CONVERGENCE OF THE ‘ RELATIVISTIC ’ HEAT EQUATION TO THE HEAT EQUATION AS

The speed of light c is the highest admissible velocity for transport of radiation in transparent media, and, to ensure it, J. R. Wilson (in an unpublished work, see [16]) proposed to use a flux limiter. The flux limiter merely enforces the physical restriction that the flux cannot exceed energy density times the speed of light, that is, the flux cannot violate causality. The basic idea is to modify the diffusion-theory formula for the flux in a way that gives the standard result in the high opacity limit, while simulating free streaming (at light speed) in transparent regions. As an example, one of the expressions suggested for the flux of the (positive) energy density u is


Introduction
To limit the speed of propagation of different types of waves which are solutions of nonlinear degenerate parabolic equations some mechanisms of saturation of the flux as the gradient becomes unbounded have been proposed by different authors [16], [11], [17].
The speed of light c is the highest admissible velocity for transport of radiation in transparent media, and, to ensure it, J. R. Wilson (in an unpublished work, see [16]) proposed to use a flux limiter.The flux limiter merely enforces the physical restriction that the flux cannot exceed energy density times the speed of light, that is, the flux cannot violate causality.The basic idea is to modify the diffusion-theory formula for the flux in a way that gives the standard result in the high opacity limit, while simulating free streaming (at light speed) in transparent regions.As an example, one of the expressions suggested for the flux of the (positive) energy density u is (1.1)F = −νu ∇u u + νc −1 |∇u| (where ν is a constant representing a kinematic viscosity and c the speed of light) which yields in the limit ν → ∞ the flux F = −cu ∇u |∇u| .Observe also that when c → ∞, the flux tends to F = −ν∇u, and the corresponding diffusion equation becomes the heat equation, which has an infinite speed of propagation.
The diffusion equation corresponding to (1.1) is (1.2) u t = ν div u∇u u + ν c |∇u| and is one among the various flux limited diffusion equations used in the theory of radiation hydrodynamics [16].Indeed, the same effects can be guaranteed for a similar equation [8] (1.3) which was introduced by Y. Brenier [8].He was able to derive (1.3) from Monge-Kantorovich's mass transport theory and described it as a relativistic heat equation.Both equations, (1.2) and (1.3), interpolate (see [8]) between the usual heat equation (when c → ∞) and the diffusion equation in transparent media (when ν → ∞) with constant speed of propagation c (1.4) Let us mention that many other models of nonlinear degenerate parabolic equations with flux saturation as the gradient becomes unbounded have been proposed by Rosenau and his coworkers [11], [17], and Bertsch and Dal Passo [7], [12].We consider the solution of (1.3) with initial condition u(0, x) = u 0 (x) ∈ L ∞ (R N ) ∩ L 1 (R N ), u 0 ≥ 0. In a series of papers [3], [4], [5] we developed a theory of existence and uniqueness of entropy solutions for (1.3) and we studied the propagation of discontinuity fronts at the speed of light c.Moreover, in [6] we proved the convergence of (1.3) to equation (1.4) when ν → ∞.Our purpose in this paper is to study the asymptotic limit of equation (1.3) as c → ∞.If u c (t, x) denotes the entropy solution of (1.3) we shall prove that u c converges as c → ∞ to the solution of the classical heat equation (1.5) u t = ν∆u with initial condition u(0, x) = u 0 (x).
Let us explain the plan of the paper.In Section 2 we recall the basic existence and uniqueness result of entropy solutions of (1.3) and state the main result of convergence of solutions of (1.3) to solutions of (1.5) as c → ∞.Section 3 is devoted to the proof of this convergence result.

Preliminaries
In [3], [4] we studied the well-posedness of (1.3) for initial conditions in (L 1 (R N ) ∩ L ∞ (R N )) + .We proved that the underlying elliptic operator to (1.3) defines a nonlinear contraction semigroup in L 1 (R N ) + , we defined the concept of entropy solution proving its existence and uniqueness and we proved that entropy solutions coincide with the semigroup ones.
For the concept of entropy solution of (1.3) we refer to [4].Let us recall the basic existence and uniqueness result proved in [4].
Our main purpose is to prove the following result.
Theorem 2.2.Let u c be the entropy solution of to the solution U of the heat equation (1.5) with U (0, x) = u 0 (x).
We observe that the same result is true for equation (1.2).Since the proof is similar, we skip the details.
Observe that v(t, x) is an entropy solution of (1.3) if and only if u(t, x) = v(νt, νx) is an entropy solution of (2.2) Thus, without loss of generality we may assume that ν = 1, and, for simplicity, we shall assume it in the sequel.Since, by Theorem 2.1, T (t)u 0 = u(t) determines a nonlinear contraction semigroup in L 1 (R N ) + , we may reduce the proof of Theorem 2.2 to a dense set of functions in (L ∞ (R N ) ∩ L 1 (R N )) + with respect to the norm in L 1 (R N ).We shall consider the set of functions for some λ 0 , β > 0}, where S(R N ) denotes the space of rapidly decreasing functions in R N .Functions in S(R N ) which are positive and behave like λe −β|x| , for some λ, β > 0 and for |x| large enough, belong to A e and, thus, both A e and A are dense in L 1 (R N ) + .
We shall concentrate our efforts in proving Theorem 2.2 when u 0 ∈ A e .In that case, we shall prove that solutions satisfy a Lipschitz bound which enables to pass to the limit as c → ∞ in (2.2).

Convergence of solutions of (1.3) to solutions of heat equation
Our first purpose is to prove the following result.
Proposition 3.1.Assume that u 0 ∈ A e .Let u be the entropy solution of (2.2).Then for any t > 0, u(t) ∈ A e .Moreover, we have Observe first that, according to Proposition 3 in [5], u(t, x) > 0 for any t > 0 and any x ∈ R N .To prove the gradient bound (3.1) we reduce it to the case where c = 1.For that, we observe that u(t, x) is the entropy solution of (2.2) with u(0, x) = u 0 (x) if and only if ũ(t, x) = u t c 2 , x c is the entropy solution of (3.2) with ũ(0, x) = u 0 x c .Now, assume that we have proved that sup Writing this inequality in terms of u, we have For any R > 0, let Ω R := [−R, R] N .In order to prove the Lipschitz bound (3.1) for the entropy solutions of (3.2) we need to approximate (3.2) by with ν ΩR the unit outward normal on ∂Ω R .
We consider the following approximating problem Then there is a solution u R η,n of (3.6) which is even, ) for any T > 0.Moreover, we have the estimates for any q ∈ [1, ∞] and any t ≥ 0, and for any T a,b (r) := max(min(b, r), a) (0 < a < b) where the constant C > 0 only depends on a, b and u 0 and does not depend on η, n, R.
We denote Proof: The present lemma follows from Theorem 3.1 and Theorem 8.1 in Chapter 5 of [14].Let us sketch the proof.Let us approximate (3.6) by the following PDE: The equation (3.10) can be written as where Then A ǫ,η (w, p) is continuously differentiable in (w, p).We denote A ǫ,ηi the coordinates of A ǫ,η , A ǫ,ηi w := ∂ w A ǫ,ηi , A ǫ,ηi pj := ∂ pj A ǫ,ηi and similarly for higher order derivatives.We require some computations: Then we have the functions A ǫ,ηi are Lipschitz in (w, p) with bounds independent of ǫ and η, as soon as η remains bounded, and the functions A ǫ,ηi w , A ǫ,ηi pj are also Lipschitz with a bound depending on 1 ǫ .Then by Theorem 8.1 on Chapter 5 of [14], we have that there exists a solution u R η,n,ǫ ∈ C 1+β/2,2+β (Q T ) for some β > 0 (indeed any β < 1) where are Lipschitz (with a bound depending on 1 ǫ ) and A ǫ,ηi wpj p i , A ǫ,ηi w and A ǫ,ηi ww p i are bounded, then the uniqueness conditions of Theorem 8.1 on Chapter 5 of [14] hold, and we have a unique solution in the class of functions which are bounded together with its derivatives of first and second order.In particular since u R η,n,ǫ (• + 2R(1, . . ., 1)) and u R η,n,ǫ (x 1 , . . ., −x i , . . ., x N ), i = 1, . . ., N , are also solutions of (3.10) in this class, we deduce that u R η,n,ǫ has the same symmetries and is periodic with fundamental period [−2R, 2R] N .In particular, this implies that Using (3.13), the estimates (3.7) and (3.8) and follow easily by multiplication by suitable test functions and integration by parts in [0, 2R] N .The estimate (3.9) can be proved as in [3].The regularity of u R η,n,ǫ permits now to use Theorem 3.1 in Chapter 5 of [14].Indeed, observe that the estimates in (3.11), (3.12) are independent of ǫ, and the terms on the right hand side of (3.11), (3.12) which do not depend on |p| are constants.Then Theorem 3.1 in Chapter 5 of [14] with bounds which depend on N , u 0 ∞ , η and the constants in the right hand side of (3.11), (3.12) (which are universal).Letting ǫ → 0, we obtain that there is a weak solution of (3.10 it satisfies the estimates in (3.7), (3.8), (3.9) and (3.14) and is such To prove the last assertion, let A η (w, p) = A 0,η (w, p) and let us observe the previous computations of A ηi ww , A ηi wpj and A ηi pj p k permit to check that A ηi , A ηi w , A ηi pj are Lipschitz continuous in any set of the form {(w, p) : are locally Hölder continuous in (t, x) and the functions ) and its derivatives are Hölder continuous in (t, x).Observe that ∂ k u R η,n is a weak solution of the partial differential equation We consider this PDE in [0, T ] × B(0, R ′ ) (where B(0, R ′ ) denotes the ball centered at 0 of radius R ′ > 0) with boundary conditions Now, we consider the PDE obtained by expanding (3.15) with boundary conditions (3.16), (3.17).Since its coefficients of (3.18) are Hölder continuous in (t, x) and the initial condition is in (3.18) which is also a weak solution of (3.15) [13], [14].By uniqueness of weak solutions of (3.15) which satisfy the boundary conditions (3.16), (3.17) classically, we have that ∇u R η,n ∈ C 1+β/2,2+β loc ).This implies our statement.Lemma 3.4.Assume that u 0 ∈ A. Then we have .
Proof: We consider here u R η,n as a function in [0, T ] × R N with the symmetries stated after Proposition 3.2.By the regularity of u R η,n stated in Lemma 3.3 we can proceed to the following change of variables and the subsequent computations.Define which is even and periodic with fundamental period [−2R, 2R] N .We shall use Bernstein method to obtain an estimate on Dv ∞ .Let us write (3.20) as (3.21) where we use the Einstein convention, Differentiating (3.21) with respect to x k , and multiplying the resulting equation by v k , we obtain 1 2 As an application of the maximum principle, we obtain We denote by C 1 ct ((0, T ) × Ω) the set of functions obtained as restrictions to (0, T ) × Ω of functions φ ∈ C 1 0 ((0, T ) × R N ).Definition 3.5.Let Ω be an open bounded set in R N with Lispchitz boundary.Let u ∈ L 1 ((0, T ) × Ω) and z ∈ L 1 ((0, T ) × Ω, R N ).We say that Next lemma is obvious and we state it for convenience.
Similarly, we deduce that From the convexity of j we have that ∆ + τ j(u) ≥ p(u)∆ + τ u for t ≤ T − τ.With a similar argument as above we deduce that (3.30) Thus, (3.24) holds for such φ.Since any function φ ∈ C 1 ct ((0, T )× Ω) can be written as φ = max(φ, 0)− max(−φ, 0) = lim n ρn * max(φ, 0)−lim n ρn * max(−φ, 0), the integrability properties of u and ∇u permit to prove that (3.24) holds for any φ ∈ C 1 ct ((0, T ) × Ω).We will use the entropy inequalities for (3.3) in the proof of Proposition 3.2.For that, we consider the truncature functions of the form T l a,b (r) := T a,b (r) − l (l ∈ R) and we denote Step 1. Assume that η ∈ (0, 1].Let us prove that there is some γ > 0 such that where C > 0 is a positive constant which does not depend on η.Using its symmetry properties, we consider u R η,n as a solution of (3.6) in [0, T ]× R N .Now, observe that we may write (3.6) as Convergence of the 'Relativistic' Heat Equation 133we have that where M n is the constant in the right hand side of (3.19).Hence the assumptions of Theorem 10.1 of Chapter 3 in [14] are satisfied and (3.31) holds.
By Lemma 3.7, the entropy inequalities are satisfied (see [2]).Indeed, let S, T ∈ T + and J ST be the primitive of ST , i.e., J ST (r) = r 0 T (s)S(s) ds, let A(w, p) := wp Replacing j by J ST , u by u R n and z by z R n in (3.24), using only the inequality ≤, we obtain the entropy inequalities [2], [4] T for any truncatures S, T ∈ T + and any smooth function Let us mention that the concept of entropy solution for the problem (3.3) defined in [2], [4] requires that the equation (3.33) holds with more general test functions and using the techniques in [2], [4] it can be proved that u R n is an entropy solution of (3.3).On the other hand, the concept of entropy solution for (3.3) implies that the equation in (3.3) holds with test functions in C 1 ct ((0, T ) × [0, 2R]), that truncatures of the solution are functions of bounded variation in x, and the entropy conditions are satisfied.The last two conditions are enough to prove uniqueness using the method in [2], [4].In the present context, this amounts to say that using the uniqueness proof in [2], [4] we can prove: a) any entropy solution of (3.3) coincides with u R n and b) any two functions with the regularity of u R n satisfying the entropy conditions above must coincide.In particular, this implies that the whole sequence Step 3. Let us prove that On the other hand, we observe that n .
Step 4. We let n → ∞.Indeed, since ũR 0n → ũR 0 in L 1 ([0, 2R] N ) and the entropy solution of (3.3) gives a contractive semigroup in L 1 ([0, 2R] N ) [2], then u R n (t, x) converges as n → ∞ to the entropy solution u R (t, x) of (3.3) with initial datum ũR 0 .Let us observe that the estimates we have permit to obtain easily the same conclusion.The estimate analogous to (3.8) for u R n , the estimate (3.35) and the boundedness of its right hand side prove that sup , is bounded independently of n.Now, the argument in Step 1 proves that the Hölder bound (3.31) holds for u R n with a bound independent of n.Hence, after extraction of a subsequence if necessary, we have that u R n → u R uniformly in [0, T ] × [0, 2R] N as n → ∞ for some u R ∈ C([0, T ] × [0, 2R] N ).Now, by Step 3, we may pass to the limit in (3.35) and we obtain (3.38) sup Moreover, by extraction of a subsequence, we may assume that Using Lemma 3.6 and the techniques in [2] one can prove the following facts: (3.40) As in Step 2, by Lemma 3.7, the entropy inequalities are satisfied (see [2]).
The same observations made in Step 2 can be done in this case.Let us only mention that it can be proved that u R is an entropy solution of (3.3) in the sense of [4] (see also [2]).But we only need to observe that solutions of (3.3) satisfying the same regularity conditions as u R and the entropy inequalities are unique.This implies that the whole sequence The entropy solution of (3.3) is obtained by a suitable translation of u R , i.e., u R (x) = u R (x + R(1, . . ., 1)).
Convergence of the 'Relativistic' Heat Equation 137Proof: Let us prove that if α ≥ 2N − 1, then U (t, x) = e −αt−β |x| 2 2 is a subsolution of (3.2).This follows from U t = −αU and the inequality div where C > 0 depends on Q but it does not depend on R. Now, we observe that letting η → 0 + , n → ∞ in this order in the estimate (3.8) we obtain that These estimates imply that, by extracting a subsequence, if necessary, we may assume that u R (t, x) → u(t, x) locally uniformly in [0, T ] × R N for some function u and We may also assume that For that, following [2], [4], we prove that for any φ(t, x) ∈ C ∞ 0 ((0, T ) × R N ), φ ≥ 0, and any g ∈ 1 ([0, T ] × R N ).Let us fix a function φ and a function g in the previous classes.For each R > 0 such that supp φ ⊆ (0, T ) × (−R, R) N , we write the inequalities where the domain of integration is considered to be R N by our choice of R > 0.
Observe that, since u R → u as R → ∞ locally uniformly in [0, T ]×R N and ∇u R → ∇u weakly * in L ∞ ([0, T ] × R N , R N ), we have Finally, using the PDE satisfied by u R (see (3.40)), (3.43) and Lemma 3.7 we have Collecting all the above inequalities, we obtain (3.45).
Since we may take a countable set of functions in C 1 ([0, T ] × R N ) dense in C 1 ([0, T ] × B k ) for any ball B k centered at 0 of radius k ∈ N, we have that the above inequality holds for all (t, x) ∈ S where S ⊆ (0, T ) × R N is such that L N ((0, T ) × R N \ S) = 0, and all g ∈ ∪ k C 1 ([0, T ] × B k ).Now, fixed (t, x) ∈ S, and given y ∈ R N , there is g ∈ ∪ k C 1 ([0, T ] × B k ) such that ∇g(t, x) = y.Then (z(t, x) − A(u(t, x), y)) • (∇u(t, x) − y) ≥ 0 ∀ y ∈ R N and ∀ (t, x) ∈ S, and, by an application of the Minty-Browder method in R N , it follows that z(t, x) = A(u(t, x), ∇u(t, x)) a.e.(t, x) ∈ Q T , and (3.44) follows.
Step 3. Concluding the proof.By Steps 1 and 2 we have that for some constant C > 0. This estimate, together with the entropy inequalities, permit to prove that if u is the entropy solution of (3.2) with initial datum u(0, x) = u 0 , then u = u.In other words, u is the entropy solution of (3.2).Since u 0 ∈ A e , then u 0 (x) ≥ λ 0 e −β |x| 2 2 for some λ 0 , β > 0 and, by Lemma 3.9 we have that u(t, x) ≥ λ 0 e −αt−β |x| 2 2 for any α ≥ β(2N − 1).Thus, given R 0 , u R is bounded away from zero in [0, T ] × [−R 0 , R 0 ] N for R large enough, and 1 u R → 1 u locally uniformly in [0, T ] × R N .This implies that ∇u R (t,x) u R (t,x) converges weakly * in L ∞ to ∇u(t,x) u(t,x) and, hence We conclude the proof of Theorem 2.2 with the following lemma.

Lemma 3 . 11 . 2 ≤
Assume that u 0 ∈ A e .Let u c be the solution of (2.2).As c → ∞, u c converges to the solutionU ∈ C([0, T ], L 1 (R N )) of the heat equation u t = ∆u with U (0, x) = u 0 (x).Proof: As in Step 2 of the proof of Proposition 3.2, let us prove that there is some γ > 0 such that for any compact subset Q ⊆ R N we have(3.51)|u c (t, x) − u c (s, x)| ≤ C|t − s| γ ∀ 0 ≤ s, t ≤ T, ∀ x ∈ Q,where C > 0 is a positive constant depending on Q which does not depend on c.For that, observe that we may write (2.2) asu t = div(a c (t, x)∇u),wherea c (t, x) a c (t, x) ≤ 1where M is the constant in the right hand side of (3.1).Hence the assumptions of Theorem 10.1 of Chapter 3 in[14] are satisfied and (3.51) holds.The estimates (3.42), (3.1), (3.51) for u c and the convergence a c (t, x) → 1 uniformly as c → ∞, imply that u c converges as c → ∞ to the solution U ∈ C([0, T ], L 1 (R N )) of the heat equation with U (0, x) = u 0 (x).