AN INHOMOGENEOUS EVOLUTION EQUATION INVOLVING THE NORMALIZED INFINITY LAPLACIAN WITH A TRANSPORT TERM

. In this paper, we prove the uniqueness and stability of viscosity solutions of the following initial-boundary problem related to the random game named tug-of-war with a transport term where denotes the normalized inﬁnity Laplacian, ξ : Q T → R n is a continuous vector ﬁeld, f and g are continuous. When ξ is a ﬁxed ﬁeld and the inhomogeneous term f is constant, the existence is obtained by the approximate procedure. When f and ξ are Lipschitz continuous, we also establish the Lipschitz continuity of the viscosity solutions. Further-more we establish the comparison principle of the generalized equation with the ﬁrst order term with initial-boundary condition where H ( x,t,p ) : Q T × R n → R is continuous, H ( x,t, 0) = 0 and grows at most linearly at inﬁnity with respect to the variable p . And the existence result is also obtained when H ( x,t,p ) = H ( p ) and f is constant for the generalized equation.

1. Introduction. In this paper, we are interested in the quasilinear degenerate parabolic normalized infinity Laplacian equation with a transport term, where Q T = Ω × (0, T ), Ω ⊂ R n is a bounded domain, ξ : Q T → R n is a vectorvalued function, f is a function in Q T and the normalized infinity Laplacian The infinity Laplacian was first studied in relation with the absolutely minimizing Lipschitz extension problem, see [6] for a survey on this subject. The infinity harmonic functions are equivalent to the property of the comparison with cone functions, see the survey [9] and the references therein. Due to the high degeneracy and singularity of the operator, the regularity of solutions is still an open problem as far as we know, see [16,17] etc. The normalized infinity Laplacian is closely related to the game theory named tug-of-war [35,36]. It has many applications in image processing [1,8] and optimal mass transportation problems [14,18].
The study of the singular inhomogeneous normalized infinity Laplace equation with a transport term (1) in this work was inspired by recent works, [23,34] and [2,3,4,10,20,24,38,39], on the normalized counterpart with a transport term by the game theory, and on the parabolic equations associated with infinity Laplacian by the partial differential equation theory.
In [23] the elliptic normalized infinity Laplacian with a transport term −∆ N ∞ u − ξ, Du = 0, in Ω, u = g, on ∂Ω (2) was first studied and the existence of viscosity solutions as the continuous value of the modified tug-of-war game is obtained by probabilistic approach when ξ is a Lipschitz vector field. While for a continuous gradient vector field ξ, the existence and uniqueness of viscosity solutions were obtained by the p−Laplace approximation. Let us briefly recall from [23] the two-player, random-turn, tug-of-war game with a transport term. Set G is the final payoff function defined in a narrow strip around the boundary ∂Ω. The tug-of-war game with a transport term is played with two stages. First we toss an unfair coin, which has head probability 0 < C(ε) < 1, and tail probability 1 − C(ε). If we have obtained a head, we then toss a new (fair) coin and the winner moves the token to any new position x 1 ∈ B ε (x 0 ). But if in the first (unfair) coin toss we obtain a tail, the token is moved to x 0 + ξ(x 0 )ε, where ξ(x) : Ω → R n is the vector field that appears (that is assumed to be Lipschitz).
Note that there is no strategies of the players involved if we get a tail in the first coin toss. The game continues until the first time the token arrives to x τ ∈ R n \ Ω and then Player I earns G(x τ ), and thus Player II earns −G(x τ ), where G is the extension of g from ∂Ω to a small strip Γ ε = {x ∈ R n \ Ω : dist(x, ∂Ω) < ε ξ ∞ } and gives the final payoff of the game. Since the normalized infinity Laplacian was first introduced from the point of game theory named tug-of-war in [35], some kinds of modified tug-of-war have received a lot of attention. A biased tug-of-war was introduced in [37] and some results were established via a comparison property with exponential cones, see also [5,27]. A tug-of-war with noise was studied in [36] by the game theory, [12,22,31,32] by the methods of the partial differential equations, and [12,13] in image processing.
In this paper we are devoted to prove the wellposedness and Lipschitz regularity of the viscosity solutions to the following problem And in fact we establish the uniqueness for a slightly more general equation with a first order term. Therefore, we consider where H : Q T × R n → R is a continuous function, H(x, t, 0) = 0 for any point (x, t) ∈ Q T and grows at most linearly at infinity with respect to the variable p, that is, there exists M > 0 such that |H(x, t, p)| ≤ M |p|. Now let us see two special cases for H(x, t, p).
Notice that (1) and its generalized version (4) not only are degenerate parabolic, singular and not in divergence form but also have wide applications to image processing and mass transportation etc. They constitute a class of equations with particular properties. And our current work also helps to build a further connection between the partial differential equation theory and the random differential game theory about the infinity Laplacian. Now our first result is Theorem 1.1. Let H be as above, f be continuous in Q T , and let g ∈ C(R n+1 ) .
Suppose also that f ≥ 0 or f ≤ 0 in Q T . Then there exists at most one viscosity solution u ∈ C(Q T ) to the following initial-boundary problem Based on the comparison principle and the 1-homogeneity of the parabolic operator P u = u t − ∆ N ∞ u − ξ, Du , we can obtain the following stability result.
is the viscosity solution of the following problems When H(x, t, p) = H(p) and the inhomogeneous term is constant, the equation is translation invariant. Therefore we can also establish the following existence theorem by the approximate procedure. Theorem 1.3. Let H be as above and H(x, t, p) = H(p), f be constant, and let g ∈ C(R n+1 ). Then there exists a viscosity solution u ∈ C(Q T ) to the initialboundary problem (5).
For f ≡ 0 and H ≡ 0, the uniqueness and existence results were obtained by Juutinen and Kawohl [20]. The evolution equations involved in the infinity Laplacian have been studied in a number of works. See [2,3,4,10,24,38,39] for details. In these references they all consider the homogeneous parabolic equation involving infinity Laplacian. Here we are interested in the inhomogeneous equation (4) and we prove the comparison principle when the inhomogeneous term does not change its sign, i.e. f ≥ 0 or f ≤ 0. For the uniqueness, the method we use is the classical perturbation argument for viscosity solutions. Due to the difficulty of the first order and inhomogeneous terms we must construct suitable double variables function. And thanks to the parabolic term the uniqueness is valid for f nonnegative or nonpositive. Note that the uniqueness was proved for f < 0 or f > 0 [5,21,28,29,30,26,35] for the elliptic case, and counterexamples show that the wellposedness fails if f changes its sign [29,35]. From this point of view, our uniqueness result is optimal. As for the existence of viscosity solutions, due to the high degeneracy and singularity of (1) we adopt the regularized approximation method. The key point is to establish the uniform estimates to the solutions of the regularized equations. The barrier function arguments are also employed to establish the equi-continuity of approximate solutions and we follow the argument in [3,12,20,38,39]. Note that in order to deal with the first order and inhomogeneous terms we must construct suitable barrier functions and calculate carefully.

Remark 1.
When H(x, t, p) = H(p) and f is constant in (5), the equation is translation invariant. Then due to the translation invariance and the approximate procedure, we can establish the existence result. And for the general first order term and f (x, t), we will study later in a different paper. Especially, during the approximation procedure we can also obtain the Lipschitz regularity in time and the Hölder continuity in space variable.
By Theorem 1.3, we can immediately obtain the existence for the problem with a transport term: Theorem 1.4. Let ξ be a fixed field, f be constant, and let g ∈ C(R n+1 ). Then there exists a viscosity solution u ∈ C(Q T ) to the initial-boundary problem (3).
Besides the wellposedness results we also obtain the Lipschtiz estimate to the viscosity solutions of the infinity Laplacian equation with a transport term (1) by approximation method. It should be pointed out that the Lipschitz regularity in time can be obtained by barrier method, but the Lipschitz property in space variable can not be obtained by the same method, and we adopt the Bernstein's method to deal with it.
for almost every (x, t) ∈ Q T .
When ξ ≡ 0, this slightly extends the result in [20], because of regarding the extra source terms ξ, Du and f . The proof is based on the Bernstein's method similar to the argument in [12] or [20]. But in order to deal with the additional first order and inhomogeneous terms we must estimate carefully. We clarify that in fact one can also obtain the Lipschitz estimate to the viscosity solutions of the generalized equation (4) without much additional cost.
The rest of the paper will be organized as follows. In Section 2, we give the equivalent definition of viscosity solutions to the equation (4). We give the comparison principle of the viscosity solutions of the initial-boundary problem of (5) and some stability results of the (1) in Section 3. In Section 4, we prove the existence of the viscosity solutions to the problem (5) by approximation method when H(x, t, p) = H(p) and the inhomogeneous term is constant. In Section 5, we furthermore establish the Lipschitz regularity in space variable of the viscosity solutions to the equation (1) by the Bernstein's method.
2. Viscosity solutions. In this section, we give the definition of the viscosity solutions to (4). In this paper, Q T = Ω × (0, T ) denotes the space-time cylinder with the parabolic boundary We denote by D 2 ϕ(x, t) the second order Hessian matrix with space variable for a smooth function ϕ. λ max D 2 ϕ(x, t) and λ min D 2 ϕ(x, t) denote the largest and the smallest of the eigenvalues to the symmetric matrix D 2 ϕ(x, t) respectively.
Due to the singularity of the infinity Laplacian, we should give a reasonable explanation when the gradient vanishes. Here we use the definition of viscosity solutions based on semi-continuous extension, and we refer to the reader to [15,19] etc.
Definition 2.1. Suppose that H is as above, f ∈ C(Q T ), and u : Q T → R is upper semi-continuous. If for every (x 0 , t 0 ) ∈ Q T and any C 2 (Q T ) test function ϕ such that u − ϕ has a strict local maximum at point (x 0 , t 0 ), that is u(x 0 , t 0 ) = ϕ(x 0 , t 0 ) and u(x, t) < ϕ(x, t) in a neighborhood of (x 0 , t 0 ), there holds then we say that u is a viscosity sub-solution of (4).
Similarly, suppose u : Q T → R is lower semi-continuous. If for every (x 0 , t 0 ) ∈ Q T and any C 2 (Q T ) test function ϕ such that u − ϕ has a strict local minimum at point (x 0 , t 0 ), that is u(x 0 , t 0 ) = ϕ(x 0 , t 0 ) and u(x, t) > ϕ(x, t) in a neighborhood of (x 0 , t 0 ), there holds then we say that u is a viscosity super-solution of (4).
If u ∈ C(Q T ) is both a viscosity sub-solution and a viscosity super-solution, then we say that u is a viscosity solution of (4).

Remark 2.
If u is twice differentiable with respect to x at point (x, t), then the normalized infinity Laplacian of u at (x, t) is defined to be the closed interval When Du(x, t) = 0, ∆ N ∞ u(x, t) contains only one real number and it is clear In fact, if the gradient of a test function vanishes, one may assume that D 2 ϕ = 0, and thus λ max D 2 ϕ = λ min D 2 ϕ = 0. This means that if Dϕ = 0, then ∆ N ∞ ϕ = 0. The proof of this fact is based on the well-known perturbation argument in [15,20,21] etc. But in order to deal with the difficulty of the first order term and the inhomogeneous term we should perturb twice. In fact the following lemma exhibits an equivalent definition to Definition 2.1. The idea of such a statement, together with the strategy of the proof, come from [20], however, some of the details are quite different due to the first order and inhomogeneous terms.
is an upper semi-continuous function with the property that for every (x 0 , t 0 ) ∈ Q T and any C 2 (Q T ) test function ϕ such that u − ϕ has a strict local maximum at point (x 0 , t 0 ), the following holds: Then u is a viscosity sub-solution of (4).
Proof. We consider the case for f ≥ 0 and leave the details for f ≤ 0 to the reader. We argue by contradiction. Suppose that u is not a viscosity sub-solution but satisfies the assumption of the lemma. Then there exist (x 0 , t 0 ) ∈ Q T and ϕ ∈ C 2 (Q T ) such that u−ϕ has a strict local maximum at point ( and By first choosing a small enough δ > 0 and then large enough j, we have (x j , t j ), (y j , s j ) ∈ Q T , and Denote then ϕ − φ has a local minimum at (y j , s j ). By (9) and continuity of f and for j large enough and δ > 0 sufficiently small. Since we also have Similarly, set then u δ − ψ has a local maximum at (x j , t j ). We consider the two cases: either x j = y j or x j = y j for all j large enough.
On the other hand, since u δ − ψ has a local maximum at (x j , t j ), we have where we have used the assumption of the lemma. Due to (11), the continuity of f and f ≥ 0 in Q T , we obtain that (14) contradicts to (13). Case 2: If x j = y j , we use jets and the parabolic maximum principle for semicontinuous functions [11,33]. Then by the parabolic theorem of sums for w j there exist n × n symmetric matrices X j , Y j such that Y j − X j is positive semi-definite and and η j → 0 as j → ∞. Using (12) and the fact that Since u δ − ψ has a local maximum at (x j , t j ) and the assumption of the lemma, we have (16) With the help of (15) and (16), we have for large enough j and δ sufficiently small that We get a contradiction. Now we have completed the proof.
Remark 3. When f ≡ 0, the above Lemma 2.2 is still valid if we modify some details in the proof. In fact in this case we can take δ = 0 in the function of double variables (10).
Remark 4. Lemma 2.2 is also valid for f ≤ 0 if we replace 1−δ by 1+δ throughout the argument.

Remark 5.
A similar result is valid for viscosity super-solutions.
3. Comparison principle. In this section we prove the comparison principle of viscosity solutions of the initial-boundary problem of (5). The method is also the perturbation argument similar to the method in Section 2. Notice that we should consider the δ−perturbation in order to deal with the inhomogeneous and the first order terms. For more comparison results about the degenerate partial differential equations one can see [3,12,19,20].
Theorem 3.1. Suppose that H is as above, f ∈ C(Q T ), and f ≥ 0. Suppose also that u is a viscosity sub-solution and v a viscosity super-solution to (4) and Furthermore, if u and v are viscosity solutions of (4), then Proof. Notice that we can also assume that v is a strict viscosity super-solution. Otherwise We argue by contradiction. Suppose that there exists ( We double the variables as above and denote by (x j , t j , y j , s j ) the maximum point of w j inQ T ×Q T . Since (x 0 , t 0 ) is a strict local maximum for u−v, there exists a strict local maximum (x δ 0 , t δ 0 ) for u δ −v and small enough δ > 0 such that (x δ 0 , t δ 0 ) → (x 0 , t 0 ) as δ → 0. By first choosing a small enough δ > 0 and then large enough j, we have (x j , t j ), (y j , s j ) ∈ Q T , and Denote Then v − ϕ has a local minimum at (y j , s j ) and u δ − φ has a local maximum at (x j , t j ). We consider the two cases: either x j = y j for all j large enough or x j = y j infinitely often.
Case 1: If x j = y j , then Dϕ(y j , s j ) = 0 and D 2 ϕ(y j , s j ) = 0. By the definition of strict viscosity super-solutions, we obtain where we have used the equivalent definition of strict viscosity super-solutions. Since u δ − φ has a local maximum at (x j , t j ) and Dφ(y j , s j ) = 0, D 2 φ(x j , t j ) = 0, we have where we have used Lemma 2.2. Due to (19), the continuity of f and f ≥ 0 in Q T , we obtain that (21) contradicts to (20).
Case 2: If x j = y j , by the parabolic theorem of sums for w j again there exist n × n symmetric matrices X j , Y j such that Y j − X j is positive semi-definite and where η j = j|x j − y j | 2 (x j − y j ) and η j → 0 as j → ∞. By the definition of strict viscosity super-solutions, we get Since u δ − φ has a local maximum at (x j , t j ) , we have by the definition of viscosity sub-solutions. With the help of (22) and (23), we have Letting j → ∞, we have , where we have used the continuity of the functions f , H and H(·, 0) = 0. This is impossible because f ≥ 0 in Q T . This finishes the proof. Remark 6. Theorem 3.1 is also valid for f ≤ 0 in Q T , if we replace 1 − δ by 1 + δ throughout the argument.
The uniqueness Theorem 1.1 follows immediately as a direct consequence of the comparison principle Theorem 3.1.
Based on the comparison principle and the 1-homogeneity of the parabolic operator P u = u t − ∆ N ∞ u − ξ, Du , we can get the following stability result. Then Especially, if g 1 = g 2 = g ∈ C(∂ p Q T ), then Then v j is a viscosity solution of the following initial-boundary problem for j = 1, 2. By the comparison principle and Remark 6, we obtain which means the desired inequality.
With the above theorem in hand we can show the stability property of the solutions to (3), i.e. Theorem 1.2.
Proof of Theorem 1.2. By Theorem 3.2, we obtain .
By the triangle inequality we have Hence 4. Existence theorem. The main purpose of this section is to prove the existence of viscosity solutions to (4) with the initial and boundary data g. The method we adopt is the classical approximation procedure and we follow the argument in [3,20,38,39]. For clarity we will present them schematically. Due to the degeneracy and singularity of the equation, we start with the approximation where L ε,δ u = ε u + 1 This means that the coefficient matrix A ε,δ (p) = a ε,δ ij (p) = εI + p⊗p |p| 2 +δ 2 is uniformly elliptic. Since the approximate equation (25) is uniformly parabolic, the existence of a smooth solution u ε,δ is guaranteed by classical results in [25] under the condition that initial-boundary data g is smooth. In the following we will prove the uniform estimates for u ε,δ independent of 0 < ε ≤ 1, 0 < δ ≤ 1 √ 3 . Then by the compactness method and the stability property of viscosity solutions we obtain that the limit function of u ε,δ as ε, δ → 0 is a viscosity solution of (4). These estimates we require will be obtained by the standard barrier methods.  H(x, t, p) as above, f be continuous in Q T , and let g ∈ C 2,1 (Q T ). Suppose that u ε,δ is a smooth solution satisfying Then there exists a constant C ≥ 0 depending on g t ∞ , Dg ∞ , D 2 g ∞ , ξ ∞ and f ∞ but independent of 0 < ε < 1 and 1 < δ < 1 such that Moreover, if g is only continuous in x and bounded in t , then the modulus of continuity of u ε,δ on Ω × [0, T ] ( for small T ) can be estimated in terms of g ∞ , f ∞ and the modulus of continuity of g(x, 0) in x. Proof.
Step 1. Suppose first that g ∈ C 2,1 (Q T ) and we consider the upper barrier function w + (x, t) = g(x, 0) + λt, where λ > 0 is to be determined. We have Clearly, w + (x, 0) = g(x, 0) for all x ∈ Ω. Moreover, for x ∈ ∂Ω and t > 0, Thus, by the classical comparison principle, we obtain Similarly, by considering also the lower barrier function we obtain the symmetric inequality, and hence the Lipschitz estimate for 0 < t < T and Step 2. Suppose now that g is only continuous in x and let µ(·) be its modulus of continuity. Let us fix a point x 0 ∈ Ω by choosing 0 < ρ < dist(x 0 , ∂Ω), then 2406 FANG LIU we have |g(x, 0) − g(x 0 , 0)| < µ whenever |x − x 0 | < ρ. Let us consider the smooth functions It is easy to check that g − ≤ g ≤ g + on the parabolic boundary ∂ p Q T . Thus if u ± are the unique classical solutions to (25) with boundary and initial data g ± , respectively, we have u − ≤ u ε,δ ≤ u + in Q T by the classical comparison principle again. Since g ± are smooth, we can use estimate (28) to conclude that Then by calculation we get and With these two inequalities (29) and (30) we have This inequality concludes the proof.
Noting that the equation (25) H(x, t, p) = H(p) and f is constant, g ∈ C 2,1 (Q T ) and u = u ε,δ is as in Theorem 4.1, then there exists a constant C ≥ 0 depending on g t ∞ , ξ ∞ and D 2 g ∞ but independent of 0 < ε ≤ 1 such that for all x ∈ Ω and t, s ∈ (0, T ). Moreover, if g is only continuous, then the modulus of continuity of u on Q T can be estimated in terms of g ∞ and the modulus of continuity of g.
Proof. Let u(x, t) = u(x, t + σ), σ > 0. Then both u and u are smooth solutions to (25) =Cσ by the classical comparison principle and Theorem 4.1. This implies the Lipschitz estimate asserted above, and the proof for the case when g is only continuous is analogous.

Remark 7.
It should be pointed out that we need not invoke the translation invariance property of the equation in the proof of the boundary regularity at t = 0. But in order to obtain the interior Lipschitz regularity with respect to t we add the condition that H(x, t, p) = H(p) and f is constant to guarantee the translation invariance property of the equation. Theorem 4.3 (Hölder regularity at the lateral boundary). Let f is continuous in Q T , and let g ∈ C 2,1 (Q T ) . Suppose that u ε,δ is a smooth solution satisfying Then for each 0 < α < 1, there exists a constant C * ≥ 1 depending on α, g ∞ , g t ∞ , f ∞ and Dg ∞ but independent of ε and δ sufficiently small such that for all (x 0 , t 0 ) ∈ ∂Ω × (0, T ) and x ∈ Ω ∩ B r (x 0 ), where r depending only on α and M . Moreover, if g is only continuous, then the modulus of continuity of u ε,δ in x can be estimated in terms of g ∞ , f ∞ , M and the modulus of continuity of g in x. Proof.

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where C * ≥ 1, λ > 0 are to be determined. Then a straightforward computation gives We have shown that w + is a super-solution of (25).
provided C * ≥ r −α (2 g ∞ + f ∞ T ), and in the last inequality we have used the comparison principle.
Step 3. To prove w + ≥ u ε,δ on ∂ b Q * . Case 1. If t * = t 0 , then Q * = (Ω∩B r (x 0 ))×(0, t 0 ), and notice that since u ε,δ = g on the bottom of this cylinder, . Using the comparison principle again, we have Step 4. In conclusion, we have shown that w + ≥ u ε,δ on ∂ p Q * , if we choose Therefore, we have w + ≥ u ε,δ in Q * by the comparison principle. In particular, . Using the lower barrier we get the symmetric inequality. When g is only continuous, the argument is analogous to the previous case. This finishes the proof.
Using again the translation invariance of the equation and the comparison principle, we can extend the Hölder estimate to the interior of the domain, cf. [20,38,39] etc.
Theorem 4.4 (full Hölder regularity in space). Let f be constant, and let g ∈ C 2,1 (Q T ). Suppose that u ε,δ is a smooth solution satisfying

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Then for each 0 < α < 1, there exists a constant C ≥ 1 , depending on α, g ∞ , g t ∞ and Dg ∞ but independent of ε and δ sufficiently small such that for all (x, y) ∈ Ω. Moreover, if g is only continuous, then the modulus of continuity of u ε,δ in x can be estimated in terms of g ∞ and the modulus of continuity of g in x.
Notice that we can not obtain the Lipschitz estimate of solutions to the regularized equation (25) by the barrier method. But it is interesting that we can get the Lipschitz estimate when we remove the viscous term from the approximate equation (25), cf. [12,20,38,39].
Theorem 4.5 (Lipschitz regularity at the lateral boundary when ε = 0). Let f be continuous in Q T , and g ∈ C 2,1 (Q T ) . Suppose that u = u δ is a smooth solution satisfying Then for each 0 < α < 1, there exist constants C * ≥ 1 and r > 0 depending on g ∞ , g t ∞ , f ∞ and Dg ∞ but independent of δ sufficiently small such that for all (x 0 , t 0 ) ∈ ∂Ω × (0, T ) and x ∈ Ω ∩ B r (x 0 ). Moreover, if g is only continuous, then the modulus of continuity of u on ∂Ω × (0, T ) can be estimated in terms of g ∞ , f ∞ and the modulus of continuity of g. Proof.
in Q T and v ≥ g on ∂ p Q T . Hence, we have ≥u(x, t), Step 3. To prove w + ≥ u on ∂ b Q * . Case 1. If t * = t 0 , then Q * = (Ω ∩ B r (x 0 )) × (0, t 0 ), and notice that since u = g on the bottom of this cylinder, Case 2. If t * = t 0 − 1, then Q * = (Ω ∩ B r (x 0 )) × (t 0 − 1, t 0 ). Using the comparison principle again, we have Step 4. In conclusion, we have shown that w + ≥ u on ∂ p Q * and w + is a supersolution of (35), if we choose K ≥ λ + f ∞ + M. Therefore, the comparison principle implies w + ≥ u in Q * . In particular, we get the symmetric inequality. Assume now that g is only continuous. Let us fix a point (x 0 , t 0 ) ∈ Ω × (0, T ) and for a given µ > 0, choose 0 < τ < t 0 such that |g( It is easy to check that otherwise, we have g − ≤ g ≤ g + on the parabolic boundary ∂ p Q T . Thus if u ± are the unique classical solutions to (35) with boundary and initial data g ± , respectively, we have u − ≤ u ≤ u + in Q T by the classical comparison principle again. Since g ± are smooth, we can use estimate (33) to conclude that Theorem 4.6. Let H(x, t, p) = H(p) and f be constant and let g ∈ C 2,1 (Q T ). Suppose that u δ is a smooth solution satisfying (32). Then there exists a constant C * ≥ 1 depending on g ∞ , g t ∞ and Dg ∞ but independent of δ sufficiently small such that for all (x, y) ∈ Ω) and t ∈ (0, T ) . Moreover, if g is only continuous, then the modulus of continuity of u δ in x on Ω × (0, T ) can be estimated in terms of g ∞ and the modulus of continuity of g in x and t.
The existence theorem follows now easily by piecing out the results in Theorems 4.2, 4.4 and 4.6, and using the standard compactness arguments and the stability properties of viscosity solutions.
Proof of Theorem 1.3: If g ∈ C 2,1 (Q T ) and u ε,δ is the unique smooth solution to Theorems 4.2 and 4.4 and the comparison principle imply that the family of functions {u ε,δ } is equi-continuous and uniformly bounded. Therefore, up to a subsequence, u ε,δ → u δ as ε → 0 and u δ is the unique viscosity solution to (32) by the stability properties of viscosity solutions. Next by Theorems 4.2 and 4.6 we conclude that u δ → u uniformly as δ → 0 and u is a viscosity solution to (4) with initial and boundary data g by the stability properties of viscosity solutions again. The existence for a general continuous data g follows by approximating the data by smooth functions and using Theorems 4.2 and 4.6 and the stability properties of viscosity solutions again.
Remark 8. Theorems 1.1 and 1.3 are also valid for unbounded domain Ω. In fact, we can let Ω r = Ω ∩ B r (0), Q r = Ω r × (0, T ] and g r : where χ r (x) = χ( x r ) and χ ∈ C ∞ c (R n ) satisfies χ(x) = 1 if |x| ≤ 1 2 , χ(x) = 0 if |x| ≥ 1. Then in Q r there exists a unique viscosity solution with initial-Dirichlet data g r . From the assumptions on g r , the estimates for u r and the stability property of the viscosity solutions we can obtain the result by letting r → ∞.
5. Lipschitz regularity. In this section, we first derive the interior gradient estimate for smooth solutions of the approximating equation where L ε,δ is as in (26). Then we use the compactness method to establish the interior Lipschitz estimate for viscosity solutions of equation (1). The idea comes from [12,20], but the details are quite different due to the transport and inhomogeneous terms.

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is a smooth solution of the approximating equation (37) in Q T , then there exists a positive constant C depending only on n, which is independent of ε ∈ (0, 1] and δ ∈ (0, 1 √ 3 ], such that the estimate Proof. For simplicity we denote u ε,δ by u. We construct an auxiliary function of the form where v = |Du| 2 + δ 2 , κ > 0 to be determined, and ζ is a smooth positive cut-off function that vanishes on ∂ p Q T . Let (x 0 , t 0 ) be a point where w attains its maximum inQ T . We first assume (x 0 , t 0 ) / ∈ ∂ p Q T . At the maximum point (x 0 , t 0 ), we have where a ε,δ ij is the operator defined in (27). The equality in (40) shows that at (x 0 , t 0 ). Since (x 0 , t 0 ) / ∈ ∂ p Q T , the denominator ζ(x 0 , t 0 ) = 0 so that the equality (41) makes sense at (x 0 , t 0 ). We next calculate the inequality in (40). Since the matrix a ε,δ ij (Du(x, t)) is positive definite for all points (x, t), by a direct calculation, we have at (x 0 , t 0 ), where we have used the (37) in the last equality. We analyze all the terms on the right hand side of (42). From the definition of a ε,δ ij , we have By (41), we have in the last inequality we have used Young's inequality and 0 < ε < 1.
By direct calculation it is easy to show that v t = 1 We differentiate the equation (37) with respect to x k to obtain (u k ) t − D k (a ε,δ ij (Du))u ij − a ε,δ ij (Du)(u k ) ij = ξ i,k u i + ξ i u ik + f k , where ξ i,k = ∂ξi ∂x k . Then we multiply u k v in (46) and sum with respect to k from 1 to n to get Noting that D k (a ε,δ ij (Du)) = D k εδ ij + Substituting (48) into (47), then we obtain for almost every (x, t) ∈ B r (x, t). Due to the stability properties of the viscosity solutions [11], we obtain thatũ satisfies in the viscosity sense. By the comparison principle Theorem 3.1, we haveũ = u in B r (x, t). Hence, we have for a.e. (x, t) ∈ B r (x, t). Then by Lebesgue-Besicovitch Differentiation Theorem, we obtain |Du(x, t)| ≤ lim (dist((x, t), ∂ p Q T )) for almost every point (x, t) ∈ Q T . The Lipschitz estimate asserted above follows.