title: Small dispersion approximation of shock wave dynamics
authors: Misha Perepelitsa
text: 
  
  Learning All Credible Bayesian Network Structures for Model Averaging
      Zhenyu A. Liao z6liao@uwaterloo.ca
   
  Charupriya Sharma c9sharma@uwaterloo.ca
   
  David R. Cheriton School of Computer Science
   
  University of Waterloo
   
  Waterloo, ON N2L 3G1, Canada
  James Cussens james.cussens@bristol.ac.uk
  
  Department of Computer Science
  
  University of Bristol
  
  Bristol, BS8 1QU, United Kingdom
  Peter van Beek vanbeek@uwaterloo.ca
  
  David R. Cheriton School of Computer Science
   
  University of Waterloo
   
  Waterloo, ON N2L 3G1, Canada
      December 30, 2023
  ====================================================================================================================================================================================================================================================================================================================================================================================================================================================================
  
  
  
  
  
  We introduce a dispersion approximation of weak, entropy solutions of multidimensional scalar conservation laws using 
  variational kinetic representation, where equilibrium densities satisfy  Gibb's entropy minimization principle for a piecewise linear, convex entropy. For such solutions, we show that  small scale discontinuities, measured by the entropy increments, propagate with characteristic velocities, while the large scale, shock-type discontinuities propagate with speeds  close to the speeds of classical shock waves. In the zero-limit of the scale parameter, approximate solutions converge  to a unique, entropy solution of a scalar conservation law.
  
  
  
  Shock waves, entropy solutions, kinetic equations
  
  
  35L60, 35L65
  
  
  
  
  
  
  
  
  § INTRODUCTION
   Consider the Cauchy problem for a quasilinear system 
  
      ∂_t U+∑_i=1^d∂_x_i F_i(U)=0,   (x,t)∈ℝ^d+1_+,
      
      U(x,0)=U_0(x),   x∈ℝ^d,
  
  where U:ℝ^d+1_+→ℝ^m,  F_i:ℝ^m→ℝ^m. The main difficulty in constructing weak solutions for quasilinear systems (<ref>) is the lack of apriori estimates on solutions in norms that control oscillations. This limits the application of such methods as  viscosity or relaxation approximations of (<ref>) for which pointwise convergence of approximate solutions is hard to establish.
  
  The difficulty is well illustrated on an example of a shock wave.
  For systems with a convex entropy, weak solutions are typically restricted to verify entropy dissipation balance:
  
      ∂_tη(U)+÷_x  q(U)=r,   r≤ 0,
  
  which provides  apriori estimate on the total entropy at time t  and total dissipated entropy up to time t in terms of the entropy of the initial data. This type of control is however too weak. For example, for a shock wave contained inside an interval [a,b], the total dissipated entropy  ∫_0^t∫_a^b r dx is cubic in the strength of the shock,  see theorem 8.5.1 of Dafermos <cit.>. Thus, in a regime of increasing number of small shock waves, the entropy does not control the oscillations as measured by sum of all shock wave strength.
  
  In this  paper we explore the possibility of constructing approximate solutions of (<ref>)
  for which  an entropy inequality implies strong compactness, at the price of distorting certain small scale details of the original solutions. More specifically, we will seek approximate, weak  solutions of (<ref>) with  large shocks  propagating with speeds close to the speeds computed from the original system (<ref>), and  discontinuities, for which the change in the entropy is smaller than a certain threshold value ,  are transported with charachteristic velocities. Thus, the approximation involves small scale dispersion effects. 
  
  In this paper, we'll show how this type of approximation can be implemented  for scalar conservation laws in multi-dimensions:
  
      ∂_t ρ+÷_x A(ρ)=0,         (x,t)∈ℝ^d+1_+,
  
  
  Our approach is based on the variational kinetic representation of entropy weak solutions of (<ref>) developed by Brenier <cit.>, Brenier and Corrias <cit.>, Giga and Miyakawa <cit.>, and Lions et al. <cit.>. According to the theory,  an admissible, non-negative solution ρ(x,t) is represented as a moment of an “equilibrium” kinetic density f_eq: 
  
      ρ(x,t)=∫ f_eq(x,t,v) dv,   f_eq(x,t,v)=𝕀_[0,ρ(x,t)](v),
  
  with f_eq solving  a kinetic equation
  
      ∂_t f+A'(v)·_x f=∂_v m,
  
  where m is non-negative Radon measure on ℝ^d+2_+. Conversely, any solution of (<ref>) constrained by condition (<ref>) for some ρ(x,t) defines an admissible weak solution of conservation law in (<ref>), see <cit.>. 
  Moreover, for any strictly convex function η, and a.e. (x,t), f_eq(x,t,v) is the unique minimizer of the problem
  
      min{∫η'(v)f̃(v) dv : f̃(v)∈[0,1], ∫f̃ dv=ρ(x,t)}.
  
  Solutions of (<ref>) can be obtained as limits of solutions of a relaxation problem 
  
      ∂_t f+A'(v)·_x f=h^-1(M_f-f),
  
  where M_f is the minimizer of (<ref>) with ρ=∫ f dv.
  
  
  The approximate solutions, with the properties described above, will be obtained from the same variational kinetic formulation (<ref>) and (<ref>), in which a strictly convex function η is replaced by a continuous, piecewise linear approximate entropy η_. With the new entropy function, the minimization problem admits multiple solutions, with indeterminacy on  small –scales. A particular minimizer M_f will be selected so that L^1 norm of f-M_f can be estimated by the entropy increment ∫η_'(v)(f-M_f) dv.
  
  Our main result, theorem <ref> describes the  kinetic functions  obtained from this kinetic relaxation approach.
  Such kinetic functions  verify equation (<ref>) where, in addition, the  right-hand side  is a signed Radon measure, with the total variation controlled by a single entropy:
   
      ||∂_v m||≤2/∫η_(ρ_0) dx.
   
  Furthermore, we show that moments ρ=∫ f dv, and ϕ=∫ A'(v)f dv, solve the  balance equation
  
      ∂_tρ+÷_xϕ=0,
  
  and ϕ(x,t)=A(ρ(x,t))+O(^2). In particular, if there is a co-dimension one discontinuity of ρ with values ρ^+,ρ^-, (such discontinuities do develop in the solutions), such that |ρ^+-ρ^-|>,  then it propagates with the velocity
  
      σ=A(ρ^+)-A(ρ^-)/ρ^+-ρ^-+O().
  
  The kinetic function f, as well as its moments, depend in the scale parameter . In theorem <ref> we show that in the limit of →0, ρ=ρ^(x,t)  converges to an admissible solutions of (<ref>).
  
  
  In summary, we describe a new type of approximation of scalar conservation laws with properties distinct from the well-known viscosity approximation of Kruzhkov <cit.>, kinetic relaxation approximation of Brenier <cit.> and Giga and Miyakawa <cit.>, or semi-linear relaxation of Katsoulakis and Tzavaras <cit.>. The model approximates the dynamics of large shock waves and controls small scale oscillations, by means of entropy balance.
  
  
  
  
  
  
  
  
  
  
  
  § MAIN RESULT
  
  Let A∈ C^2(ℝ)^d. Without the loss of generality, we will assume that ρ_0 is non-negative and bounded, so that all kinetic functions are defined for the range of the kinetic variable v∈ [0,L], for some L>0. Let >0. Define a piecewise constant function η_ as
  
      η_(v)=k,   v∈[k,(k+1)), k=0..⌈ L/⌉.
  
  η_ approximates the derivative of the quadratic entropy function. Here, for notational convenience, we use η_ to denote the derivative of the entropy function described in the  introduction.
   
  
  Let f_0∈ L^1(ℝ^d×[0,L]) with values {0,1}.   For any >0 there is f∈ L^1(ℝ^d+1_+×[0,L]) with values in [0,1] and m – a non-negative Radon measure on ℝ^d+1_+×ℝ_+ such that ∂_v m is a signed Radon measure on ℝ^d+1_+×[0,L] with the following properties:
  
  
    i.(Kinetic equation) f and m verify (in distributional sense) equation
  
      ∂_t f+A'·_x f=∂_v m.
  
  Moreover,
  
      ||∂_v m||_ℝ^d+1_+×[0,L]≤2/∬η_f_0 dxdv;
  
  
  
    ii.(Optimality) for a.e. (x,t), f is a minimizer of 
  
      min{∫η_(v)f(v) dv :  f(v)∈[0,1], ∫ f dv=ρ(x,t)};
  
  
  
    iii.(Equi-continuity) for a.e. t>0, and any ξ∈ℝ^d,
  
      ∬ |f(x+ξ,t,v)-f(x,t)| dxdv ≤∬|f_0(x+ξ,v)-f_0(x,v)| dxdv.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Kinetic functions from theorem <ref> give rise to the approximate solutions of the conservation law (<ref>), as described in the next theorem.
  
  
   
  For function f from the previous theorem,  moments
  
      ρ(x,t)=∫ f(x,t,v) dv,  ϕ(x,t)=∫ A'(v)f(x,t,v) dv
  
  have the following properties.
  
  
    i. ρ,ϕ∈ L^∞(ℝ^d+1_+) and  verify  (in distributional sense) conservation law
  
      ∂_t ρ+÷_x ϕ=0.
  
  For any ψ∈ C^∞_0(ℝ^d), ∫ρ(x,t)ψ(x) dx is continuous in t and 
  
      lim_t→0+∫ρ(x,t)ψ(x) dx = ∬ f_0(x,v)ψ(x) dvdx;
  
  
  
    ii. for any two pairs of values (ρ(x,t),ϕ_i(x,t)) and (ρ(y,τ),ϕ_i(y,τ)), such that  |ρ(x,t)-ρ(y,t)|≥ c_0, it holds:
  
      ϕ_i(x,t)-ϕ_i(y,τ)/ρ(x,t)-ρ(y,τ)=A_i(ρ(x,t))-A_i(ρ(y,τ))/ρ(x,t)-ρ(y,τ)+O(),   i=1..d;
  
  
  
    iii.(Limit to Kruzhkov's solution) As a function of , ρ converges the unique, entropy solution of the conservation law (<ref>), when →0.
  
  
  
  
  
  
  The existence of an approximating pair (ρ,ϕ) with properties i. and ii. can also be established via a kinetic averaging lemma of Gérard <cit.>, using the estimate (<ref>), provided that the following  non-degeneracy condition holds:
  
      ∀ σ∈𝕊^d-1,  ∀ξ∈ℝ,   meas{ v∈ [0,L] :  A'(v)·σ=ξ}=0,
  
  where 𝕊^d-1 is the unit sphere in ℝ^d. This approach is independent of the particular structure of the right-hand side of (<ref>) as ∂_v m, or  L^1–contraction property (<ref>) that we use in the proof, both being characteristic properties of scalar conservation laws. 
  
  
  
  
  
   §.§ Proof of theorem <ref>
  
  
  For a non-negative constant ρ∈[0,L] consider a minimization problem
  
      min{∫η_(v)f(v) dv :  f(v)∈[0,1], ∫ f dv=ρ}.
  
  
  
  In the next lemma 𝕀_A(v) stands for a characteristic function of set A.
  
   Let n=⌊ρ/⌋.
  The minimum in problem (<ref>) equals
  
      {[ ∑_k=0^n-1k+  n(ρ-n),                n≥ 1,;                   0,                 n=0. ].
  
  It is achieved on minimizers
  
      f_min(v)=𝕀_[0,n](v)+f̃(v),
  
  where f̃ is an arbitrary function verifying conditions:
  
      f̃(v)∈ [0,1],  ∀ v∈[0,L];  f̃⊂ [n,(n+1)];
         ∫f̃ dv=ρ - n.
  
  
  
  η_(v) is a non-decreasing function. To minimize  functional ∫η_ f dv one needs to pick f that has all its mass as close to v=0 as possible, and is less than or equal 1. This shows the first statement. On interval [n,(n+1)],  a minimizer f_min can be arbitrarily re-arranged without changing the value of its η_ moment.   This leads to the second part of the lemma.
  
  
  Given a kinetic density f we  select a particular minimizer of (<ref>) with ρ=∫ f dv in the following way.
  If ∫_(n+1)^Lf dv > n - ∫_0^nf dv,
  we set 
  
      M_f(v) = 𝕀_[0,n+v_0](v) + f(v)𝕀_(n+v_0,(n+1))(v),
  
  where v_0∈(0,) is determined by the relation ∫ M_f dv=∫ f dv. It is the smallest number such that 
  
      ∫_0^n+v_01-f dv=∫_(n+1)^L f dv.
  
  
  If ∫_(n+1)^Lf dv ≥ n - ∫_0^nf dv,
  we set  
  
      M_f(v) = 𝕀_[0,n](v) + f(v)𝕀_(n,n+v_0)(v),
  
  where  v_0∈(0,) is uniquely determined 
  as the smallest number such that 
  
      ∫_0^n1-f dv=∫_n+v_0^L f dv.
  
  
  
  This minimizer can be thought of as a rearrangement of mass f obtained by shifting its pieces by to the locations with smaller values of η_e(v). 
  
  
  The key properties of the minimizer f_min are listed in the next lemma.
  
  
   Let f be any function with values in [0,1]  and supported on [0,L]. For M_f, defined above
  
      ∫ |f-M_f| dv ≤2/∫η_(v)(f-M_f) dv.
  
  For any non-decreasing function  η, 
  
      ∫η(v) (f(v)-M_f(v)) dv≥ 0.
  
  For any two functions f_1, f_2  with values in {0,1} and supported on [0,L],
  
      ∫ |M_f_1 - M_f_2| dv≤∫ |f_1 - f_2| dv,
  
  where M_f_1,M_f_2 are the corresponding minimizers.
  
   
  Let n be as in the previous lemma. Consider case (<ref>)
  
      ∫ |f-M_f| dv =∫_0^n+v_0 1-f dv+∫_(n+1)^L f dv
      =2∫_(n+1)^L f dv
      ≤2/∫η_(v)(f-M_f) dv,
  
  where the last inequality holds since all mass of f on interval [(n+1),L] has been removed from that interval.
  Similarly, in case (<ref>)
  
      ∫ |f-M_f| dv =∫_0^n 1-f dv+∫_n+v_0^L f dv
      =2∫_n+v_0^L f dv
      ≤2/∫η_(v)(f-M_f) dv.
  
  
  For  a non-decreasing function η, (<ref>) follows from the definition of M_f. 
  
  
  To prove (<ref>) it suffices to show that
  
      ∫ f_1 f_2 dv≤∫ M_f_1M_f_2 dv,
  
  since functions take only values 0 or 1. Let n_1,v_1,0 and n_2,v_2,0 be the corresponding values of n and v_0 from (<ref>), (<ref>) for functions f_1 and f_2.
  
  Consider the case n_1> n_2 first. Here
  
      ∫ M_f_1M_f_2 dv =∫_0^(n_2+1)M_f_2 dv=∫ f_2 dv≥∫ f_1 f_2 dv.
  
  Next, consider the case n_1=n_2 (=n).
  Suppose that representation (<ref>) applies to both functions f_1,f_2, and assume  v_1,0≥ v_2,0.
  Then,
  
      ∫ M_f_1M_f_2 dv ≥∫_n+v_1,0^(n+1)f_1f_2 dv
      +∫_0^n+v_1,0f_2 dv +∫_(n+1)^L f_2 dv≥∫ f_1 f_2 dv.
  
  Suppose that representation (<ref>) applies to both functions f_1,f_2, and assume  v_1,0≥ v_2,0.
  Then,
  
      ∫ M_f_1M_f_2 dv ≥∫_0^n f_2 dv+∫_n+v_2,0^Lf_2 dv+∫_n^n+v_2,0f_1f_2 dv≥∫ f_1f_2 dv.
  
  Suppose that (<ref>) applies to function f_1 and (<ref>) to f_2. If v_1,0≥ v_2,0 then
  
      ∫ M_f_1M_f_2 dv ≥∫_0^n+v_2,0 f_1 dv+∫_n+v_1,0^Lf_1 dv+∫_n+v_2,0^n+v_1,0f_1f_2 dv≥∫ f_1f_2 dv.
  
  If v_1,0<v_2,0 then
  
      ∫ M_f_1M_f_2 dv ≥∫_0^L f_1 dv≥∫ f_1f_2 dv.
  
  The contraction property (<ref>) is proved now.
  
  
  
  
  
  
  
  Now we consider a discrete-time approximation, with time step h>0 and t_n=nh, n=0,1,2... Given f_n-1(x,v) the next period kinetic function 
  
      f_n(x,v) = M_f̂_̂n̂,  f̂_̂n̂(x,v) = f_n-1(x-A'(v)h),
  
  with f_0 being the initial data.
  A continuous time approximate is defined as
  
      f^h(x,v,t)={[ f_n-1(x-A'(v)(t-nh)),        t∈[(n-1)h,nh),;             f_n(x,v),                 t=nh. ].
  
  
  It can be easily seen that in dimension one, if initial data f_0 is such that f_0(x,v)=1, for 0≤ v≤ k and f_0(x,v)=0 for v>((k+1) then f_n is evolved by simple translation with kinetic velocities v, leading to dispersion effect. On the other hand if initial data, for example, has a form
  	
      f_0(x,v) = {[ 𝕀_[0,v_1](v),          x<0,; 𝕀_[0,v_2](v),          x>0, ].
   
  with v_1-v_2> and A(v)=v (corresponding to Burger's equation) then f_n evolves as a classical shock wave in a discrete-time approximation. 	
  
  
  The following properties of f^h follow easily from its definition and properties established in lemma <ref>.
  
  
  It holds:
  
  
    i. for any (x,v,t), f^h∈{0,1};
  
    ii. for any (x,t), supp f^h⊂ [0,L];
  
    iii. for any t>0,
  
      ∬ f^h(x,v,t) dvdx≤f_0(x,v) dvdx;
  
  
      ∬η_(v)f^h(x,v,t) dvdx≤η_(v)f_0(x,v) dvdx;
  
  
    iv. f^h is a weak solution of the equation
  
      ∂_tf^h+A'(v)·_xf^h=R^h,
  
  where
  
      R^h=∑_n=1^∞δ(t-nh)(f_n(x,v) - f_n-1(x-A'(v)h));
  
  
    v. for any t>0 and any ξ∈ℝ^d,
  
      ∬ |f^h(x+ξ,t,v)-f^h(x,t)| dxdv ≤∬|f_0(x+ξ,v)-f_0(x,v)| dxdv.
  
  
  
  Next, we estimate the interaction term R^h in equation (<ref>)
  For any t>0,
  
      ∬ R^h dvdx≤∑_n=1^∞δ(t-nh)∬|(f_n(x,v) - f_n-1(x-A'(v)h))| dxdv;
  
  and 
  
      ∫_0^∞∬ |R^h| dvdxdt≤2/∬η_(v)f_0(x,v) dvdx.
  
  
  
  The first inequality is obvious. Using equation (<ref>) we find that
  
      ∑_n=1^∞∬η_(v)(f_n(x,v)-f_n-1(x-A'(v)h,v)) dxdv≤∬η_(v)f_0(x,v) dxdv.
  
  Since f_n=M_f_n-1(x-A'(v)h,v), using inequality (<ref>) we get
  
      ∑_n=1^∞∬ |f_n(x,v)-f_n-1(x-A'(v)h,v))|,dxdv≤2/∬η_(v)f_0(x,v) dxdv,
   
  from which the second inequality of the lemma follows.
  
  
  With the information from the last two lemma, we consider compactness properties of f^h as h→0. There is f with a.e. values in [0,1] and a signed Radon measure m̃ such that on a suitable subsequence h_k→0,
  
      f^h_k→ f   L^∞(ℝ^d+1_+×[0,L]),
      
       R^h_k→m̃  ℳ_loc(ℝ^d+1_+×[0,L]),
  
   for a.e. t>0,
  
      ∬ f(x,v,t) dvdx≤∬ f_0(x,v) dvdx,
      ∬η_(v)f(x,v,t) dvdx≤∬η_(v)f_0(x,v) dvdx,
   
  and inequalities (<ref>) and (<ref>) hold.
  
  Inequality (<ref>) implies that ⟨m̃,η(v)ψ(x,t)⟩≤ 0 for any continuously non-decreasing function η, and any non-negative ψ∈ C^∞_0(ℝ^d+1_+). Thus, m̃=∂_v m for a non-negative Radon measure. 
  
  
  To complete the proof of theorem <ref> it remain to establish
  (<ref>). For that we first show that v–moments of f^h are compact in L^p norms.
  
  
  Let ω(v) be a measurable, bounded function on [0,L]. Then, the set of moments
  
      {∫ω(v)f^h(x,v,t) dv}   L^p_loc(ℝ^d+1_+), p∈[0,+∞).
  	
  
  
   Denote by ρ^h_ω=∫ω(v)f^h(x,v,t) dv.
  ρ^h_ω is bounded in L^∞(ℝ^d+1_+). It follow from part v. of lemma <ref> that for any ξ∈ℝ^d, and any T>0, and p∈[1,+∞),
  
      ρ^h_ω(x+ξ,t)-ρ^h_ω(x,t)_L^∞((0,T); L^p(ℝ^d))→0,   |ξ|→0,
   
  uniformly in h. It follow from equation (<ref>) that for any T>0 and p∈[0,+∞), 
  
      {∂_t ρ^h_ω}  ℳ((0,T);L^p(ℝ^d)) + L^∞((0,T); W^-1,p_loc(ℝ^d)).
  
  Under these conditions, compactness lemma 5.1 of Lions <cit.> ensures that on a suitable sequence of values of h→ 0, (ρ^h_ω)^2→ (ρ_ω)^2 in distributional sense, where ρ_ω is a limiting point of ρ^h_ω in *-weak topology of L^∞(ℝ^d+1_+). This implies the statement of the lemma. 
  
  
  A little bit more can be said about moments ρ^h=∫ f^h(x,v,t) dv. Indeed, 
  
      {∂_tρ^h }  L^∞((0,T); W^-1,p_loc(ℝ^d)), p∈[1,∞).
  
  Thus, ρ^h converges  for a limiting point ρ, in C([0,T]; W^-1,p_loc(ℝ^d)). This shows, in  particular, that  ρ(x,0)=∫ f_0(x,v) dv.
  
  
  We consider the moments of f^h from the set ω∈{1,η_(v),A_1(v),..,A_d(v)} and select a sequence h=h_k→0 on which f^h and ρ^h_ω converge in the topologies described above to their limiting values.
  
  
  
   To finish the proof of theorem <ref> it remain to establish (<ref>).
   
   
   Consider a piece-wise constant in time interpolation of functions f_n(x,v):
  
      f̃^h(x,t,v)=
      f_n-1(x,v),    t∈[(n-1)h,nh),   n=1,2,3...
  
  For any  ψ∈ C^∞_0(ℝ^d+1_+×[0,L]), using the definition of function f^h we find that
  
      ∭ψ(f^h-f̃^h) dvdxdt
      =∑_n=1^∞∫_(n-1)h^nh(ψ(x,t,v)-ψ(x+A'(v)(t-(n-1)h),t,v)f_n-1 dvdxdt=O(h).
  
  Thus  f̃^h converges the same f in *-weak topology of L^∞(ℝ^d+1_+×[0,L]). Furthermore, there is another sequence f̂^h constructed by taking suitable convex linear combinations of a finite number of elements of {f̃^h} that converges to f in L^p_loc() and a.e. (x,t,v).
  
  Let ρ̂^h=∫f̂^h dv. For each (x,t),  f̂^h(x,t,v) is a minimizer of the problem (<ref>) with ρ=ρ̂^h(x,t). Since this problem depends continuously on the value of the constraint ρ̂^h and the latter converges a.e. (x,t) to ρ(x,t), then the limit of the minimizers f̃^h is a minimizer corresponding to ρ.
  
  
  
  
  
  
   §.§ Proof of theorem <ref>
  
  
  Part i. of the theorem <ref> was established in proving theorem <ref>. Part ii. follows from from (<ref>) and lemma <ref>. Indeed, let ρ, and ϕ be given by (<ref>), and (x,t) is such that  f(x,t,·) is the minimizer of (<ref>). Let n and f̃ be as in lemma <ref>. We can write for any i=1..d,
  
      ϕ_i(x,t)=∫ A_i'(v)f(x,t,v) dv=
      A_i(ρ(x,t))+∫_n^(n+1) A_i'(v)(f̃-𝕀_[0,ρ](v)) dv
      =A_i(ρ(x,t))+∫_n^(n+1)( A_i'(v)-A_i'(n))(f̃-𝕀_[0,ρ](v)) dv
      =A_i(ρ(x,t))+O(^2),
  
  which establishes (<ref>).
  
  To show part iii. of the theorem we consider the sequence of kinetic functions f^ and their moments ρ^e=∫ f^ dv, ϕ^_i=∫ A_i'(v)f^ dv from theorem <ref> in the limit →0. Given the uniform bounds on the sequence  f^, continuity estimate (<ref>) and equation one can repeat the arguments of the proof of theorem <ref> to establish that v–moments of f^ are pre-compact in L^p_loc(ℝ^d+1_+) and (one a subsequence) converge to a pair (ρ,ϕ) – a solution of (<ref>), while f^ itself converges weakly to a function that f that verifies the kinetic equation (<ref>) (but not (<ref>)) and which a.e. (x,t) is a minimizer of problem (<ref>) with function η(v)=v, in place of η_. This new problem
  
      min{∫η(v)f(v) dv :  f(v)∈[0,1], ∫ f dv=ρ(x,t)}
  
  has a unique minimizer in the form f(x,v,t)=𝕀_[0,ρ(x,t)](v).
  Thus, ϕ(x,t) =A(ρ(x,t)) a.e. (x,t) and ρ is a unique entropy solution of the conservation law (<ref>). The uniqueness implies that the sequence ρ^ converges to ρ in the limit of →0.
  
  
  
  
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