1 차계수보존방정식해들의상호변화 횟수의비증가성 Nonincreasing number of alternate changes of solutions for a one dimensional scalar conservation law

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1 석사학위논문 Master s Thesis 1차계수보존방정식해들의상호변화횟수의비증가성 Nonincreasing number of alternate changes of solutions for a one dimensional scalar conservation law 윤창욱 ( 尹彰旭 Yoon, Chang-Wook) 수리과학과 Department of Mathematical Sciences KAIST 2010

2 1 차계수보존방정식해들의상호변화 횟수의비증가성 Nonincreasing number of alternate changes of solutions for a one dimensional scalar conservation law

3 Nonincreasing number of alternate changes of solutions for a one dimensional scalar conservation law Advisor : Professor Kim, Yong Jung by Yoon, Chang-Wook Department of Mathematical Sciences KAIST A thesis submitted to the faculty of the Korea Advanced Institute of Science and Technology in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematical Sciences Daejeon, Korea Approved by Professor Kim, Yong Jung Advisor

4 1 차계수보존방정식해들의상호변화 횟수의비증가성 윤창욱 위논문은한국과학기술원석사학위논문으로학위논문심사 위원회에서심사통과하였음 년 6 월 1 일 심사위원장김용정 ( 인 ) 심사위원권길헌 ( 인 ) 심사위원김홍오 ( 인 )

5 MMAS 윤창욱. Yoon, Chang-Wook. Nonincreasing number of alternate changes of solutions for a one dimensional scalar conservation law. 1차계수보존방정식해들의상호변화횟수의비증가성. Department of Mathematical Sciences p. Advisor Prof. Kim, Yong Jung. Text in English. Abstract In this paper, solutions of one dimensional conservation law u t + f(u) x = 0 are treated. This PDE governs various one- dimensional phenomena involving fluid dynamics and models the foundation and propagation of shock waves. In general, the conservation law has no classical solutions due to discontinuities. i.e. shock waves. So, vanishing viscosity method is used for observing the behavior of solutions rather than handling the original equation. By adding a second order derivative with small coefficient, we can construct a second-order semilinear parabolic equation which has smooth solutions. With some theorems about regularity of solutions, zero sets and maximum-principle, nonincreasing property of intersection points w.r.t time t of the approximated equation can be proved. And as the viscosity coefficient tends to zero, we obtain the similar property of solutions as limits of solutions of parabolic equation. That is, the number of alternate changes of two solutions is nonincreasing in t. Here the alternate changes means the sign changes of difference of two solutions. i

6 Contents Abstract Contents i iii 1 Introduction 1 2 Survey on the scalar conservation law Discontinuity of solutions for the conservation law The concept of entropy Nonincreasing number of zeros of a solution for a perturbed equation Regularity of solutions for perturbed equation Zero set of a solution of a linear parabolic equation Maximum principle in unbounded domains Application of theorems Limiting process Existence, uniqueness of an entropy solution Entropy condition Main theorem Conclusion and further research 19 Summary (in Korean) 20 References 21 iii

7 1. Introduction Consider the one-dimensional scalar conservation law { ut + f(u) x = 0 in x R, t > 0 u(x, 0) = u 0 (x) x R where u 0 L, f C 2 (R) and f > 0(strictly convex). (1.1) It is typical that this equation admits discontinuous solutions; i.e. shock waves, so that the equation must be understood in some generalized sense. Hence we will survey the basic notion of solutions for (1.1) in section 2. By the discontinuity, it is hard to observe the behavior of solutions. perturbed equation for ɛ > 0 { u ɛ t + f(u ɛ ) x = ɛu ɛ xx in x R, t > 0 u ɛ (x, 0) = u 0 (x) x R So we consider a (1.2) Here, the new term ɛu ɛ xx represents a small diffusion. This technique adding a diffusion term is called the vanishing viscosity method. Since the initial condition is just L, it can not be guaranteed that a solution of perturbed equation is sufficiently smooth. So, in section 3 we will investigate the regularity of solutions of (1.2). Then (1.2) has smooth solutions so that it should be easier to understand. As models from physics based on conservation laws, these equations usually inherit scaling invariance. In view of this geometric nature, zero set argument is a good tool of the asymptotic analysis of parabolic PDE. It turned out that the structure and time-evolution of intersections of two different solutions can reveal the actual asymptotic behavior of general solutions. In other words, some crucial properties of general solutions could be described by using intersection comparison with solutions. Let v,w be given that are solutions of (1.2) with initial condition v 0,w 0, respectively. i.e. Set e ɛ = v ɛ w ɛ. From (1.3), we get { v ɛ t + f(v ɛ ) x = ɛv ɛ xx, v ɛ (x, 0) = v 0 (x) w ɛ t + f(w ɛ ) x = ɛw ɛ xx, w ɛ (x, 0) = w 0 (x) { e ɛ t + f (v ɛ )e ɛ x + f (v ɛ ) f (w ɛ ) v ɛ w wxe ɛ ɛ = ɛe ɛ ɛ xx e ɛ (x, 0) = e ɛ 0(x) = v 0 (x) w 0 (x) (1.3) (1.4) 1

8 Then, equation (1.4) seems to be parabolic in some sense. In section 3, zero set theory on the parabolic PDEs will be introduced to be applied to our problem. Maximum principle is also treated. At the end of section 3, we will prove that the number of zeros of (1.4) (intersections of v ɛ, w ɛ ) can not be increasing in t > 0. By the nonlinearity of conservation law, it needs new concept to discriminate the meaningful solution of (1.1) from other many candidates. In section 4, we will study about entropy condition of conservation law and as ɛ goes to 0, solutions of (1.1) is obtained as limits of solutions of (1.2) by using theorems in section 3. Then, we shall obtain the main result as desired. 2

9 2. Survey on the scalar conservation law The goal of this chapter is to investigate some basic background about the scalar conservation law in one space dimension. 2.1 Discontinuity of solutions for the conservation law Again, consider (1.1) { ut + f(u) x = 0 in x R, t > 0 Let us see the characteristics Along such a curve, u(x, 0) = u 0 (x) dt ds = 1, x R dx ds = f (u) du ds = u dt t ds + u dx x ds = u t + f (u)u x = 0 So, u is constant along the characteristics. Since the slope of the characteristics is 1/f (u), the characteristics are straight line, having slope determined by u 0 (x). Thus, if there are points x 1 < x 2 with 1 f (u 0 (x 1 )) < 1 f (u 0 (x 2 )) then the two characteristics will meet in t > 0. Thus the solution must be discontinuous. Since we can not in general find a smooth solution of (1.1), we must adopt a less regular function u solving this problem. We multiply (1.1) by φ, integrate over t > 0, and using integration by parts, we get (uφ t + f(u)φ x )dxdt + u 0 φ(x, 0)dx = 0 (2.1) t 0 R Definition 1. We say that u L 1 loc (R (0, )) is an weak solution of (1.1) provided that (2.1) holds for all φ C 1 0(R (0, )). The explanation of this section gives us basic notion of a weak solution for a scalar conservation law. R 3

10 2.2 The concept of entropy Since a weak solution of (1.1) is not a classical solution in usual, we should verify that what kind of discontinuity is admissible. Let Γ be a smooth curve across which u has a jump discontinuity. Then the following equality σ[u] = [f(u)] holds along the discontinuity curve, where [u] = u l u r = jump in across the curve Γ, [f(u)] = f(u l ) f(u r ) =jump in f(u), σ = speed of discontinuity. This equality is called Rankin-Hugonit condition. But weak solutions are not unique in general. So we must have some mechanism to pick out the physical relevant solution, and guarantee the uniqueness of a weak solution. The idea is to consider a additional conservation law. Let η, q be given C 1 functions such that η (u)f (u) = q (u) provided that (1.1) holds for a any smooth function u. By multiplying (1.1) by η (u), we get u t η (u) + η (u)f(u) x = 0 η(u) t + η (u)f (u)u x = 0 η(u) t + q(u) x = 0 So, η, q automatically satisfy an additional conservation law η(u) t + q(u) x = 0 for any smooth u. But this additional conservation law is too bad to select a proper solution among all weak ones if u is not smooth enough. Therefore, we should replace the equality by a inequality which is possible to select a proper weak solution as followings. η(u) t + q(u) x 0 and η is convex Definition 2. The pair of smooth functions in C 1, (η, q) is called and entropy-entropy flux pair of the conservation law (1.1) such that η (u)f (u) = q (u) and η is convex for all u. 4

11 Definition 3. u is called an entropy solution of (1.1) if u satisfies η(u)φ t + q(u)φ x dxdt 0 0 R for φ C c (R (0, )), φ 0 for each entropy-entropy flux pair (η, q) such that η(u) t + q(u) x 0 is fulfilled Indeed, these definitions are motivated by the vanishing viscosity methods. Since we regard a unique solution u as a limit of u ɛ, the inequality comes from taking limit. In the process of using the viscosity method, we also assume for a moment that some kind of convergence of u ɛ to u has already been proved. These will be treated later on section

12 3. Nonincreasing number of zeros of a solution for a perturbed equation In this chapter, we will observe some theorems in order to apply them to (1.2),(1.4), and get a theorem about nonincreasing property for a solution of (1.4) 3.1 Regularity of solutions for perturbed equation The aim of this section is to present a theory of existence, uniqueness, and regularity of a weak solution to the problem (1.2) Let us return to (1.2) { u ɛ t + f(u ɛ ) x = ɛu ɛ xx in x R, t > 0 u ɛ (x, 0) = u 0 (x) x R Consider the following Theorem. Theorem 1. Let f C m (R) and let u 0 W m,2 (R) L (R), m N. Then there exists a unique weak solution u ɛ of the problem (1.3) satisfies, for all T > 0, u ɛ L 2 (0, T ; W m+1,2 (R)) C(0, T ; W m,2 (R)) Further, for k N, 2k m we have k u ɛ t k L2 (0, T ; W m+1 2k,2 (R)) C(0, T ; W m 2k,2 (R)) while for 2k = m + 1 we have k u ɛ t k L2 (0, T ; L 2 (R)) norm Note that L p (I; X) means the space of all measurable functions u : Ī X for which the u L p (I,X) T 0 u(t) p X dt 1 p, p < 6

13 Similarly, C k (I, X) contains all continuous functions u : Ī X for which all time-derivatives, up to order k can be continuously extended to Ī. In particular, u C k (I,X) By Sobolev s imbedding theorem, if u ɛ k s=0 sup s u(t) t s X L 2 (0, T ; W m+1,2 (R)) C(0, T ; W m,2 (R)), there exists a imbedding from W m,2 (R) to C m 2 B (R) for m 2. Similarly, if u ɛ t L 2 (0, T ; W m+1 2,2 (R)) C(0, T ; W m 2,2 (R)), then W m 2,2 (R) can be also imbedded to C B (R) for m 3. Existence of this embedding to a continuous space means that u ɛ,u ɛ t can be redefined to a new smooth function as far as f is sufficiently smooth. Theorem 1 does not consider the convexity of f. So, we get a restricted result such that in order to enhance the regularity of u ɛ, more regular initial condition u 0 is required. It is proved by Olienik that the convexity of f makes u ɛ smooth enough with just L -condition of u 0 as following. Theorem 2. Let u 0 L (R), f C 2 (R) and f > 0,. Then there exists a unique solution u ɛ of (1.3) which has all continuous derivatives. And u ɛ satisfies the initial condition in the following sense. If u 0 (x) is continuous at x = x 1, then R (φ(x, t)u ɛ (x, t) φ(x, 0)u 0 (x))dx 0 as t 0 lim t 0,x x 1 u ɛ (x, t) = u 0 (x 1 ) Then, it can be guaranteed that the solution u ɛ does not change its sign with jump. 3.2 Zero set of a solution of a linear parabolic equation In this section, we state results about zero sets for linear parabolic equation. Let I, J be bounded intervals in R. Consider a linear parabolic equation u t = a(x, t)u xx + b(x, t)u x + c(x, t)u (3.1) in D = I J, where a µ > 0 in D. Let u be a solution of (2.1). Define the number of zeros of u(, t) to be supremum over all k such that there exists x 1 < x 2 < < x k with u(x i, t) u(x i+1, t) < 0 (i = 1, 2,, k 1) 7

14 Let z(t) denote this supremum at t. Let u C (D) C( D) be a solution of (3.1) with C -coefficients a, b, c. Then any zero of u has finite multiplicity. With these facts, the following theorem holds. Theorem 3. If u(, t 0 ) has a multiple zero at x 0 with multiplicity m, then z(t) drops at t 0 and for sufficiently small ɛ > 0, z(t 0 ɛ) z(t 0 + ɛ) = { m if m is even m 1 if m is odd Proof. Without loss of generality, assume that u does not change its sign on I J, and u(x, 0) has a multiple zero at 0 with multiplicity m. Then we have u(x, o) = A x m +O(x m+1 ). By Taylor s expansion, u(x, t) = u(x, 0) + u t (x, 0)t + 1 n! Dn t u(x, 0)t n + O(t n+1 ) (3.2) where n = m m 1 2, if m is even, n = 2, if m is odd. From (3.1), we get u t (x, 0) = a(x, 0)u xx (x, 0) + b(x, o)u x (x, o) + c(x, 0)u(x, o) = m! a(0, 0) A (m 2)! xm 2 + O(x m 1 ) u tt (x, 0) = a(0, 0) 2 A. D n t u(x, 0) = a(0, 0) n A So, it follows from (3.2) that u(x, t) = A j=0 m! (m 4)! xm 4 + O(x m 3 ) m! (m 2n)! xm 2n + O(x m 2n+1 ) n a j 0 m! (m 2j)! xm 2j t j + O( x m+1 + x m 1 t + + x m 2n+1 t n + t n+1 ) By substituting x = z a(0, 0)( t) for t < 0, it follows that 1 A a(0, 0) m 2 ( t) m 2 u(x, t) = Pm (z) + O( t) P m (z) = n ( 1) n m! (m 2j)!j! zm 2j j=0 where right-hand side is the Hermite polynomial which has exactly m simple zeros. With a similar expansion for u x (x, t), it turned out that u(x, t) has only m simple zeros 8

15 {x i (t)} such that x i (t) = z i t + O( t) 0 as t 0 for t < 0 and ux (x i (t), t) 0. Therefore, there are m simple zero curves meeting each other at (0, 0). Similarly, by substituting x = z a(0, 0)t for t > 0, it follows that 1 A a(0, 0) m 2 t m 2 u(x, t) = Qm (z) + O( t) Q m (z) = n j=0 m! (m 2j)!j! zm 2j If m is even, Q m has positive minimum at z = 0. If m is odd, Q m (0) = 0, Q m is strictly increasing. Therefore, there is only one zero curve if m is odd, and no zeros if m is even. Furthermore, it turned out that a similar conclusion is valid for more general equation by Angenent. Consider u t = a(x, t)u xx + b(x, t)u x + c(x, t)u x R, 0 < t < T (3.3) where a, a 1, a t, a x, a xx L b, b t, b x L c L with a priori bound u(x, t) Ae Bx2 for some constants A, B. Let u be a solution of (3.3). Then the main results concerning u(x, t) are the following. Theorem 4. For each t (0, T ), the zero set of u(x, t) Z t = {x R u(x, t) = 0} is a discrete subset of R Theorem 4 says that there is no accumulation point of zeros. In other words, all zeros are isolated. Theorem 5. If at (x 0, t 0 ) both u and u x vanish then there is a neighborhood N = [x o ɛ, x 0 + ɛ] [t o δ, t 0 + δ] of (x 0, t 0 ) such that 1) u 0 on the sides of N (i.e. u(x 0 ± ɛ, t) 0 for t t 0 δ). 2) u(, t + δ) has at most one zero in [x o ɛ, x 0 + ɛ]. 3) u(, t δ) has at least two, but at most finite zeros in [x o ɛ, x 0 + ɛ]. 9

16 Roughly speaking, at t = t 0, if u(, t) has a multiple zero, then for all t 1 < t 0 < t 2, z(t 1 ) > z(t 2 ). Note that if u(, t 0 ) have a zero of multiplicity n(< ) at x 0, then there exists n single zero curves for t < t 0 near (x 0, t 0 ). And Z = {0 < t < T u x (x, t) + u(x, t) > 0 for all x R} is dense in (0, T ). (i.e. a multiple root appears in a moment.) Furthermore, let u : [x 0, x 1 ] [0, T ] R be a classical continuous solutions of (2.2) such that u(x i, t) 0 (i = 0, 1, 0 t T ) Then Theorem 4, 5 are also satisfied. In particular, in Theorem 4, discrete is changed by finite. 3.3 Maximum principle in unbounded domains Consider the nonlinear parabolic differential equation in one space variable. P [u] := u t f(t, x, u, u x, u xx ) Let G be a unbounded domain, lying in the strip 0 < t < T, and let Γ be a parabolic boundary of G which does not intersect with t = T. Theorem 6. Let f(t, x, z, o, r) be nondecreasing in r and let P [constant] = 0, i.e. f(t, x, z, 0, 0) = 0 Furthermore, we assume that there are positive constants K, L, c such that f(t, x, z, p, 0) L x p for x K, p c. If u is bounded and continuous in Ḡ and u t,u x,u xx exist in G, then for P [u] = 0 in G, Maximum-minimum principle for u holds. Here the maximum(minimum) principle for u is the statement that supremum(infimum) of u on G and on Γ are equal. Proof. Assume that P [u] 0, T <. Let M be the supremum of u on Γ. Define v := M + e Lt (ɛ + δh(x)), h(x) = max( x K, 0). Claim : u v in G for any ɛ, δ > 0. 10

17 Since u is bounded while v increases in x, u is strictly smaller that v with large x. Assume that there exists a point in G at which u > v holds. So, there exists (x 0, t 0 ) G such that (1) u < v for t < t 0 (2) u = v and u t v t at (x 0, t 0 ) (3) u(x, t 0 ) v(x, t 0 ) Now choose δ > 0 such that δk > ɛ, δe Lt < c. Case (a) : x 0 > K. Let d = v u, then d has a minimum at (x 0, t 0 ) with respect to x. Hence, at (x 0, t 0 ), Since P [u] 0, But by (2), Which is contradiction. Case (b) : x 0 K. d x = 0 u x = v x = ±δe Lt0 d xx 0 u xx v xx = 0 u t f(t 0, x 0, u, u x, u xx ) f(t 0, x 0, u, v x, 0) L x 0 δe Lt 0 u t v t = Le Lt 0 (ɛ + δ( x 0 K) > L x 0 δe Lt 0 Then v(x, t) = M + e Lt 0 ɛ. So, such (x 0, t 0 ) in (a) dose not depend on δ. In other words, (x 0, t 0 ) is the same point for all small δ > 0. By letting δ 0 in (3), we obtain where the equality holds for x = x 0. Hence u x = 0, u xx 0 at (x 0, t 0 ). Then u(x, t 0 ) lim δ 0 v(x, t 0 ) = M + ɛe Lt 0 u t f(t 0, x 0, u, u x, u xx ) f(t 0, x 0, u, 0, 0) = 0 But from (2), u t v t = ɛle Lt0 > 0 at (x 0, t 0 ) Which is contradiction. Therefore, the claim is true. Since δ, ɛ is arbitrary, the maximum principle is proved. Similarly, minimum principle for P [u] 0 is also proved. By combining two results, we can conclude that the maximum-minimum principle holds for P [u] = 0. 11

18 3.4 Application of theorems By using previous theorems, we can get a new theorem. Theorem 7. Assume that v 0, w 0 L (R), f C 2 (R) and f > 0. Then the number of intersections of solutions v ɛ,w ɛ for (1.2) with v 0, w 0 respectively, is nonincreasing in t > 0. Proof. Consider (1.4) e ɛ t + f (v ɛ )e ɛ x + f (v ɛ ) f (w ɛ ) v ɛ w ɛ wxe ɛ ɛ = ɛe ɛ xx Due to the continuity of v ɛ and w ɛ, the solution e ɛ of (1.4) also can not change its sign with discontinuity (in section 3.1). So, in order to understand the behavior of e ɛ, it is sufficient to observe e ɛ near its zeros. Since we assume that e ɛ,v ɛ,w ɛ are given, Theorem 4, 5 can be applicable to (1.4) in a point of view that (1.4) is regarded as a second order linear partial differential equation. Before using Theorem 4 and 5, we should check whether each coefficients of (1.4) have no defect. Assume that (x 0, t 0 ) be a zero of e ɛ (i.e. v ɛ (x 0, t 0 ) = w ɛ (x 0, t 0 ) ). First, lim f (v ɛ ) = f (v ɛ (x 0, t 0 )) (x,t) (x 0,t 0) lim f (v ɛ ) x = f (v ɛ (x 0, t 0 ))vx(x ɛ 0, t 0 ) (x,t) (x 0,t 0 ) lim f (v ɛ ) t = f (v ɛ (x 0, t 0 ))vt(x ɛ 0, t 0 ) (x,t) (x 0,t 0 ) Since we already know the regularity of v ɛ, v ɛ x, v ɛ t, these make sense. Also, f (v ɛ ) f (w ɛ ) lim (x,t) (x 0,t 0) v ɛ w ɛ wx ɛ = f (v ɛ (x 0, t 0 ))vx(x ɛ 0, t 0 ) Since the boundedness conditions of the each coefficients of (1.4) are satisfied near a zero set, we can conclude that the number of zeros of e ɛ are noncincreasing in t, by Theorem 4, 5. To complete the proof, we exclude the case that there exist no zero curve from infinity. Suppose that there is a zero curve Γ : x = γ(t) of e ɛ such that γ as t t 0. Consider a region G = {(x, t) : t 0 < t < t 1, x < γ(t)} which is the left side of Γ. The sign of e ɛ in G doesn t matter. Now we modify (1.4) such that Let e ɛ t ɛe ɛ xx + f (v ɛ )vx ɛ f (w ɛ )vx ɛ vx ɛ wx ɛ e ɛ x = 0 b ɛ (x, t) = f (v ɛ )v ɛ x f (w ɛ )v ɛ x v ɛ x w ɛ x 12

19 If (x 0, t 0 ) be a zero of e ɛ. (i.e. v ɛ (x 0, t 0 ) = w ɛ (x 0, t 0 ).) then lim (x,t) (x 0,t 0 ) bɛ (x, t) = f (v ɛ (x 0, t 0 )) Since b ɛ is bounded near Γ, we may assume that b ɛ is bounded in G. To apply theorem 6, we set P [e ɛ ] = e ɛ t f(t, x, e ɛ, e ɛ x, e ɛ xx) = 0 f(t, x, e ɛ, e ɛ x, e ɛ xx) = ɛe ɛ xx b ɛ e ɛ x Then P [constant] = 0 and f(t, x, z, 0, r) is nonincreasing r. Moreover, since b ɛ is bounded in Γ, there exist K, L, c such that f(t, x, z, p, 0) = b ɛ p L p x in G where x K, p c. Later, a theorem(actually, theorem 8) will be introduced such that u ɛ u 0 Hence the boundedness for solutions is fulfilled. All conditions for theorem 6 are satisfied. Then by Maximum principle, e ɛ = 0 in G. Which is contradiction. In the same way, the assumption γ + as t t 0 also leads to a contradiction. This completes the proof of theorem 7. 13

20 4. Limiting process In the previous chapter, we showed that for any ɛ > 0, there exists a unique classical solution u ɛ of perturbed equation (1.2) provided that f is sufficiently regular. Now we are interested in the limiting process as ɛ Existence, uniqueness of an entropy solution In this section, under the assumptions that u 0 L (R), f C 1 (R)(f may not be convex), we will show that the existence and uniqueness of entropy solution. For the existence, the following theorems are already known. Theorem 8. Let u 0 L (R) and f C 1 (R). Then there exists a unique solution u ɛ L of (1.2) such that and 0 R u ɛ (t) L u 0 L for almost all t > 0 [u ɛ φ t + ɛu ɛ φ xx + f(u ɛ )φ x ]dxdt + R u 0 (x)φ(0, x)dx = 0 Theorem 8 imply the existence of weak solutions u ɛ of (1.2). Since the theorem does not assume the convexity of f, it is generalized version of theorem 2. Theorem 9. Let u 0 L (R) and f C 1 (R). And let u ɛ be solutions of the previous theorem. Then there exists a subsequence u ɛ k of u ɛ an u L (R) which satisfies u ɛ k u a.e. as ɛ k 0 Theorem 9 shows the existence of u as the limit of weak solutions u ɛ. Then the following theorem holds. Theorem 10. Let u 0 L (R) and f C 1 (R). Then (1.1) has and entropy solution u L (R) such that u(t) L u 0 L Proof. Let η be a C 2 entropy. Multiplying (1.2) by η (u ɛ ), we get η(u ɛ ) t + q(u ɛ ) x = ɛη (u ɛ )u ɛ xx = ɛη(u ɛ ) xx ɛη (u ɛ )(u ɛ x) 2 14

21 where η (u)f (u) = q (u). Since η is convex, ɛη (u ɛ )(u ɛ x) 2 0 Then, η(u ɛ ) t + q(u ɛ ) x ɛη(u ɛ ) xx The weak formula of this inequality for φ Cc reads [η(u ɛ ) t + q(u ɛ ) x ]φdxdt η(u ɛ )ɛη(u ɛ ) xx φdxdt 0 R 0 R By integration by parts, we get η(u ɛ )φ t + q(u ɛ )φ x dxdt ɛ 0 R 0 R η(u ɛ )φ xx dxdt Under assumptions that u ɛ u almost everywhere and u ɛ C, let ɛ 0. Then, by Dominated convergence theorem, we obtain, η(u)φ t + q(u)φ x dxdt 0 0 R Thus u is an entropy weak solution to (1.2) The theory of entropy observed until now is applicable to the multi-dimensional case such that { ut + divf(u) = 0 in x R d, t > 0 u(x, 0) = u 0 (x) x R d where f = (f 1, f 2,, f d ) C 1 (R) d, f j C 1 (R), u : R d R. Oleinik proved a theorem about more specific condition(f is convex). Theorem 11. Let u(x, t) be the entropy solution of (1.1). And f is convex. Then u ɛ u a.e. as ɛ Entropy condition Under the assumption f > 0, we could say one more thing about discontinuity. Suppose that at some point of discontinuity u has different left and right limits, u l and u r. If we observe characteristics from the discontinuity along backward direction in time, they will 15

22 not meet any others. Since the solution u takes the constant t 0 = u 0 (x 0 ) along the projected characteristic In view of this formula, we can deduce that Since f > 0, we get an inequality t(s) = (f (u 0 (x 0 ))s + x 0, s) f (u l ) > σ > f (u r ) u l > u r This means that the jump must always be downward as we increase in x. Exlample 1. Assume that u 0 of (1.1) is a piecewise-constant initial function such that { ul if x < 0 u 0 = u r if x > 0 which is called Riemann s problem. Then, if u l > u r, the unique entropy solution is { x ul if t u 0 = < σ x u r if t > σ where σ is defined as above. And, if u l < u r, the unique entropy solution is x u l if t < f (u l ) u 0 = G( x t ) if f (u l ) < x t < f (u r ) x u r if t > f (u r ) where G = (f ) 1. i.e. in the first case, u l, u r are separated by a shock wave and in the second case, the gap between u l, u r is filled with a smooth function which is called a rarefaction wave. More precisely, here is a theorem about entropy condition. Theorem 12. Let u 0 L (R), f C 2 (R) and f > 0. Then, there exists a solution u of (1.1) as following. There is a constant E > 0, such that u(x + a, t) u(x, t) a < E t for all a > 0, t > 0. x R 16

23 This inequality interprets the regularity of a solution u in the sense that u(, t) has a locally bounded total variation for any t > 0. To confirm this claim, let k be a constant such that k > E t, and let v(x, t) = u(x, t) kx. For a > 0, v(x + a, t) v(x, t) = u(x + a, t) u(x, t) ka a( E t k) < 0 for all x R Since v is nonincreasing, v has locally bounded total variation. The same thing also true for kx. Hence, the claim that u has locally bounded total variation holds. So, we can conclude that u(.t) has at most a countable number of discontinuities for any t > 0, even if the initial condition u 0 is just in L. 4.3 Main theorem Now we will state the main theorem of this paper and prove it. Theorem 13. Let v 0, w 0 L (R), f C 2 (R) and f > 0. For solutions v, w of (1.1) with v 0, w 0, the number of alternate changes of v, w is nonincreasing in t. Proof. By Theorem 2 and 8 11, there exists a unique entropy solution v, w for (1.1) and sequences of smooth solutions {v ɛ }, {w ɛ } for (1.2) which converges to v, w almost everywhere, respectively. Let e(x, t) = v(x, t) w(x, t), e ɛ (x, t) = v ɛ (x, t) w ɛ (x, t). We already know that v, w has only jumps with downward direction. So e may have jumps with upward direction. If v has a jump at (x, t ), then e have a jump with downward direction. Similarly, if w has a jumps at (x, t ), then e has a jump with upward direction. Finally, if both v, w have jumps at (x, t ), set v l (x, t ) w l (x, t ), v r (x, t ) w r (x, t ). Then e has a jump from (v l w l )(x, t ) to (v r w r )(x, t ). Even though e allows both type of jumps, e can not have an isolated discontinuity. Under this property of e, suppose that if e has a jump at (x, t ) (i.e. e l (x, t ) e r (x, t )), then there exists (x, t ) in any neighborhood of (x, t ) at which e has the same sign to e l (x, t ) or e r (x, t ) and e ɛ (x, t ) e(x, t ), because the number of shocks is at most countable. So, without loss of generality we may assume that e(x, t 1 ) = v(x, t 1 ) w(x, t 1 ) > 0 for all x I = (x 1, x 2 ) e(x i, t) = v(x i, t) w(x i, t) > 0 for x i I, i = 1, 2, t [t 1, t 2 ] e(y, t 1 ) > 0, e(y, t 2 ) < 0 for some y I 17

24 Since v ɛ, w ɛ of (1.2) with v 0, w 0 pointwisely converge to v, w, respectively, there exists a small ɛ > 0 such that e ɛ = v ɛ w ɛ has the same sign to e at (x i, t j ), (y, t j ), i = 1, 2, j = 1, 2. Then, at least two zeros occur at some (y 1, t 2 ), (y 2, t 2 ) for x 1 < y 1 < y < y 2 < x 2. Hence the number of zeros of e is increasing in t. Which is contradiction to Theorem 7. If one consider the Riemann problem(example 1), a solution of the scalar conservation propagate along the characteristic and discontinuity of that also moves like a traveling wave in case of u l > u r. So, we can make sure that two solutions of the scalar conservation law could have an interval in which values of these solutions are equivalent under some proper initial conditions. In view of this fact, thinking about the number of intersections is meaningless. Instead we could say similar argument such that the number of alternate changes of two different solutions of the scalar conservation law in nonincreasing in t. 18

25 5. Conclusion and further research In the heat equation u t = u xx which is the simplest example of second order linear parabolic equation, long time behavior of solutions has dissipation effect. So, it seems to be natural that the number of zeros of a solution is nonincreasing in t. Actually, for more generalized parabolic case, nonincreasing property of zero sets has already proved. The last theorem says that we can construct the similar theory of nonincreasing property of solutions for the scalar conservation law. It may play a key role to reveal the steepness of the fundamental solution. More study about the steepness should be accomplished further. And the result is remarkable for the reason that nonincreasing property of solutions is admissible in the nonlinear PDEs. On the other hand, there is no theory about multi-dimensional case as yet. So, building a theory for generalized problem can be a good research. 19

26 요약문 1차 계수 보존 방정식 해들의 상호변화 횟수의 비증가성 본 논문에서는 1차 계수 보존 방정식 ut + f (u)x = 0 을 다룬다. 이 편미분방정식은 점성이 없거나 상수 엔트로피를 가지는 유체 역학등 쇼크가 생성, 확산되는 여러 물리 현상을 설 명하고 모델링한다. 일반적으로 보존 방정식은 쇼크라 불리는 불연속점이 발생하므로 방 정식 자체가 가지는 미분의 개념을 그대로 적용시키기 어렵다. 따라서 본래 방정식에 작은 점성계수가 포함된 2차 미분항을 삽입한 뒤 점성계수를 0으로 수렴시켜 극한을 취해보는 vanishing viscosity method를 사용한다. 2차 미분항을 추가한 방정식은 충분히 미분이 가 능한 형태의 해를 가지면서 parabolic 방정식과 유사한 형태의 방정식이 된다. parabolic 편 미분 방정식에 대해서는 초기 값이 서로 다른 두 해의 교차점이 시간이 지날 수록 증가하지 않는다는 사실이 Angenent에 의해 이미 증명되어 있다. 이 정리와 maximum-principle을 사 용하면 2차 미분항을 추가한 방정식에 대해서도 같은 결론을 내릴 수 있다. 여기서 점성계 수를 0으로 수렴시킬때 극한으로서 얻을 수 있는 해에 대해서도 비슷한 맥락의 결과를 얻 을 수가 있는데, 단 보존 방정식에서는 초기값에 따라 두 해의 값이 일치하는 구간이 생길 수가 있으므로 교차점의 갯수가 증가하지 않는 것이 아닌 두 해의 상호 대소관계가 변화하 는 구간의 갯수가 증가하지 않는다 라고 결론 지을 수 있다. 20

27 References [1] Joel Smoller. (1983) Shock Waves and Reaction-Diffusion Equations. Springer-Verlag. New York. Heidelberg Berlin. [2] Lowrence C. Evans. (1998) Partial Differential Equations. Amer. Math. Soc. Providence, Rhode Island. [3] J. Malek, J. Necas, M. Rokyra and M. Ruzicka. (1996) Weak and Measure-valued Solutions to Evolutionary PDEs. Champion & Hall. [4] Robert A. Adams and John. J. F. Fourier. (2003) Sobolev Spaces. Academic Press. [5] Victor A. Galaktionov. (2004) Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications. Champion & Hall. [6] S. Angenent. (1988) The zero set of a solution of a parabolic equation. J. reine angew. Math. 390, [7] Redheffer, R. M. and W. Walter. (1974) The total variation of solutions of parabolic differential equations and a maximum principle in unbounded domains. Math. Ann. 209, [8] O. A. Oleinik. (1963) Discontinuous Solutions of Non-Linear Differential Equations. Amer. Math. Soc. Transl. II. Ser,

28 감사의 글 부족한 제가 이 논문을 무사히 완성할 수 있도록 도움을 주신 모든 분들께 감사의 마음을 전하고 싶습니다. 우선 논문을 쓰는 동안 자상하고 세심하게 지도해 주시고 본받아 마땅한 학자로서의 자세를 보여주신 김용정 교수님께 감사드립니다. 그리고 바쁘신 와중에도 적 극적으로 논문심사를 맡아주신 권길헌 교수님, 김홍오 교수님께도 감사드립니다. 우수하 고 모범적인 사람들과 같이 한다는 것이 저에게 얼마나 연구하기에 좋은 환경이었는지를 일깨워준 연구실 식구들에게도 감사의 말을 전합니다. 2년 간의 석사 생활이 고단하지 않 고 즐거웠던 것은 곁에 좋은 사람들이 많이 있었기 때문입니다. 제가 이렇게 인복이 많나 싶을 정도로 항상 의지가 된 대학원 선후배, 동기 들에게도 고맙다는 말을 전하고 싶습니 다. 또한 제가 흔들리지 않고 계속 공부를 할 수 있도록 옆에서 응원해주신 지인분들께도 감사드립니다. 힘들 때마다 가장 큰 위로가 되어주고, 끊임없이 격려해주었던 경민이에게 도 고마움을 전하고 싶습니다. 마지막으로 제가 선택한 길에 대해 무조건적인 지지를 해주 시는 사랑하는 나의 가족, 부모님 동생에게도 감사드립니다.

29 이력서 이 름 : 윤창욱 생 년 월 일 : 1983년 9월 4일 출 생 지 : 충남 천안시 구성동... 본 적 지 : 충남 천안시 신방동... 주 소 : 대전 유성구 궁동... 주 소 : wook3994@kaist.ac.kr 학 천안북일고등학교 고려대학교 수학과 (B.S.) 경 력 력 한국과학기술원 수리과학과 조교