On Nonlinear Stochastic Balance Laws

arXiv:1111.5217v1 [math.AP] 22 Nov 2011

ON NONLINEAR STOCHASTIC BALANCE LAWS GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN Abstract. We are concerned with multidimensional stochastic balance laws. We identify a class of nonlinear balance laws for which uniform spatial BV bounds for vanishing viscosity approximations can be achieved. Moreover, we establish temporal equicontinuity in L1 of the approximations, uniformly in the viscosity coefficient. Using these estimates, we supply a multidimensional existence theory of stochastic entropy solutions. In addition, we establish an error estimate for the stochastic viscosity method, as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions. Various further generalizations of the results are discussed.

1. Introduction We are concerned with the well-posedness and continuous dependence estimates for the stochastic balance laws ∂t u(t, x) + ∇ · f (u(t, x)) = σ(u(t, x)) ∂t W (t),

with initial data:

x ∈ Rd , t > 0,

(1.1)

u(0, x) = u0 (x), x ∈ Rd . (1.2) We denote by ∇ and ∆ the spatial gradient and Laplacian, respectively. Equation (1.1) is a conservation law perturbed by a random force driven by a Brownian motion W (t) = W (t, ω), ω ∈ Ω, over a stochastic basis (Ω, F , {Ft }t≥0 , P ), where P is a probability measure, F is a σ-algebra, and {Ft }t≥0 is a right-continuous filtration on (Ω, F ) such that F0 contains all the P –negligible subsets. The initial function u0 (x) is assumed to be a random variable satisfying î ó E ku0 kpLp (Rd ) + |u0 |BV (Rd ) < ∞, p = 1, 2, · · · . (1.3) Regarding the flux f = (f1 , · · · , fd ) : R → Rd , we assume that fi ∈ C 2 (R), i = 1, . . . , d, and that each fi has at most polynomial growth in u, i.e., |fi (u)| ≤ C (1 + |u|r )

for some finite integer r ≥ 0.

(1.4)

In this paper we focus mainly on the class of noise functions σ for which there exists a constant C > 0 such that σ(0) = 0,

|σ(u) − σ(v)| ≤ C|u − v| ∀u, v ∈ R.

(1.5)

This can be generalized to wider classes for different results in terms of existence, stability, and continuous dependence, respectively; see Section 6 for more details. Date: October 8, 2011. 2000 Mathematics Subject Classification. Primary: 35R60, 60H15, 35F25, 35L60; Secondary: 35L03, 60G15. Key words and phrases. Stochastic balance law, vanishing viscosity method, entropy solution, existence, uniqueness, stability, BV estimates, error estimates, continuous dependence. 1

2

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

One reason for requiring σ(0) = 0 is that it follows from the L1 –contraction principle that E[ku(t, ·)kL1 (Rd ) ] is finite. Similarly, the Lipschitz continuity of σ(u) is required for the existence and uniform Lp estimates of solutions. Stochastic partial differential equations arise in a number of problems concerning random-phenomena occurring in biology, physics, engineering, and economics. In recent years, there has been an increased interest in studying the effect of stochastic forcing on solutions of nonlinear stochastic partial differential equations. Of specific interest is the effect of noise on discontinuous waves, since these are often the relevant solutions; an issue of particular importance concerns the well-posedness (existence, uniqueness, and stability) of discontinuous solutions. The fundamental fluid dynamics models are based on the compressible NavierStokes equations and Euler equations. However, abundant experimental observations suggest that the chaotic nature of many high-velocity fluid dynamics phenomena calls for their stochastic formulation. Indeed, in these flows with large Reynolds numbers, microscopic perturbations get amplified to macroscopic scales giving rise to unsteady flow patterns that deviate significantly from those predicted by the classical Navier-Stokes/Euler models, and more viable models seem to be the stochastic Euler or Navier-Stokes equations. In the present paper we are interested in nonlinear hyperbolic equations with stochastic forcing, so-called stochastic balance laws. These balance laws can be viewed as a simple caricature of the stochastic Euler equations. Some efforts have been made in the analysis of nonlinear stochastic balance laws. When σ ≡ 0, (1.1) becomes a nonlinear conservation law for which the maximum principle holds. A satisfactory well-posedness theory is now available (cf. [5]). In [10], a one-dimensional stochastic balance law was analyzed for u0 in L∞ and compactly supported σ = σ(u), which ensures an L∞ bound. A splitting method was used to construct approximate solutions, and it was shown that a subsequence of these approximations converges to a (possible non-unique) weak solution. For general σ, the maximum principle is no longer valid. Indeed, even for L∞ initial data u0 , the solution is no longer in L∞ generically. For σ = σ(t, x) in Ct (Wx1,∞ ) and with compact support in x, Kim [12] established the existence and uniqueness of entropy solutions in the one-dimensional case; see also [22]. For more general σ = σ(x, u) depending on u and for multidimensional equations in the Lp framework, the uniqueness of strong stochastic entropy solutions was first established in Feng-Nualart [9], but the existence result was restricted to one dimension; see the recent paper Debussche-Vovelle [6] for multidimensional results via a kinetic formulation1. For the Lp theory of deterministic conservations laws, see [21]. One of our main observations is that uniform spatial BV bounds are preserved for stochastic balance laws with noise functions σ(u) satisfying (1.5). This yields the existence of strong stochastic entropy solutions in Lp ∩ BV , as well as in Lp , for multidimensional balance laws (1.1). Furthermore, we develop a “continuous dependence” theory for stochastic entropy solutions in BV , which can be used, for example, to derive an error estimate for the vanishing viscosity method. Whenever σ = σ(x, u) has a dependency on the spatial position x, BV estimates are no longer available, but we show that the continuous dependence framework can be used to derive local fractional BV estimates, which in turn can be used, as before via a temporal equicontinuity estimate, to establish a multidimensional existence result. 1We became aware of this paper after our main results were obtained.

STOCHASTIC BALANCE LAWS

3

Besides providing an existence result in a multidimensional context by standard methods, one reason for singling out the class of nonlinear balance laws defined by (1.5) is that it makes a natural test bed for numerical analysis, without having to account for all the added technical complications in a pure Lp framework. Moreover, by assuming σ(a) = σ(b) = 0 for some constants a < b, one ensures that the solution remains bounded between a and b if the initial function u0 does so. Consequently, it is possible to identify a class of stochastic balance laws for which Lp ∩ BV , or even L∞ ∩ BV , supplies a relevant and technically simple functional setting, tailored for the construction and analysis of numerical methods. For other related results, we refer to Sinai [19] and E-Khanin-Mazel-Sinai [7] for the existence, uniqueness, and weak convergence of invariant measures for the one-dimensional Burgers equation with stochastic forcing which is periodic in x, as well as the structure and regularity properties of the solutions that live on the support of this measure. We also refer to Lions-Souganidis [16] for Hamilton-Jacobi equations with stochastic forcing and the so-called “stochastic” viscosity solutions. We employ the vanishing viscosity method to establish the existence of stochastic entropy solutions. To this end, consider the stochastic viscous conservation law ∂t uε (t, x) + ∇ · f (uε (t, x)) = σ(uε (t, x))∂t W (t) + ε∆uε (t, x)

(1.6)

for any fixed ε > 0, with initial data uε (0, x) = uε0 (x), where

uε0 (x)

x ∈ Rd ,

(1.7)

is a standard mollifying smooth approximation to u0 (x) with ïZ ò ïZ ò p E |uε0 (x)| dx ≤ E |u0 (x)|p dx Rd

Rd

d

and, if u0 ∈ BV (R ), ïZ E R

Rd

ò ïZ |∇uε0 (x)| dx ≤ E

Rd

ò |∇u0 (x)| dx .

|∇2 uε0 |

|∇2 uε0 (x)| dx Rd



In addition, E is integrable for each fixed ε. < ∞, i.e., With regard to the viscous equation (1.6), we should replace (f , σ) by appropriate smooth approximations (f ε , σ ε ). However, mainly to ease the presentation throughout this paper, we will not do that but instead simply assume that (f , σ) are sufficiently smooth (cf. [9]) in order to ensure the validity of our calculations. At times, we will do the same with the initial data. The existence of global smooth solutions to (1.6)–(1.7) is established in [9], along with the following uniform estimates for p ≥ 1 and T > 0: ô ñ Z T î ó p ε 2 ε k∇u (t, ·)kL2 (Rd ) dt < ∞. (1.8) sup sup E ku (t, ·)kLp (Rd ) + sup E ε ε>0 0≤t≤T

ε>0

0

The solution satisfies uε (t, x) =

Z

Gε (t, x − y)u0 (y) dy

Rd

−

Z tZ 0

Rd

+

Z tZ 0

Gε (t − s, x − y)∇ · f (uε (t, y)) dy ds Rd

Gε (t − s, x − y)σ(uε (s, y)) dy dW (s),

(1.9)

4

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

where Gε (t, x) is the heat kernel: Gε (t, x) =

|x|2 1 − 4εt , e (4πεt)d/2

t > 0.

Using (1.3) and (1.8)–(1.9), it follows that, for each fixed ε > 0,   for any finite T > 0, E k(∇, ∆)uε kL1 ((0,T )×Rd ) < ∞ ε

(1.10)

2 ε

that is, ∇u and ∇ u are integrable for each fixed ε > 0. With different methods, we will later prove an ε-uniform spatial BV estimate. The remaining part of this paper is organized as follows: In Section 2, we prove the uniform spatial BV bound for stochastic viscous solutions uε (t, x). Based on the BV bound, we establish the equicontinuity of uε (t, x) in t > 0, uniformly in the viscosity coefficient ε > 0, in Section 3. With these uniform estimates, we establish the existence of stochastic entropy solutions in Lp ∩ BV , as the vanishing viscosity limits for problem (1.6)–(1.7) with initial data in Lp ∩ BV , in Section 4. Combining this existence result with the L1 -stability theory in Feng-Nualart [9] leads to the well-posedness in Lp for problem (1.1)–(1.2). We further establish estimates for the “continuous dependence on the nonlinearities” for BV stochastic entropy solutions in Section 5, which also leads to an error estimate for (1.6)–(1.7). Various further generalizations of the results are discussed in Section 6. 2. Uniform spatial BV –estimates As indicated in Section 1, we have known the regularity and the uniform Lp – estimate (1.8) (p ≥ 1) for the viscous solutions uε (t, x) of (1.6)–(1.7). In this section, we establish the uniform L1 -estimate for ∇uε , that is, the uniform BV -estimate of uε (t, x) in the spatial variables x. Before we do that, let us indicate why BV estimates do not seem to be available when the noise coefficient function σ = σ(x, u) depends on the spatial position x, even if that dependence is C ∞ (see Section 6 for fractional BV estimates). To this end, it suffices to consider the simple stochastic differential equation: du = σ(x, u) dW (t),

u(0) = u0 (x),

x ∈ R,

where we have dropped nonlinear transport effects and restricted to one spatial dimension. The spatial derivative v = ∂x u satisfies dv = (σu (x, u)v + σx (x, u)) dW (t). Let η be a C 2 –function. By Ito’s formula,
2
1 dη(v) = η ′ (v) σu (x, u)v + σx (x, u) dW (t) + η ′′ (v) σu (x, u)v + σx (x, u) dt. 2 Integrating in x and taking expectations, it follows that ò ò ïZ ïZ η(v(0)) dx η(v(t)) dx = E E ñZ t Z ô
2 1 ′′ +E η (v) σu (x, u)v + σx (x, u) dx ds . 2 0

Modulo an approximation argument, we can take η(·) as |·|. Unless σx ≡ 0, the second term on the right-hand side does not seem to be controllable (this term vanishes when σx ≡ 0).

STOCHASTIC BALANCE LAWS

5

Let us now continue with the derivation of the BV estimate for (1.6). We will need a C 2 –approximation of the Kruzkov entropy. Let η¯ : R → R be a C 2 –function satisfying η¯(0) = 0, and

η¯(−r) = η¯(r),

η¯′ (−r) = −¯ η ′ (r),

η¯′′ ≥ 0,

  when r < −1, −1, ′ η¯ (r) = ∈ [−1, 1], when |r| ≤ 1,   +1, when r > 1.

(2.2)

For any ρ > 0, define the function ηδ : R → R by r ηρ (r) = ρ¯ η ( ). ρ Then ′′ M2 ηρ (r) ≤ 1|r| 0, there exists a constant C = Cp such that, if Mt is a continuous martingale and t a stopping time, then ñ ô î ó p/2 p E sup |Ms | ≤ Cp E hM it , s≤t

where hM it is the quadratic variation of Mt .

Theorem 2.1 (Spatial BV estimate). Suppose that (1.3)–(1.5) hold. Let uε (t, x) be the solution of (1.6)–(1.7). Then, for t > 0, ïZ ò ïZ ò ïZ ò E |∇uε (t, x)| dx ≤ E |∇uε0 (x)| dx ≤ E |∇u0 (x)| dx . Rd

Rd

Rd

Proof. Taking the derivative of (1.6) with respect to xi , 1 ≤ i ≤ d, we obtain
∂t (uεxi ) + ∇ · f ′ (uε (t, x))uεxi = σ ′ (uε (t, x))uεxi ∂t W (t) + ε∆(uεxi ). Applying Ito’s formula to ηρ (uεxi ) yields

∂t ηρ (uεxi ) = ηρ′ (uεxi )σ ′ (uε )uεxi ∂t W (t)

We observe that


+ ηρ′ (uεxi ) ε∆uεxi − ∇ · (f ′ (uε )uεxi )
2 1 + ηρ′′ (uεxi ) σ ′ (uε )uεxi . 2

Ä 2 ä εηρ′ (uεxi )∆(uεxi ) = ε ∇ · (ηρ′ (uεxi )∇uεxi ) − ηρ′′ (uεxi ) ∇uεxi
= ε ∆ηρ (uεxi ) − ηρ′′ (uεxi )|∇uεxi |2 ≤ ε∆ηρ (uεxi ),

(2.6)

(2.7)

by using the convexity of ηρ , and interpreting ∆ηρ (uεxi ) in the distributional sense. Here we have used that ∇uεxi , 1 ≤ i ≤ d, are integrable (cf. (1.10)) so that they vanish at infinity, which leads to the vanishing boundary terms in (2.7).

6

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

Integrating (2.6) with respect to x, using (1.10) and (2.7), and noting that Z Z t η ′ (uεxi )σ ′ (uε )uεxi dW (s)dx Rd

0

is a martingale, we arrive at ïZ ò ïZ ò E ηρ (uεxi (t, x)) dx − E ηρ (uεxi (0, x)) dx Rd Rd " Z Z t

≤E −

Rd

0

+

1 2

ηρ′ (uεxi )∇ · (f ′ (uε )uεxi ) dx ds

Z tZ 0

ηρ′′ (uεxi ) σ ′ (uε )uεxi


2

dx ds .

Now we send ρ → 0 in (2.8). By the dominated convergence theorem, ò ïZ ε ux (t, x) dx E i

Rd

≤E

ïZ

Rd

ñZ t Z ò ε ux (0, x) dx − lim E i ρ→0

+ lim

ρ→0

:= E

(2.8)

#

ïZ

Rd

1 E 2

ñZ t Z 0

Rd

0

Rd

ηρ′ (uεxi )∇

ηρ′′ (uεxi ) σ ′ (uε )uεxi

ò ε ux (0, x) dx + I1 + I2 . i


2

dx ds

′

· (f (u

ε

)uεxi ) dx ds

ô

ô

For the I1 term,

ñZ Z ô t
ε ′ ε ′ ε ∇ · f (u )ηρ (uxi )uxi dx ds |I1 | = lim E ρ→0 d 0 R ñZ Z ô t ′ ε ε ε ′′ ε + lim E ηρ (uxi )uxi ∇uxi · f (u ) dx ds ρ→0 d 0 R ñZ t Z ô ε 1 u χ[−ρ,ρ] (uε ) ∇uε |f ′ (uε )| dx ds . ≤ C lim E xi xi xi ρ→0 ρ 0 Rd

Notice that ε 1 u χ[−ρ,ρ] (uε ) → 0 for a.e. (t, x) almost surely as ρ → 0, xi xi ρ and ε 1 ux χ[−ρ,ρ] (uεx ) ∇uεx |f ′ (uε )| i i i ρ ä Ä 2 2(r−1) , ≤ C ∇uε + |uε | xi

where the right-side term of the inequality is integrable and independent of ρ > 0. Then the dominated convergence theorem implies that |I1 | = 0. Next we consider I2 . By condition (1.5) and estimate (2.4), we have ′′ ε ηρ (ux )(σ ′ (uε )uεx )2 = ηρ′′ (uεx ) |uεx 2 (σ ′ (uε ))2 i i i i ≤ C uεx 1{|uε | 0 and ε u x 1 ε for a.e. (t, x) almost surely as ρ → 0, i ||u | 0. Theorem 3.1 (Temporal L1 –continuity). Suppose that (1.3)–(1.5) hold. Let uε (t, x) be the solution of (1.6)–(1.7). Let D ⊂ Rd be a bounded domain in Rd and T > 0 finite. Then, for any small ∆t > 0, there exists a constant C > 0 independent of ∆t such that

E

ñZ

T −∆t

Z

0

ε

D

ε

|u (t + ∆t, x) − u (t, x)| dxdt

≤ C(∆t)1/3 → 0

ô

as ∆t → 0.

(3.1)

Proof. Fix ∆t > 0. For t ∈ [0, T − ∆t], set wε (t, ·) := uε (t + ∆t, ·) − uε (t, ·). Then, for any ϕ ∈ L∞ (0, T ; C0∞ (D)), we have Z wε (t, x)ϕ(t, x) dx D

= =

t+∆t

Z Z

t D t+∆t Z

Z

t

−ε

Z

 ∂s uε (s, x)ds ϕ(t, x) dx

f (uε (s, x)) · ∇ϕ(t, x) dx ds

D t+∆t

t

+

Z

t

Z

(3.2)

∇uε (s, x) · ∇ϕ(t, x) dx ds

D t+∆t Z

σ(uε (s, x))ϕ(t, x) dx dW (s).

D

For each t ∈ [0, T − ∆t], take δ > 0, set D−δ := {x ∈ D : dist(x, ∂D) ≥ δ}, and denote by χD−δ (·) its characteristic function. Let J ∈ Cc∞ (Rd ) be the standard mollifier defined by Ä ä ® C exp |x|21−1 if |x| < 1, J(x) = (3.3) 0 if |x| ≥ 1, R where the constant C > 0 is chosen so that Rd J(x)dx = 1. For each δ > 0, we take Z ϕ := ϕδ (t, x) = δ −d

Rd

J( x−y δ ) sgn (w(t, y)) χD−δ (y) dy

in (3.2). It is clear that kϕδ kL∞ (D) + δ k∇ϕδ kL∞ (D) ≤ C, uniformly in t, for some constant C > 0 independent of δ > 0.

8

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

Integrating (3.2) in t from 0 to T − ∆t yields Z

T −∆t

0

Z

=

0

−

Z

|wε (t, x)| dxdt

D T −∆t Z t+∆t Z

Z

0

+

Z

+

Z

t D T −∆t Z t+∆t Z t

T −∆t

0 T −∆t

0

:=

4 X

ÇZ

t

Z

f (uε (s, x)) · ∇ϕδ (t, x) dx ds dt

D t+∆t 

ε∇uε (s, x) · ∇ϕδ (t, x) dx ds dt Z

å  σ(u (s, x))ϕδ (t, x) dx dW (s) dt ε

D


wε (t, x) wε (t, x) − ϕδ (t, x) dxdt

D

Ijδ .

j=1

We examine these parts separately. Thanks to the polynomial growth of f and (1.8),   ∆t ∆t kf kL1 (D×(0,T )) ≤ C(T, D) . E I1δ ≤ C δ δ For the term I2δ , we have

  E I2δ ≤ C

E

"Z

0

T −∆t Z

Ç hZ × E

Ç hZ ≤ C∆t E

0

≤ C(T, D)

∆t , δ

D

ÇZ

T −∆t 0

T −∆t Z

t+∆t

√ ε|∇uε (s, x)| ds

t

dx dt

1

Z

D

2

ε|∇ϕδ | dx ds 1

2

D

å2

|∇ϕδ | dx ds

iå 2

iå 2

where the second inequality follows from the energy estimate (1.8):

ñ Z sup E ε ε>0

0

T

k∇u

ε

(t, x)k2L2 (Rd )

ô

dt < ∞.

#! 12

STOCHASTIC BALANCE LAWS

9

For the term I3δ , by the Burkholder-Davis-Gundy inequality applied to the marR t+∆t R
tingale 0 ≤ ∆t 7→ t σ(uε (s, x))ϕδ (t, x) dx dW (s), we have D

Z   E I3δ ≤ C

T −∆t

Ç Z E

t+∆t

T −∆t

Ç ñZ ≤C E

∆t Z

0

0

Z

t

0

Ç ñZ √ ≤ C ∆t E Ç ñZ √ ≤ C ∆t E √ ≤ C ∆t,

0

Z

D

Z


2 σ(u (s + t, x) dx dt ds


2 σ(u (t, x) dx dt ε

D

å 12  ds  dt


2 σ(u (s, x)ϕδ (t, x) dx ds dt ε

D

ã2

ε

ε

D

T 0

t+∆t Z

T −∆t Z

T

σ(uε (s, x))ϕδ (t, x) dx

D

t

0

Ç ñZ ≤C E

ÅZ

2

|u (t, x)| dx dt

ôå 21

ôå 21

ôå 12

ôå 12

  where we have used that supε>0 E kuε (t)k22 < ∞, uniformly in t > 0. This L2 –bound also implies

E

ñZ

0

T

Z

ε

D\D−2δ

≤C E

√ ≤ C δ.



|u (t, x)| dx dt

kuε k22


21

Ç

E



Z

T 0

ô Z

D\D−2δ

dx dt



å 12

10

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

Hence,   E I4δ ñZ ≤ 2E

T −∆t

Z

0

+E √

"Z

0

≤C δ "Z

+E

√

0

D−2δ

T −∆t Z

≤C δ + CE √

D\D−2δ

T −∆t Z

ñZ

T −∆t 0

≤ C δ + CE



Rd

Z

J(z)

√ 1 ≤ Cδ 2 + 4δ ≤ C δ,

Rd

δ −d J( x−y δ ) |w(t, x)|

D−2δ

ñZ

ô

Z |w(t, x)| − w(t, x)

Z

D−2δ

|w(t, x)| dx dt

Z

δ

Rd

Z

0

T

−d

δ −d J( x−y δ ) sgn

− w(t, x) sgn(w(t, y)) dy dx dt

J( x−y δ )|w(t, x)

Z

ε

D−2δ


w(t, y) dy dx dt

− w(t, y)| dy dx dt

ε

#

#

ô

|u (t, x) − u (t, x − δz)| dx dt dz

ô

where the third inequality follows from |a| − a sgn(b) ≤ 2|a − b| for any a, b ∈ R. ε The fifth inequality follows, to BV ©in x. ¶ since u belongs 1 1 ∆t Setting ρ(∆t) = inf δ>0 C1 δ + C2 (∆t) 2 + C3 δ 2 , it follows that Z T −∆t Z |w(t, x)| dxdt ≤ ρ(∆t). 0

D

2

The function ρ(·) reaches the infimum at δ = C(∆t) 3 , and hence Z T −∆t Z 1 |w(t, x)| dx dt ≤ C(∆t) 3 → 0 as ∆t → 0. 0

D



Remark 3.1. Since Brownian sample paths are α-H¨older continuous for every α < 12 , a fractional order in the temporal L1 –continuity in (3.1) is expected. The proof of Theorem 3.1 uses an idea due to Kruzkov [13].

4. Well-Posedness Theory in Lp Before we introduce the relevant notions of generalized solutions, let us define what is meant by an entropy-entropy flux pair (η, q), or more simply an entropy pair, namely a C 2 function η : R → R such that η ′ , η ′′ have at most polynomial growth, with corresponding entropy flux q defined by q′ (u) = η ′ (u)f ′ (u). An entropy pair is called convex if η ′′ (u) ≥ 0. Definition 4.1 (Stochastic entropy solutions). A {Ft }t≥0 –adapted, L2 (Rd )–valued stochastic process u = u(t, x; ω) is a stochastic entropy solution of the balance law (1.1) with initial data (1.2) provided that the following conditions hold:

STOCHASTIC BALANCE LAWS

11

(i) for p = 1, 2, · · · ,

ó î p sup E ku(t)kLp (Rd ) < ∞,

for any T > 0;

0≤t≤T

(ii) for any convex entropy pair (η, q) and any 0 < s < t, Z ã ÅZ η(u(s, x))ϕ(x) dx − η(u(t, x))ϕ(x) dx − Rd

Rd

+

Z tZ s

+

Rd tZ

q(u(τ, x)) · ∇ϕ dx dτ


2 1 ′′ η (u(τ, x)) σ(u(τ, x)) ϕ dx dτ s Rd 2 Z t ÅZ ã η ′ (u(τ, x))σ(u(τ, x))ϕ dx dW (τ ) ≥ 0, + Z

s

Rd

for all ϕ ∈ Cc∞ (Rd ), ϕ ≥ 0, where

Rt s

(· · · ) dW (τ ) is an Ito integral.

To motivate the next definition, let us make a formal attempt to derive the L1 – contraction property for stochastic entropy solutions. To this end, consider smooth (in x) solutions to the one-dimensional problems: du + ∂x f (u) dt = σ(u) dW,

u|t=0 = u0 ,

dv + ∂x f (v) dt = σ(v) dW,

v|t=0 = v0 .

Subtracting the two stochastic conservation laws yields d(u − v) = − [∂x (f (u) − f (v))] dt + [σ(u) − σ(v)] dW. Let η(·) be an entropy. An application of the chain rule (Ito’s formula) now yields "
dη(u − v) = −∂x η ′ (u − v)(f (u) − f (v))
+ η ′′ (u − v) f (u) − f (v) ∂x (u − v) #
2 1 ′′ + η (u − v) σ(u) − σ(v) dt 2
+ η ′ (u − v) σ(u) − σ(v) dW,

where the last term is a martingale. Choosing η(·) = |·| yields η ′′ (·) = δ0 and the two “η ′′ terms” vanish. Consequently, after integrating and taking expectations, we arrive at the L1 –contraction (conservation) principle: ïZ ò ïZ ò E |u(t) − v(t)| dx = E |u0 − v0 | dx . Of course, for non-smooth solutions, the Ito formula is not available and we should instead derive the L1 –contraction principle from the (stochastic) entropy inequalities via Kruzkov’s method. Attempting precisely that, we write the entropy condition for u(t) = u(t, x; ω) with the entropy η(u(t) − v(s, y; ω)), where v(s, y; ω) is being treated as a constant with respect to (t, x). Similarly, write the entropy condition for v(s) = v(s, y; ω) for the entropy η(v(s) − u(t, x; ω)), with u(t, x; ω) being constant with respect to

12

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

(s, y). Take η(·) = |·|, and then q(u, v) = sgn(u − v)(f (u) − f (v)). After adding together the two entropy inequalities, we formally obtain (dt + ds )|u − v| h
≤ − (∂x + ∂y ) sgn(u − v)(f (u) − f (v)) î ói 1 2 2 dt ds + δ(u − v) (σ(u)) + (σ(v)) 2 + sgn(u(t, x) − v(s, y))σ(u(t, x)) dW (t) ds

− sgn(u(t, x) − v(s, y))σ(v(s, y)) dW (s) dt.

Depending on t < s or t > s, one of the last two terms are not adapted, and this causes a problem for the Ito integral. In particular, by taking the expectation of the above îinequality, only one ó of the last two terms vanishes. Moreover, to write 2 2 1 δ(u − v) (σ(u)) + (σ(v)) in the favorable form: 2 1 2 δ(u − v) (σ(u) − σ(v)) , 2 we are missing the cross term 2σ(u)σ(v). These difficulties can be effectively handled by the notion of “strong” stochastic entropy solutions.

Definition 4.2 (Strong stochastic entropy solutions). An {Ft }t≥0 –adapted, L2 (Rd )– valued stochastic process u = u(t) = u(t, x; ω) is a strong stochastic entropy solution of the balance law (1.1) with initial data (1.2) provided u is a stochastic entropy solution, and the following additional condition holds: (iii) for each {Ft }t≥0 –adapted, L2 (R)–valued stochastic process u ˜ = u ˜(t) = u˜(t, x; ω) satisfying ó î p for any T > 0, p = 1, 2, · · · , sup E k˜ u(t)kLp (Rd ) < ∞ 0≤t≤T

and for each entropy function S : R → R, with Z S ′ (˜ u(r, x) − v)σ(˜ u(r, x))ϕ(x, y) dx, S(r; v, y) := Rd

where r ≥ 0, v ∈ R, y ∈ Rd , and ϕ ∈ Cc∞ (Rd × Rd ), there exists a deterministic function ∆(s, t), 0 ≤ s ≤ t, such that ô ñZ Z t S(τ ; v = u(t, y), y) dW (τ ) dy E Rd

≤E

s

ñZ t Z s

ô

∂v S(τ ; v = u˜(τ, y), y)σ(u(τ, y)) dy dτ + ∆(s, t), Rd

m

where ∆(·, ·) is such that, for each T > 0, there exists a partition {ti }i=1 of [0, T ], 0 = t0 < t1 < · · · < tm = T , so that lim

max|ti+1 −ti | i

m X

∆(ti , ti+1 ) = 0.

i=1

The notion of strong stochastic entropy solutions is due to Feng-Nualart [9], who proved the L1 –contraction property for these solutions:     for t > 0, (4.1) E ku(t) − v(t)kL1 (Rd ) ≤ E ku0 − v0 kL1 (Rd )

STOCHASTIC BALANCE LAWS

13

where u(t) is any stochastic entropy solution with u|t=0 = u0 and v(t) is any strong stochastic entropy solution with v|t=0 = v0 . In (4.1), the entropy |·| can be replaced by (·)+ , yielding the L1 –comparison principle. Feng-Nualart [9] employed the compensated compactness method to prove an existence result in the one-dimensional context. The following theorem provides the existence of strong stochastic entropy solutions for a class of multidimensional equations. Theorem 4.1 (Existence in Lp ∩ BV ). Suppose that (1.3)–(1.5) hold. Then there exists a strong stochastic entropy solution u of the balance law (1.1) with initial data (1.2), satisfying î ó   E |u(t, ·)|BV (Rd ) ≤ E |u0 |BV (Rd ) for any t ≥ 0. (4.2) ó î Proof. For fixed ε > 0, we mollify u0 by uε0 ∈ C ∞ so that E kuε0 k2H s (Rd ) is finite for any s > 0, and ó ó î î E kuε0 kpLp (Rd ) + |uε0 |BV (Rd ) ≤ E ku0 kpLp (Rd ) + |u0 |BV (Rd ) < ∞,

for any p = 1, 2, · · · , and uε0 (x) → u0 (x) for a.e. x, almost surely as ε → 0. Now the same arguments as in Section 4 of Feng-Nualart [9] yield that there exists an Ft –adapted stochastic process uε = uε (t) ∈ C([0, ∞); L2 (Rd )) satisfying almost surely that ó î (i) E kuε (t, ·)k2H s (Rd ) < ∞ for all t > 0; (ii) ∂xi xj uε (t, ·) ∈ C(Rd ) for all i, j = 1, . . . , d;

(iii) For any ϕ ∈ Cc∞ (Rd ), ϕ ≥ 0, and 0 < s < t,

hη(uε (t, ·)), ϕi − hη(uε (s, ·)), ϕi Z t Z  1 t ′′ ε = hq(uε (τ, ·)), ∇ϕi dτ + η (u (τ, ·))(σ(uε (τ, ·))2 , ϕ dτ 2 s s Z t hη ′ (u(τ, ·))σ(u(τ, ·)), ϕi dW (τ ) + s

+ε

Z t s

t

 hη(uε (τ, ·)), ∆ϕi − η ′′ (uε (τ, ·))|∇uε (τ, ·)|2 , ϕ dτ

Z  1 t ′′ ε hq(u (τ, ·)), ∇ϕi dτ + ≤ η (u (τ, ·))(σ(uε (τ, ·))2 , ϕ dτ 2 s s Z t hη ′ (uε (τ, ·))σ(u(τ, ·)), ϕi dW (τ ) + O(ε), + Z

ε

s

where the first equality in (iii) follows from the Ito formula. Combining the results established in Sections 2 and 3, we conclude that there exist a subsequence (still denoted) {uε (t, x)}ε>0 and a limit u(t, x) such that as ε → 0, uε (t, x) → u(t, x) for a.e. (t, x), almost surely,

and the limit u(t, x) satisfies (4.2). Arguing as in Feng-Nualart [9], we can pass to the limit in the entropy inequality (iii) to conclude that the limit function u(t, x) is a stochastic entropy solution (cf. Definition 4.1). Moreover, we can prove that u is a strong stochastic entropy solution, as defined in Definition 4.2. 

14

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

Combining Theorem 4.1 with the L1 –stability result established in Feng-Nualart [9], we conclude Theorem 4.2 (Well-posedness in Lp ). Suppose (1.4) and (1.5) hold, and that u0 satisfies î ó E ku0 kpLp(Rd ) < ∞, p = 1, 2, · · · .

(i) Existence: There exists a strong stochastic entropy solution of the balance law (1.1) with initial data (1.2), satisfying for any t ≥ 0, î ó E ku(t, ·)kpLp (Rd ) < ∞, p = 1, 2, · · · . (4.3)

(ii) Stability: Let u(t, x) be a strong stochastic entropy solution of (1.1) with initial data u0 (x), and let v(t, x) be a stochastic entropy solution with initial data v0 (x). Then, for any t > 0, ò ò ïZ ïZ |u0 (x) − v0 (x)| dx . (4.4) |u(t, x) − v(t, x)| dx ≤ E E Rd

Rd

p d Proof. For the ∩∞ p=1 L (R )-valued random variable u0 , we can approximate u0 δ 1 by u0 (x) in L as δ → 0, with E[kuδ0 kpp + |uδ0 |BV ] < ∞ for fixed δ > 0. Then Theorem 4.1 indicates that there exists a corresponding family of global strong entropy solutions uδ (t, x) for δ > 0. Then the L1 –stability (contraction) result established in Feng-Nualart [9] implies that uδ (t, x) is a Cauchy sequence in L1 , which yields the strong convergence of uδ (t, x) to u(t, x) a.e., almost surely. Since î ó î ó E kuδ (t, ·)kpLp (Rd ) ≤ E kuδ0 (·)kpLp (Rd ) ≤ C, p = 1, 2, · · · ,

where C is independent of δ, one can check that u(t, x) is a strong stochastic entropy solution, and (4.3) holds. For the stability result (4.4), see [9].  5. Continuous dependence estimates The aim of this section is to establish an explicit “continuous dependence on the nonlinearities” estimate in the BV class. Let u(t) = u(t, x; ω) be a strong stochastic entropy solution of ∂t u + ∇ · f (u) = σ(u) ∂t W,

u|t=0 = u0 .

(5.1)

Let v(t) = v(t, x; ω) be a strong stochastic entropy solution of ∂t v + ∇ · ˆf (v) = σ ˆ (v) ∂t W,

v|t=0 = v0 .

(5.2)

We are interested in estimating E [ku(t) − v(t)kL1 ] in terms of u0 − v0 , f − ˆf , and σ−σ ˆ . Relevant continuous dependence results for deterministic conservation laws have been obtained in [17, 2], and in [4] for strongly degenerate parabolic equations; see also [3, 11]. We start with the following important lemma. Lemma 5.1. Suppose that (1.3)–(1.5) hold for the two data sets (u0 , f , σ) and (v0 , ˆf , σ ˆ ). For any fixed ε > 0, let u(t) = u(t, x; ω) be the solution to the stochastic parabolic problem   du + ∇x · f (u) − ε∆x u dt = σ(u) dW (t), u|t=0 = u0 . (5.3)

STOCHASTIC BALANCE LAWS

15

For any fixed εˆ > 0, let v(t) = v(t, y; ω) be the solution to the stochastic parabolic problem   dv + ∇y · ˆf (v) − εˆ∆y v dt = σ ˆ (v) dW (t), v|t=0 = v0 . (5.4) Take 0 ≤ φδ = φδ (x, y) ∈ Cc∞ (Rd × Rd ) to be of the form: φδ (x, y) =

x+y 1 J( x−y 2δ )ψ( 2 ) δd

x+y =: Jδ ( x−y 2 )ψ( 2 ),

(5.5)

where J(·) is a regularization kernel as in (3.3) and 0 ≤ ψ ∈ Cc∞ (Rd ). Moreover, given any entropy function η(·) with η(0) = 0 and η ′ (·) odd, introduce the associated entropy fluxes for u, v ∈ R: Z u Z u ˆ η ′ (ξ − v)ˆf ′ (ξ) dξ. η ′ (ξ − v)f ′ (ξ) dξ, qf (u, v) = qf (u, v) = v

v

Then, for any t > 0, ZZ ZZ η(u(t, x) − v(t, x))φδ (x, y)dx dy − η(u0 (x) − v0 (y))φδ (x, y) dx dy ˆ

≤ I f (φδ ) + I f ,f (φδ ) + I σ,ˆσ (φδ ) + I ε,ˆε (φδ ) ZZ Z t
η ′ (u(s, x) − v(s, y)) σ(u(s, x)) − σ ˆ (v(s, y)) φδ (x, y) dW (s) dx dy, + s

where f

I (φδ ) = I

f ,ˆ f

ZZ Z

(φδ ) =

t x−y qf (u(s, x), v(s, y)) · ∇ψ( x+y 2 )Jδ ( 2 ) ds dx dy,

0

ZZ Z

t 0


ˆ qf (v(s, y), u(s, x)) − qf (u(s, x), v(s, y)) · ∇y φδ (x, y) ds dx dy,

√
2Z Z Z t √ x+y I ε,ˆε (φδ ) = ε − εˆ η(u(s, x) − v(s, y))∆y Jδ ( x−y 2 )ψ( 2 ) ds dx dy 0

√
2Z Z Z t 1 √ x+y + ε + εˆ η(u(s, x) − v(s, y))Jδ ( x−y 2 )∆ψ( 2 ) ds dx dy 4 0 ZZ Z t
η(u(s, x) − v(s, y))∇y Jδ (x − y) · ∇ψ( x+y + εˆ − ε 2 ) ds dx dy, 0

I

σ,ˆ σ

(φδ ) =

ZZ Z

0

t

1 ′′ η (u(s, x) − v(s, y)) 2


2 × σ(u(s, x)) − σ ˆ (v(s, y)) φδ (x, y) ds dx dy.

Proof. Subtracting (5.4) from (5.3) and subsequently applying Ito’s formula to
η u(t) − v(t) , we obtain h

dη(u − v) = − η ′ (u − v) ∇x · f (u) − ∇y · ˆf (v) + η ′ (u − v) ε∆x u − εˆ∆y v
2 i 1 (5.6) dt + η ′′ (u − v) σ(u) − σ(v) 2
+ η ′ (u − v) σ(u) − σ(v) dW (t). Observe that

η ′ (u − v)∇x · f (u) = ∇x · qf (u, v),

ˆ η ′ (u − v)∇y · ˆf (v) = ∇y · qf (v, u),

16

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

and thus
− η ′ (u − v) ∇x · f (u) − ∇y · ˆf (v)


ˆ = −(∇x + ∇y ) · qf (u, v) + ∇y · qf (u, v) − qf (v, u) .

Next,


η ′ (u − v) ε∆x u − εˆ∆y v

= ε∆x + εˆ∆y η(u − v) − η ′′ (u − v) ε|∇x u|2 + εˆ|∇y v|2 √ √ 2
√ √ = ε∆x + 2 ε εˆ∇x · ∇y + εˆ∆y η(u − v) − η ′′ (u − v) ε∇x u − εˆ∇y v .

Inserting the last two relations into (5.6), we arrive at h
ˆ dη(u − v) = − (∇x + ∇y ) · qf (u, v) + ∇y · q(u, v) − qf (v, u)
√ √ + ε∆x + 2 ε εˆ∇x · ∇y + εˆ∆y η(u − v) √ √ 2 − η ′′ (u − v) ε∇x u − εˆ∇y v
2 i 1 dt + η ′′ (u − v) σ(u) − σ(v) 2
+ η ′ (u − v) σ(u) − σ(v) dW (t).

(5.7)

We integrate (5.7) against the test function φδ defined in (5.5), yielding ZZ ZZ η(u(t, x) − v(t, x))φδ (x, y)dx dy − η(u0 (x) − v0 (y))φδ (x, y) dx dy

≤ Ic1 + Ic2 + Id + I σ,ˆσ (φδ ) ZZ Z t
η ′ (u(s, x) − v(s, y)) σ(u(s, x)) − σ(v(s, y)) φδ (x, y) dW (s) dx dy, + s

where Ic1 Ic2

:= − :=

Id :=

ZZ Z

t

0

ZZ Z

t

0

ZZ Z

0

t

(∇x + ∇y ) · qf (u, v)φδ (x, y) ds dx dy,


ˆ ∇y · qf (u(s, x), v(s, y)) − qf (v(s, y), u(s, x)) φδ (x, y) ds dx dy,


√ √ ε∆x + 2 ε εˆ∇x · ∇y + εˆ∆y η(u(s, x) − v(s, y))φδ (x, y) ds dx dy. ˆ

Integrating by parts gives Ic2 = I f ,f (φδ ), and also Ic1 = I f (φδ ), since

x+y x−y x+y (∇x + ∇y )φ = Jδ ( x−y 2 )(∇x + ∇y )ψ( 2 ) = Jδ ( 2 )∇ψ( 2 ).

We now investigate the term Id . A calculation shows that
√ √ ε∆x + 2 ε εˆ∇x · ∇y + εˆ∆y φ(x, y)
√ √ x+y = ε∆x + 2 ε εˆ∇x · ∇y + εˆ∆y Jδ ( x−y 2 )ψ( 2 )
√ √ + Jδ (x − y) ε∆x + 2 ε εˆ∇x · ∇y + εˆ∆y ψ( x+y 2 ) + R,

STOCHASTIC BALANCE LAWS

17

and x+y ε∇y Jδ (x − y) · ∇y ψ( x+y R = 2ε∇x Jδ ( x−y 2 ) · ∇x ψ( 2 ) + 2ˆ 2 ) √ √ √ √ x+y x−y x−y + 2 ε εˆ∇x Jδ ( 2 ) · ∇y ψ( 2 ) + 2 ε εˆ∇y Jδ ( 2 ) · ∇x ψ( x+y 2 )  √ √ √ √ ˆ∇x Jδ ( x−y ˆ∇y Jδ ( x−y = 2ε∇x Jδ ( x−y 2 )+2 ε ε 2 )+2 ε ε 2 )  x+y + 2ˆ ε∇y Jδ ( x−y 2 ) · ∇y ψ( 2 )

x+y x+y = 2∇y Jδ ( x−y ˆ − ε = ∇y Jδ ( x−y ˆ− ε . 2 ) · ∇y ψ( 2 ) ε 2 ) · ∇ψ( 2 ) ε

Moreover,

√
2
√ √ √ ε∆x + 2 ε εˆ∇x · ∇y + εˆ∆y Jδ (x − y) = ε − εˆ ∆y Jδ (x − y), √
2
√ √ 1 √ ε + εˆ ∆ψ( x+y ε∆x + 2 ε εˆ∇x · ∇y + εˆ∆y ψ( x+y 2 )= 4 2 ).

Consequently, after integrating by parts, Id becomes I ε,ˆε (φδ ).



Theorem 5.1 (Continuous dependence estimates). Suppose that (1.3)–(1.5) hold for the two data sets (u0 , f , σ) and (v0 , ˆf , σ ˆ ). Let u(t) and v(t) be the strong stochastic entropy solutions of (5.1)–(5.2), respectively, for which î ó î ó E |v(t)|BV (Rd ) ≤ E |v0 |BV (Rd )

for t > 0.

In addition, we assume that either u, v ∈ L∞ ((0, T ) × Rd × Ω)

for any T > 0,

or f ′′ , f ′ − ˆf ′ , σ − σ ˆ ∈ L∞ . Then (i) there is a constant CT > 0 such that, for any 0 < t < T with T finite, E

ïZ

Rd

ò |u(t, x) − v(t, x)|ψ(x) dx

≤ CT E

ïZ

Rd

ò √ |u0 (x) − v0 (x)|ψ(x) dx + tkψkL1 (Rd ) kσ − σ ˆ kL∞ î

+ t E |v0 |BV (Rd )

óÄ

′

ˆ′

ˆ kL∞ kf − f kL∞ + kσ − σ

ä

!

,

where the constant CT > 0 is independent of |u0 |BV (Rd ) and |v0 |BV (Rd ) , and may grow exponentially in T . Moreover, ψ = ψ(x) ≥ 0 is any function satisfying |ψ| ≤ C0 , |∇ψ| ≤ C0 ψ, which includes ψ(x) = e−C0 |x| and, more generally, ψ(x) = 1 when |x| ≤ R and ψ(x) = e−C0 (|x|−R) when |x| ≥ R.

18

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

In particular, for any R > 0, this choice implies

E

ñZ

|x| 0, let ηρ : R → R be the function defined by (2.1)–(2.5). Then the function qfρ (u, v)

=

Z

v

u

ηρ′ (ξ − v)f ′ (ξ) dξ,

u, v ∈ R,

satisfies
M2 ′′ kf kL∞ ρ, ∂u qfρ (u, v) − qfρ (v, u) ≤ 2

where M2 = sup|u|≤1 |¯ η ′′ (u)|.

(5.8)

STOCHASTIC BALANCE LAWS

19

In view of Lemma 5.1 with εˆ = ε,

E

ïZ Z

≤E

ò ηρ (u(t, x) − v(t, y))φδ (x, y) dx dy ïZ Z ò −E ηρ (u0 (x) − v0 (x))φδ (x, y) dx dy ñZ Z Z t

+E

qfρ (u(s, x), v(s, y))

0

"Z Z Z

t

ˆ

− +E

t

"Z Z Z

qfρ (u(s, x), v(s, y))

1 ′′ η (u(s, x) − v(s, y)) 2 ρ

0

+ εE

ô

qfρ (v(s, y), u(s, x))

0

"Z Z Z

·

x−y ∇ψ( x+y 2 )Jδ ( 2 ) ds dx dy

t 0




· ∇y φδ ds dx dy

#

(5.9)


2 × σ(u(s, x)) − σ ˆ (v(s, y)) φδ (x, y) ds dx dy ηρ (u(s, x) −

#

x+y v(s, y))Jδ ( x−y 2 )∆x ψ( 2 ) ds dx dy

#

.

Observe that
ˆ − ∇y · qfρ (v(s, y), u(s, x)) − qfρ (u(s, x), v(s, y))
ˆ = ∇y v · ∂v qfρ (u, v) − qfρ (v, u)

(u,v)=(u(s,x),v(s,y))

,

and, thanks to (5.8),


ˆ ∂v qfρ (u, v) − qfρ (v, u)

ˆ = ∂v qfρ (v, u) − qfρ (v, u) + ∂v qfρ (u, v) − qfρ (v, u) M2 ′′ ≤ |f ′ (v) − ˆf ′ (v)| + kf kL∞ ρ. 2

Hence, after an integration by parts, ñZ Z Z ô t
ˆ f f qρ (v(s, y), u(s, x)) − qρ (u(s, x), v(s, y)) · ∇y φδ ds dx dy E 0   î ó M2 ′′ kf kL∞ ρ . ≤ t E |v0 |BV (Rd ) kψkL∞ (Rd ) kf ′ − ˆf ′ kL∞ + 2

20

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

Consequently, again thanks to (5.8) and also (2.4), we can write (5.9) as ïZ Z ò E |u(t, x) − v(t, y)|φδ (x, y) dx dy ò ïZ Z |u0 (x) − v0 (x)|φδ (x, y) dx dy −E ô ñZ Z Z t ≤E

+E

x−y qfρ (u(s, x), v(s, y)) · ∇ψ( x+y 2 )Jδ ( 2 ) ds dx dy

0

"Z Z Z

0

t

1 ′′ η (u(s, x) − v(s, y)) 2 ρ
2 × σ(u(s, x)) − σ ˆ (v(s, y)) φδ (x, y) ds dx dy

(5.10) #

Ä ä + t |v0 |BV (Rd ) kψkL∞ (Rd ) kf ′ − ˆf ′ kL∞ + O(ρ) Ä ä + O kψkL1 (Rd ) ρ + O(ε).

Sending δ → 0 and using |∇ψ(x)| ≤ C0 ψ(x), we obtain ñZ Z Z ô t x+y f qρ (u(s, x), v(s, y)) · ∇ψ( 2 )Jδ (x − y) ds dx dy lim E δ→0 0 Z t ïZ ò ≤ C2 kf ′ kL∞ E |u(s, x) − v(s, x)| ψ(x)dx ds; 0

hence, sending δ → 0 in (5.10) returns ïZ ò ïZ ò E |u(t, x) − v(t, x)| ψ(x) dx − E |u0 (x) − v0 (x)| ψ(x) dx Z t ïZ ò ≤ C2 kf ′ k∞ E |u(s, x) − v(s, x)| ψ(x)dx ds ñZ Z t0 ô
2 1 ′′ +E ηρ (u(s, x) − v(s, x)) σ(u(s, x)) − σ ˆ (v(s, x)) ψ(x) ds dx 0 2 î ó ä Ä + t E |v0 |BV (Rd ) kψkL∞ (Rd ) kf ′ − ˆf ′ kL∞ + O(ρ) Ä ä + O kψkL1 (Rd ) ρ + O(ε). Next, with our choice of ηρ , it follows that ñZ Z ô t
2 1 ′′ ηρ (u(s, x) − v(τ, x)) σ(u(s, x)) − σ ˆ (v(s, x)) ψ(x) ds dx E 0 2 ñZ Z t ô
2 M2 ≤E 1|u(s,x)−v(s,x)| 0,

(5.13)

and let uε be the solution to the parabolic problem

  duε + ∇x · f (uε ) − ε∆x uε dt = σ(uε ) dW (t),

uε |t=0 = u0 .

In addition, we assume that

either u, v ∈ L∞ ((0, T ) × Rd × Ω) for any T > 0, or f ′′ ∈ L∞ .

Then there exists a constant CT > 0 such that, for any 0 < t < T with T finite,

E

ïZ

Rd

ò î ó √ |u(t, x) − u (t, x)| dx ≤ CT E |u0 |BV (Rd ) t ε. ε

STOCHASTIC BALANCE LAWS

23

Proof. We proceed as in the proof of Theorem 5.1, starting off from Lemma 5.1 with σ ˆ = σ, ˆf = f , εˆ 6= ε, uε = u, uεˆ = v, leading to ò ïZ Z ε E u (t, x) − uεˆ(t, y) φδ (x, y) dx dy ñZ Z Z t ô ≤E

+E

x−y qfρ (uε (s, x), uεˆ(s, y)) · ∇ψ( x+y 2 )Jδ ( 2 ) ds dx dy

0

"Z Z Z

t

ηρ′′ (uε (s, x) − uεˆ(s, y))

0


2 × σ(uε (s, x)) − σ(uεˆ(s, y)) φδ (x, y) ds dx dy

Ä ä + t |u0 |BV (Rd ) kψkL∞ (Rd ) O(ρ) + O kψkL1 (Rd ) ρ " √
2 Z Z Z t √ ε − εˆ E + ηρ (uε (s, x) − uεˆ(s, y))

#

0

× √
2 1 √ ε + εˆ E + 4

"Z Z Z

"Z Z Z

0

(5.14)

#

t

0

ηρ (uε (s, x) − uεˆ(s, y)) ×


+ εˆ − ε E

x+y ∆y Jδ ( x−y 2 )ψ( 2 ) ds dx dy

x+y Jδ ( x−y 2 )∆ψ( 2 ) ds dx dy

#

t

ηρ (uε (s, x) − uεˆ(s, y)) ×

∇y Jδ ( x−y 2 )

·

∇ψ( x+y 2 ) ds dx dy

#

=: I1 + I2 + I3 + I4 + I5 + I6 . As before, |I2 | ≤ C1

Z

t

E

0

ïZ

ε

ε

|u (s, x) − u (s, y)| Jδ (x −

y)ψ( x+y 2 ) dx dy

ò

ds.

Noting that the right-hand side is independent of ρ, we can first send ρ → 0 in (5.14), and then let ψ tend to 1Rd , keeping in mind the Lp –estimates (4.2), with the outcome that I1 , I3 , I5 , I6 → 0. The resulting estimate reads ïZ Z ò ε x−y εˆ E J ( (t, x) − u (t, y) ) dx dy δ 2 u (5.15) Z t ïZ Z ò x−y ≤ C1 E |u(s, x) − v(s, y)| Jδ ( 2 ) dx ds + I, 0

where √
2 √ ε − εˆ E I=

ñZ Z Z

0

t ε

|u (s, x) − u

(s, y)|∆y Jδ ( x−y 2 ) ds dx dy

εˆ

ô

.

24

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

An integration by parts, followed by application of the spatial BV –estimate (5.13), yields √
î ó √ε − εˆ 2 |I| ≤ C 2 t E |u0 |BV (Rd ) . δ In view of this, it follows from (5.15) in a completely standard way that ïZ ò E |uε (t, x) − uεˆ(t, x)| dx Z t ïZ ò ≤ C1 E |uε (s, x) − v ε (s, x)| dx ds 0

î

+ C3 E |u0 |BV (Rd )

ó

δ+t

√

ε− δ

√
2  εˆ

.

√ √ Choosing δ = ε − εˆ gives ïZ ò î ó Ä√ √ ä E |uε (t, x) − uεˆ(t, x)| dx ≤ CT E |u0 |BV (Rd ) t ε − εˆ . Rd

Sending εˆ → 0 concludes the proof of the theorem.



Remark 5.2. Theorem 5.2 indicates that {uε (t, x)} is the Cauchy sequence in C(0, T ; L1), which directly implies its strong convergence. 6. More General Equations We now discuss briefly diverse generalizations. First of all, as in [9], the stochastic term in (1.1) can be replaced by the more general term Z σ(u(t, x); z)∂t W (t, dz), z∈Z

where Z is a metric space, σ : R × Z → R, W (t, dz) is a space-time Gaussian white noise martingale random measure with respect to a filtration {Ft } (see e.g., Walsh [24], Kurtz-Protter [14]) with   E W (t, A) ∩ W (t, B) = µ(A ∩ B)t

for measurable A, B ⊂ Z, where µ is a (deterministic) σ-finite Borel measure on the metric space Z. In particular, when Z = {1, 2, . . . , m} and µ is a counting measure on Z, then the stochastic term reduces to m X

σk (u(t, x))∂t Wk (t).

k=1

For the spatial BV and temporal L1 –continuity estimates and stability results, we can allow for more general flux functions f (t, x, u) with spatial dependence, by combining the present methods with those in [3, 11]. Next, let us discuss the case where the noise coefficient σ(x, u) has a spatial dependence, focusing on the stochastic balance law ∂t u + ∇ · f (u) = σ(x, u) ∂t W (t),

(6.1)

STOCHASTIC BALANCE LAWS

25

where the noise coefficient is assumed to satisfy σ(x, 0) = 0 and |σ(x, u) − σ(x, v)| ≤ C |u − v| , |σ(x, u) − σ(y, u)| ≤ C |x − y| |u| ,

∀ u, v ∈ R, ∀ x ∈ Rd ,

∀ u ∈ R, ∀ x, y ∈ Rd ,

(6.2)

where C is a deterministic constant. In the previous sections, we have established the existence of a strong stochastic entropy solution in the multidimensional context. The proof was based on deriving BV –estimates. However, as mentioned before, the BV –estimates are no longer available when the noise term σ depends on the spatial location x. However, it possible to derive fractional BV estimates. For fixed ε > 0, let uε (t, x) be the solution to the stochastic parabolic problem   duε + ∇x · f (uε ) − ε∆x uε dt = σ(x, uε ) dW (t), uε |t=0 = u0 , (6.3)

where we tactically assume that f , σ, u0 are sufficiently smooth to ensure the existence of a regular solution [9]. Utilizing the continuous dependence framework (Lemma 5.1) which also holds when the noise term σ depends on x, we will prove that, for any δ > 0, ò ïZ Z ε ε |u (t, x + z) − u (t, x − z)| Jδ (z)ψ(x) dx dz E Rd Rd ò ïZ Z (6.4) |u0 (x + z) − u0 (x − z)| Jδ (z)ψ(x) dx dz ≤ CT E d d R ÄR ä 1 0 < t < T, + CT δ 2 1 + kψkL1 (Rd ) , for some finite constant CT independent of ε, where Jδ is a symmetric mollifier and ψ ≥ 0 is a compactly supported smooth function. In what follows, we assume that the cut-off function ψ ≥ 0 satisfies |∇ψ(x)| ≤ C0 ψ(x),

|∆ψ(x)| ≤ C0 ψ(x),

ψ ≡ 1 on KR := {|x| < R},

for some constants C0 > 0 and R > 0. One example of such a function, at least after an easy approximation argument, is the compactly supported function ψ ∈ W 2,∞ (Rd ) defined by   when |x| ≤ R, 1 √ π−(|x|−R)
π 1 ψ(x) = eπ +1 2e sin(|x| − R + 4 ) + 1 when R ≤ |x| ≤ R + π,   0 when |x| ≥ R + π. Estimate (6.4) can be turned into a fractional BV estimate thanks to the following deterministic lemma, which is related to known links between Sobolev, Besov, and Nikolskii fractional spaces (cf., e.g., [18]); a proof can be found in the appendix. Lemma 6.1. Let h : Rd → R be a given integrable function, r, s ∈ (0, 1),Ä ψ ä∈ Cc∞ (Rd ), and {Jδ }δ>0 a sequence of symmetric mollifiers, i.e., Jδ (x) = δ1d J |x| δ , R ∞ 0 ≤ J ∈ Cc (R), supp (J) ⊂ [−1, 1], J(−·) = J(·), and J = 1.

26

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

Suppose r < s. Then there exists a finite constant C1 = C1 (J, d, r, s) such that, for any δ > 0 Z Z |h(x + z) − h(x − z)| Jδ (z)ψ(x) dx dz Rd Rd Z (6.5) −s |h(x + z) − h(x − z)| ψ(x) dx. ≤ C1 δ r sup |z| Rd

|z|≤δ

Suppose r < s. Then there exists a finite constant C2 = C2 (J, d, r, s) such that for any δ > 0 Z |h(x + z) − h(x)| ψ(x) dx sup |z|≤δ

Rd

≤ C2 δ

r

sup δ

−s

0 0, ò ïZ Z 1 |uε (t, x + z) − uε (t, x − z)| Jδ (z)ψ(x) dx dz δ− 2 E Rd Rd Z Z 1 |u0 (x + z) − u0 (x − z)| Jδ (z) ψ(x) dx dz ≤ CT δ − 2 Rd Rd Ä ä + CT 1 + kψkL1 (Rd ) (6.7) Z −s |u0 (x + z) − u0 (x)| dx ≤ 2 CT C1 kψkL∞ (Rd ) sup |z| |z|≤δ

Ä

+ CT

≤ C(T, R),

1 + kψkL1 (Rd )

Rd

ä

where (6.5) with r = 12 and s > 12 was used to arrive at the second inequality. In view of (6.6) with s = 21 and r < 12 , ò ïZ ε ε sup E |u (t, x + z) − u (t, x)| ψ(x) dx |z|≤ 2δ

Rd

1

≤ C2 δ r sup δ − 2 +

Z

Z

|uε (t, x + z) − uε (t, x − z)| Jδ (z)ψ(x) dx dz

0 0, let uε solve the stochastic parabolic problem (6.3) with deterministic initial data u0 belonging to the Besov ν space B1,∞ (Rd ) for some ν ∈ ( 21 , 1). In addition, we assume that either uε ∈ L∞ ((0, T ) × Rd × Ω) for any T > 0, or f ′′ ∈ L∞ .

STOCHASTIC BALANCE LAWS

27

Fix T > 0 and R > 0. There exists a constant CT,R independent of ε such that, for any 0 < t < T , sup E |z|≤δ

ñZ

ε

KR

ô

ε

|u (t, x + z) − u (t, x)| dx ≤ CT,R δ r

for some r ∈ (0, 21 ). Proof of (6.4). We start off from Lemma 5.1 with ˆf = f , εˆ = ε, σ ˆ = σ, v0 = u0 , v = u (which also holds when σ depends on the spatial location): E

ïZ Z

≤E

ε

ε

x+y (t, y))Jδ ( x−y 2 )ψ( 2 ) dx dy

ò

ηρ (u (t, x) − u ïZ Z ò x+y −E ηρ (u0 (x) − u0 (y))Jδ ( x−y )ψ( ) dx dy 2 2 ñZ Z Z t qfρ (uε (s, x), uε (s, y))

0

+E

ñZ Z Z

+E

t

0

qfρ (uε (s, y), uε (s, x))

−

"Z Z Z

qfρ (uε (s, x), uε (s, y))




1 ′′ ε η (u (s, x) − uε (s, y)) 2 ρ

t

ε

0

· ∇y φδ ds dx dy


2 × σ(x, u (s, x)) − σ(y, u (s, y)) φδ (x, y) ds dx dy ε

+ εE

ô

t

0

"Z Z Z

·

x−y ∇ψ( x+y 2 )Jδ ( 2 ) ds dx dy

ε

ηρ (u (s, x) − u

ε

x+y (s, y))Jδ ( x−y 2 )∆x ψ( 2 ) ds dx dy

ô

#

#

=: I1 + I2 + I3 + I4 . (6.9) Finally, denoting the left-hand side of (6.9) by LHS and utilizing (2.4), we have LHS = E

ïZ Z

ε

ε

x+y (t, y)| Jδ ( x−y 2 )ψ( 2 ) dx dy

ò

|u (t, x) − u ò ïZ Z x+y x−y |u0 (x) − u0 (y)| Jδ ( 2 )ψ( 2 ) dx dy + O(ρ) kψkL1 (Rd ) . −E

Since |∇ψ(x)| ≤ C0 ψ(x), |I1 | ≤ C

Z

t

E 0

ïZ Z

ε

|u (s, x) − u

ε

x+y (s, y)| Jδ ( x−y 2 )ψ( 2 ) dx dy

ò

ds.

Note that, thanks to (5.8) and the boundedness of f ′′ , qfρ (v, u)

=

qfρ (u, v)

+

Z

v

u


∂ξ qfρ (ξ, v) − qfρ (v, ξ) dξ = qfρ (u, v) + |u − v| O(ρ),

28

GUI-QIANG CHEN, QIAN DING, AND KENNETH H. KARLSEN

so that |I2 | ≤ C ρ E

ñZ Z Z

t

x+y |u (s, x) − u (s, y)| ∇y Jδ ( x−y 2 ) ψ( 2 ) ds dx dy ε

0

+ C ρE

ñZ Z Z

ε

t

ε

0

≤ C t kψkL∞ (Rd ) because of the estimate

ô

ρ δ

|u (s, x) − u

 +ρ ,

ε

x+y (s, y)| Jδ ( x−y 2 ) ∇ψ( 2 )

î ó sup E kuε (t)kL1 (Rd ) < ∞,





ds dx dy

ô

for any T > 0,

0≤t≤T

and we have again exploited |∇ψ(x)| ≤ C0 ψ(x). Regarding I3 , "Z Z Z t
2 M2 1|uε (s,x)−uε (s,x)|