, resolving the documented incompatibility between the quasilinear pathwise approach and the semilinear Doss–Sussman approach.

Background

A scalar conservation law is the first-order nonlinear equation $u_t + \operatorname{div} A(u) = 0$ for a scalar unknown $u(t,x)$. Its classical solutions typically develop discontinuities (shocks) in finite time, and the physically admissible weak solution is selected by the entropy condition — the requirement that a convex entropy of the solution be dissipated across shocks — introduced by Kružkov (1970). The kinetic formulation of Lions, Perthame and Tadmor (1994) rewrites the entropy formulation as a linear transport equation, with a nonnegative measure source, for the auxiliary function $f(t,x,\xi)=\mathbf{1}_{0<\xi<u(t,x)}$; this representation is the technical backbone of $L^1$-contraction, uniqueness and error estimates. The Rankine–Hugoniot condition states that a shock between the states $u_-$ and $u_+$ travels with speed $(A(u_+)-A(u_-))/(u_+-u_-)$; hence any change of unknown that modifies the flux also modifies the shock speeds.

Two stochastic frameworks. When noise enters a conservation law, two distinct structures arise. If the forcing is semilinear, $du + \operatorname{div} A(u)\,dt = \Phi(u)\,d\tilde{\mathbf{W}}$ with $\tilde{\mathbf{W}}$ a Brownian motion and the integral understood in the Itô sense (the Itô integral is defined probabilistically, as an $L^2$-limit of Riemann sums adapted to a filtration, and is not defined path by path), a well-posedness theory of entropy/kinetic solutions with an $L^1(\mathbb{R}^N\times\Omega)$-type contraction was established by Feng and Nualart (2008), Debussche and Vovelle (2010), and Bauzet, Vallet and Wittbold (2012). If instead the flux itself is modulated by noise, the natural formulation is pathwise and uses the Stratonovich product, denoted by $\circ$ (the Stratonovich integral obeys the classical chain rule and is the stochastic counterpart of a pathwise differential); the equation becomes $du + \sum_{i=1}^{N}(A_i(u))_{x_i}\circ dW^i = 0$, where $\mathbf{W}=(W^1,\ldots,W^N)$ is a continuous signal of infinite variation, a typical Brownian trajectory being the motivating example.

Rough paths. To treat such equations path by path, the driving signal is viewed as a rough path: the path is enhanced with its iterated-integral data so that integrals and the associated characteristics are defined deterministically, pathwise, and depend continuously on the driver; a geometric rough path is one whose iterated integrals satisfy the usual shuffle relations, so that the classical chain rule of the Stratonovich calculus holds; the theory is stable under local uniform convergence of paths, so pathwise solutions can be obtained as limits of solutions driven by smooth approximations of the signal.

The pathwise theory for the quasilinear problem. Lions, Perthame and Souganidis (2013) developed a complete pathwise theory for the quasilinear equation (their equation (4)): pathwise stochastic entropy solutions are defined by testing the kinetic formulation against test functions transported along the stochastic characteristics (the flow of the differential equation $dX = a(\xi)\circ dW$ built from $a=A'$), a solution notion closed under local uniform limits of the driving paths. They prove existence, uniqueness, continuous dependence, and a pathwise $L^1$-contraction. This is the conservation-law analogue of the pathwise theory of stochastic viscosity solutions of Lions and Souganidis (1998) for fully nonlinear first- and second-order stochastic PDE.

The Doss–Sussman transformation and the obstruction. For semilinear multiplicative stochastic dependence — a forcing term $\Phi(u)\circ d\tilde{\mathbf{W}}$ — the classical device, going back to Doss and Sussman and used by Lions and Souganidis (2000) for stochastic Hamilton–Jacobi equations, is a change of unknown $u=\Psi(v,t)$ where $\Psi$ solves the ordinary differential equation $d\Psi = \Phi(\Psi)\circ d\tilde{\mathbf{W}}$ with $\Psi(v;0)=v$; the new unknown $v$ then satisfies a conservation law with a modified, time-dependent flux. In Section 6 of their 2013 paper Lions, Perthame and Souganidis show that for conservation laws this route fails: the modified flux changes the Rankine–Hugoniot speed of shocks, so the transformation does not preserve shocks unless the forcing is linear. Their one-dimensional semilinear Burgers example takes $u_t + \tfrac12(u^2)_x = \Phi(u)$ with initial data $u^0=1$ for $x<0$ and $u^0=0$ for $x>0$, and a nonlinearity with $\Phi(0)=\Phi(1)=0$ and $\Phi>0$ on $(0,1)$; the entropy solution is the shock $x=t/2$, whereas after the transformation the shock of $v$ travels at speed $\int_0^1\Psi(w;t)\,dw > 1/2$. In addition, the properties of the Itô integral preclude a pathwise treatment of the Itô version of the semilinear problem. The approaches of the quasilinear and the semilinear theories therefore, in the authors' words, 'appear to be incompatible', and the well-posedness of the semilinear problem (9) is posed as an open question.

Problem Statement

Consider, on $\mathbb{R}^N\times(0,\infty)$ with initial datum $u^0\in(L^1\cap L^\infty)(\mathbb{R}^N)$, the scalar conservation law with semilinear multiplicative rough-path dependence (problem (9) of Lions, Perthame and Souganidis (2013)):

$$du + \sum_{i=1}^{N} (A_i(u))_{x_i}\circ dW^i = \boldsymbol{\Phi}(u)\circ d\tilde{\mathbf{W}}, \qquad u = u^0 \ \text{on}\ \mathbb{R}^N\times\{0\},$$

where $\mathbf{A}=(A_1,\ldots,A_N)\in C^2(\mathbb{R};\mathbb{R}^N)$ is the flux, $\boldsymbol{\Phi}=(\Phi_1,\ldots,\Phi_m)\in C^2(\mathbb{R};\mathbb{R}^m)$ is the forcing with $\boldsymbol{\Phi}$ genuinely nonlinear (i.e. not linear, which is the regime in which the source's counterexample shows the Doss–Sussman route fails), and $\mathbf{W}=(W^1,\ldots,W^N)$, $\tilde{\mathbf{W}}$ are continuous driving signals (geometric rough paths, in the genuinely rough regime of infinite-variation paths; $N$- and $m$-dimensional Brownian motions in the stochastic special case), $\circ$ denoting the Stratonovich/pathwise product. The task is to settle the well-posedness of this problem: define a mathematically precise notion of solution — a pathwise (rough) entropy/kinetic solution, or another justified concept — which is (i) meaningful for the Stratonovich/rough-path equation as written (not a reinterpreted Itô problem), (ii) stable under smooth-path approximation of the drivers, and (iii) entropy-admissible, i.e., it selects the physically relevant shocks; and prove, for that notion, existence, uniqueness, and continuous dependence on the initial data and on the drivers in the form of an $L^1$-type contraction (pathwise, or in $L^1(\Omega\times\mathbb{R}^N)$, as the theory dictates). A complete theory must reduce to the established theories in the degenerate cases: forcing absent (where it must agree with the pathwise stochastic entropy solutions of the quasilinear theory of the source paper), linear forcing (where the Doss–Sussman route is known to preserve shocks), and smooth drivers (classical entropy theory). Equivalently, a proof that problem (9) is not well posed within a precisely stated natural class of entropy-type solution concepts satisfying requirements (i)–(iii) above also settles the question; a class of solution concepts that fails requirements (i)–(iii) lies outside the scope of the question. The precise solution concept, the structural assumptions supplementing $C^2$ smoothness (boundedness, nondegeneracy, growth), and the class of driving paths are not prescribed by the question; no restriction on dimension, noise dimension, or geometry is imposed.

References

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