Diffusive limit of the generalized Carleman system with superlinear density-dependent collision rate $ρ^α$ for $1 < α < 2$

The generalized Carleman kinetic model describes a one-dimensional gas of particles moving with equal speeds in opposite directions with densities $u(t,x)$ and $v(t,x)$, colliding at a rate proportional to a power $ρ^α$ of the macroscopic density $ρ=u+v$. Under the diffusive scaling with mean free path $ε$, the expected limit $ε→0$ of $ρ^ε$ is the nonlinear diffusion equation $∂_t ρ = (1/2)∂_x(ρ^{-α}∂_x ρ)$. This limit is rigorous for all $α ≤ 1$ (classical for $α=1$; Lions–Toscani 1997 for $α<1$) and, in the superlinear range $α>1$, only partially: Salvarani–Vázquez (2005) obtained an iterated 'lifted' limit and showed the limit is trivial for $α ≥ 2$; Salvarani–Toscani (2009) proved genuine convergence only for $1<α<4/3$ with admissible (essentially bounded, weighted-moment) data, identifying $α=4/3$ as the exact barrier where the entropy/compensated-compactness method fails; Golse–Salvarani (2007) established the bounded-domain inflow-boundary theory only for $|α| ≤ 1$. The open problem is to complete (or refute) the diffusive limit in the remaining superlinear window — for each of three independent sub-ranges, by either proving convergence or giving a rigorous refutation: the Cauchy problem for every $α ∈ [4/3,2)$, the unrestricted limit for integrable data with $1<α<4/3$, and the inflow initial-boundary value problem for $α ∈ (1,2)$ — where the expected limit, if it exists, is the maximal mass-conserving solution of a very fast diffusion equation with singular diffusivity.

Background

Discrete-velocity kinetic models replace the Boltzmann equation of gas dynamics by finitely many coupled hyperbolic equations. The simplest family, going back to Carleman (1957), describes a fictitious one-dimensional gas of particles moving with speed $c$ along the $x$-axis, either to the right with density $u(t,x)$ or to the left with density $v(t,x)$, exchanging direction upon collision:

$∂_t u + c∂_x u = k(u,v,x)(v-u), \; ∂_t v - c∂_x v = k(u,v,x)(u-v),$

where $k ≥ 0$ is the collision (interaction) rate. The choice $k = u+v$ (binary collisions) is Carleman's model; $k = 1$ gives the Goldstein–Taylor model, which is linear and equivalent to the telegrapher's equation (the damped wave equation). The 'generalized Carleman' family takes the power-law rate $k_α = (u+v)^α = ρ^α$, where $ρ = u+v$ is the macroscopic mass density and $α$ is a real parameter interpolating between these cases ($α=1$: Carleman, $α=0$: Goldstein–Taylor).

The diffusive (parabolic) scaling rescales space and time so that $x^2/t$ is invariant, introducing a small mean free path $ε$ (the average distance travelled between two collisions):

$∂_t u^ε + (1/ε)∂_x u^ε = (1/ε^2)\,(ρ^ε)^α\,(v^ε - u^ε), \; ∂_t v^ε - (1/ε)∂_x v^ε = (1/ε^2)\,(ρ^ε)^α\,(u^ε - v^ε).$

Writing the rescaled flux $j^ε = (u^ε - v^ε)/ε$, the system is equivalent to $∂_t ρ^ε + ∂_x j^ε = 0$ and $ε^2∂_t j^ε + ∂_x ρ^ε = -2(ρ^ε)^α j^ε$; formally dropping the $ε^2$-term yields the nonlinear diffusion equation

$∂_t ρ = (1/2)\,∂_x(ρ^{-α}∂_x ρ) = (1/(2(1-α)))\,∂_{xx}\,ρ^{1-α} \; (α ≠ 1), \qquad ∂_t ρ = (1/2)\,∂_{xx}\ln ρ \; (α=1).$

Depending on $α$ the limit equation belongs to different regimes of nonlinear diffusion: for $α<0$ it is of porous-medium type (exponent $1-α>1$, degenerate diffusivity vanishing at vacuum); for $0 ≤ α < 1$ the fast diffusion equation (exponent in $(0,1)$); for $α=1$ the logarithmic diffusion equation; and for $α>1$ a very fast diffusion equation with negative exponent $1-α ∈ (-1,0)$, whose diffusivity $ρ^{-α}$ is singular at vacuum.

The rigorous justification of this limit has a long history. For the Carleman equation ($α=1$), Kurtz (1973) proved $L^1$ convergence of $ρ^ε$ to the logarithmic diffusion equation by nonlinear semigroup theory, exploiting that the system is $L^1$-accretive (dissipative), i.e. order-preserving and contractive in $L^1$; McKean (1975) removed the restriction of zero initial flux; Fitzgibbon (1982) treated bounded intervals with specular (reflecting) boundary conditions. Pulvirenti–Toscani (1996) handled the whole range $0 ≤ α ≤ 1$. Lions–Toscani (1997) — the source of the present problem — gave a unified treatment of all $α<1$ in one dimension and of a six-velocity model in three dimensions, combining entropy methods (uniform control of a Lyapunov functional, the entropy $H_α = -∫(u^{2-α}+v^{2-α})dx$, and its dissipation) with compensated compactness — specifically the div-curl lemma of Murat and Tartar, which recovers convergence of nonlinear pairings of two vector fields from uniform $L^2$ bounds plus their differential constraints, and thereby enough strong compactness of $ρ^ε$ to pass to the limit in the nonlinear flux relation.

For $α>1$ the problem changes character. The collision operator loses the $L^1$-dissipativity (accretivity) that underlies both the semigroup route and the entropy route, and the target equation itself becomes delicate: Esteban–Rodríguez–Vázquez (1988), analyzing this singular-diffusivity equation, showed it is non-unique for exponents $α ∈ [1,2)$, the natural candidate being the maximal mass-conserving solution, whose tails at infinity are so heavy that $∫ρ^{2-α}dx$ diverges; and for $α ≥ 2$ nonnegative solutions with integrable data do not exist (Vázquez 1992), so the diffusive limit degenerates and mass disappears. Salvarani–Vázquez (2005) improved the $α ≤ 1$ theory to plain $L^1$ data and studied $α>1$ by a 'lifting' approximation, obtaining an iterated (double) limit rather than an unrestricted one; Salvarani–Toscani (2009) proved the first genuine convergence result for superlinear rates, $1<α<4/3$, for admissible data (nonnegative, essentially bounded, with a weighted moment $∫(u_0+v_0)(1+x^2)^δ dx < ∞$ where $(α-1)/(2(2-α)) < δ < 1/4$, a window that exists only for $α<4/3$), and showed the barrier at $α=4/3$ is real: the entropy $H_α$ is bounded below only for $α<4/3$, and an explicit family of self-similar solutions (solutions invariant under the equation's scaling symmetry) of a time-modified variant of the system loses the uniform $L^2$ bound on the flux exactly at $α = 4/3$.

The boundary-value thread: the early convergence results are posed on the whole line, on periodic domains, or with specular reflection. For bounded domains with nonhomogeneous inflow boundary conditions — prescribing the densities of particles entering the domain, $u^ε(t,0) = φ^-(t)$ and $v^ε(t,1) = φ^+(t)$ — the first and so far only nonlinear theory is due to Golse–Salvarani (2007), via a relative entropy of $(u^ε,v^ε)$ with respect to a boundary-matching profile, and it covers exactly $α ∈ [-1,1]$; the authors state that they 'have restricted our analysis to the case $|α| ≤ 1$, as it leads to a unified treatment based on the dissipative nature of the problem'. Miyoshi–Tsutsumi (2016) treated homogeneous absorbing boundary conditions for a doubly nonlinear rate family $(u+v)^a|u-v|^b$ with $a ≤ 1$. Thus both threads — superlinear exponents and physical boundary conditions — stop exactly at the boundary of the dissipative range, and their surviving open parts coincide.

Problem Statement

Consider, for a fixed exponent $α ∈ (1,2)$, the scaled generalized Carleman system with superlinear power-law collision rate:

$∂_t u^ε + (1/ε)∂_x u^ε = (1/ε^2)\,(ρ^ε)^α\,(v^ε - u^ε), \qquad ∂_t v^ε - (1/ε)∂_x v^ε = (1/ε^2)\,(ρ^ε)^α\,(u^ε - v^ε), \qquad ρ^ε = u^ε + v^ε,$

posed (i) on the whole line $x ∈ \mathbb{R}$ with nonnegative initial data $u^0, v^0 ∈ L^1(\mathbb{R})∩L^∞(\mathbb{R})$ (or the maximal data class for which the question is tractable), and (ii) on the interval $(0,1)$ with nonnegative initial data $u^0, v^0 ∈ L^1(0,1)∩L^∞(0,1)$, supplemented with nonnegative inflow boundary data $u^ε(t,0) = φ^-(t)$, $v^ε(t,1) = φ^+(t)$, where $φ^± ∈ W^{1,∞}([0,T])$ are given boundary traces on a fixed time horizon $[0,T]$.

The problem is to complete the rigorous diffusive limit in the superlinear range — that is, to prove or refute, for the stated sub-ranges:

(a) Untouched range, $α ∈ [4/3, 2)$, Cauchy problem: determine whether $ρ^ε$ converges as $ε → 0$ and whether the limit is the maximal mass-conserving solution of the singular-diffusivity (very fast diffusion) equation

$∂_t ρ = (1/2)\,∂_x(ρ^{-α}∂_x ρ), \qquad ρ(0,\cdot) = u^0 + v^0,$

the maximal mass-conserving solution being the natural candidate because the equation is non-unique for $α ∈ [1,2)$. A rigorous negative result — an exhibit of data for which the limit fails to exist, loses mass, or solves a different problem — settles this range equally.

(b) Data generality, $1 < α < 4/3$, Cauchy problem: prove or refute the convergence as an unrestricted limit $ε → 0$ for plain nonnegative $L^1$ data, removing both the admissibility restrictions (essential boundedness and the weighted-moment condition) of Salvarani–Toscani (2009) and the lifted (iterated) approximation of Salvarani–Vázquez (2005).

(c) Boundary value problem, $α ∈ (1,2)$: establish (or refute) the nonlinear diffusion limit on $(0,1)$ with the inflow boundary conditions above, the limit solving the corresponding Dirichlet problem with initial datum $ρ(0,\cdot) = u^0 + v^0$ and boundary data $ρ(t,0) = 2φ^-(t)$, $ρ(t,1) = 2φ^+(t)$, whose well-posedness in the singular-diffusivity regime must itself be addressed.

Sub-problems (a)–(c) are three independent, separately refutable parts of this one superlinear window. They share one objective — the rigorous kinetic-to-diffusion passage for superlinear density-dependent collision rates — and one set of obstructions (loss of $L^1$-dissipativity of the collision operator, entropy functional unbounded below for $α ≥ 4/3$, non-uniqueness and singular diffusivity of the target equation). The endpoints of the window are settled and frame the problem: $α = 1$ is the classical Carleman limit (logarithmic diffusion; Kurtz 1973 and successors), and for $α ≥ 2$ the limit is trivial (no nontrivial nonnegative solutions of the target equation exist; Vázquez 1992), so the meaningful open window is exactly $(1,2)$.

References

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