D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math., 11 (1973), pp. 368--386.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....t) j x = X k (t)g, each of which satisfies the Rankine Hugoniot and the Lax conditions: X k = f(u(X k ; t) u(X k ; t) 3.23) u(X k (t) Gamma; t) X k (t) f (u(X k (t) t) 3. 24) We note in passing that many practical initial data lead to finite number of shocks (see, e.g. [17, 22]) and in this case one has a global L error bound of order ffl, 2.7) Next we consider the characteristic variables, It follows that their L convergence rate from their limiting value ffu Sigmav with v = f(u) is also or der (ffl) Moreover, Theorem 3.1 implies the Lip ....

D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math., 11 (1973), pp. 368-386.

....S(t) x, t) x = X k (t) each of whP h satisfiesth Rankine Hugoniot and th Lax conditions: X # k = f(u(X k ,t) u(X k ,t) 3.24) f # (u(X k (t) t) X # k (t) f # (u(X k (t) t) 3. 25) We note in passingths many practical initial data lead to finite number of sh cks (see, e.g. [17, 22]) and inth8 case one he a global L 1 error bound of order #, 2.7) Next we considerth chJ6RS2AI8PIJ variables, # #u # v # : It followsthI thIJ L 1 convergence rate from thm2 limiting value # #u vwith v = f(u) is also order (#) Moreover,Th2885 3.1 impliesth Lip boundedness # ....

D G. Schaeffer, A regularity theorem for conservation laws, Adv. Math., 11 (1973), pp. 368-- 386.

....solution itself, essentially nothing is obtained for its derivative. In this work, we will investigate the convergence of the first derivative of the approximate solutions. We will assume that the 2 inviscid solution u has finitely many discontinuities, which is the generic situation, [15, 22]. By properly choosing a weighted function, we will obtain an O(ffl) bound for u ffl Gammau in a weighted W 1;1 space. More precisely, we will show that the following estimate holds: Z R ae(x; t) i ju ffl x Gamma u x j ju ffl Gamma uj j dx C ffl ; where ae is a distance ....

D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math., 11 (1973), pp. 368-386.

.... 0) i.e. jg(u) Gamma g(v)j Lju Gamma vj: 1.4) In general, the problem (1.1) 1.2) does not possess a global smooth solution even if the initial value is C 1 smooth. The structure of entropy solutions has been studied by many authors, e.g. Dafermos [1] Lax [6] Oleinik [9] and Schaeffer [12]. The entropy solutions consisting of finite number of shock or rarefaction discontinuities form an important solution class. They are the only solutions that can be computed numerically. Therefore, it is useful to study the solution structure, and in particular to identify the initial conditions ....

....set. Dafermos [1] has shown that in case that both the (convex) flux and the initial condition are infinitely smooth the solution is C 1 almost everywhere apart from the shock set which must be closed. Thus, the shock set cannot be everywhere dense but shocks may still accumulate. Schaeffer [12] proved that if f 2 C 1 satisfies (1.3) and u 0 satisfies certain conditions, then the shock set is finite. However, the conditions for the initial data are so abstract that in practice we are unable to verify them. For homogeneous scalar conservation laws, Tadmor and Tassa [13] and Li and Wang ....

D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math. 11, 1973, pp. 368--386.

....we discuss in x4 the complement of the C 1 smoothness part of the entropy solution, that is, we determine the size of the set of shocks. Theorem 4.1 asserts that this set is equivalent to the set of negative minima of a(u 0 ) 0 . Thus Theorem 4. 1 complements Schaeffer s regularity theorem [11], by realizing the first category set of infinitely smooth initial conditions , fu 0 g, which evolve into entropy solutions with infinitely many shock discontinuities. In summary we conclude that if a(u 0 ) has a finite number of decreasing inflection points, then only a finite number of shocks ....

....simplify the picture : Dafermos [2] has shown that in case that both the (convex) flux and the initial condition are infinitely smooth the solution is C 1 a.e. apart from the shock set which must be closed. Thus, the shock set cannot be everywhere dense but shocks may still accumulate. Schaeffer [11] has proved that, generically, the shock set is finite when the initial condition is infinitely smooth. He has shown that if f 2 C 1 satisfies (1.2) there exists a subset, Omega Gamma of the first category in Schwartz space, S( such that if u 0 2 S( Gamma Omega then u 2 C 1 ( Theta ....

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D.G. Schaeffer, "A regularity theorem for conservation laws", Adv. in Math., 11 (1973), 368-386.

....t f(u) x = 0; x 2 R; t 0; 1.3) which is subject to the same initial conditions u(x; 0) u 0 (x) 1.4) We will investigate the pointwise convergence rate of u ffl towards u, assuming that the inviscid solution u has finitely many shocks or rarefaction waves. This is the generic situation, [17], 20] It is well known that u ffl ( Delta; t) converges strongly in L 1 to u( Delta; t) where u( Delta; t) is the unique, so called entropy solution of (1.3) 1.4) The L 1 convergence rate in this case is upper bounded by ku ffl ( Delta; t) Gamma u( Delta; t)k L 1 const Delta p ....

....= X k (t)g, each of which satisfies the RankineHugoniot and the Lax conditions: X 0 k = f(u(X k ; t) u(X k ; t) 3.1) f 0 (u(X k (t) Gamma; t) X 0 k (t) f 0 (u(X k (t) t) 3. 2) We note in passing that many practical initial data lead to finite number of shocks (see, e.g. [17, 20]) Owing to the convexity of the flux f , the viscosity solutions of (1.1) satisfy a Lip stability condition, similar to the familiar Oleinik s E condition [14] which asserts an a priori upper bound for the Lip seminorm of the viscosity solution u ffl (x; t) Gamma u ffl (y; t) x ....

D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math., 11 (1973), pp. 368-386.

....s u(x; T )j Const p Delta (1 ju (p s) j loc ) s 2 p s 2 Delta p p s 2 : Next, let us consider the case of smooth initial data, u(x; 0)fflS. Then, there exists a dense subset of S such that the corresponding entropy solution of (3. 1) with C 1 convex flux f) is piecewise smooth ,[13], and we are able to recover the pointwise values of s x s u(x; T ) with error as close to as desired, if we take p large enough in (4.9) 902 EITAN TADMOR We close this section with a brief description of the spectral post processing technique [3] which enables the pointwise recovery ....

D. G. Schaeffer, "A regularity theorem for conservation laws," Adv. in Math., 11 (1973), pp. 368-386.

....k (t)g, each of which satisfies the Rankine Hugoniot and the Lax conditions: X 0 k = f(u(X k ; t) u(X k ; t) 3.24) f 0 (u(X k (t) Gamma; t) X 0 k (t) f 0 (u(X k (t) t) 3. 25) We note in passing that many practical initial data lead to finite number of shocks (see, e.g. [17, 22]) and in this case one has a global L 1 error bound of order ffl, 2.7) Next we consider the characteristic variables, p ffu ffl Sigma v ffl : It follows that their L 1 convergence rate from their limiting value p ffu Sigma v with v = f(u) is also order (ffl) Moreover, Theorem ....

D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math., 11 (1973), pp. 368-386.

.... a(u 0 (x) d dx i a(u 0 (x) j ; a(u 0 (x) d 2 dx 2 i a(u 0 (x) j ; u 0 (x) du 0 (x) dx ; u 0 (x) d 2 u 0 (x) dx 2 : The behavior and structure of entropy solutions for scalar convex conservation laws have been studied for many years, see for example [2, 11, 15, 17]. It is well known that if the initial function is piecewise C 2 smooth then the entropy solutions consist of at most countable number of C 2 smooth regions. Tadmor and Tassa [18] proved that if the initial speed has a finite number of decreasing inflection points then it bounds the number ....

....If a(u 0 ) has a continuum of negative minimal points it is considered as one minimum. The assumption (A4) indicates that a(u 0 ) has a finite number of decreasing inflection point. It can be shown that under the assumptions (A1) A3) and (A4) only a finite number of shock will occur (see, e.g. [17, 18]) The corresponding entropy solution consists of a finite number of C 2 smooth pieces. The shocks are disjoint, except for common endpoints arising from collisions, and (without loss of generality) we assume that no more than two shocks ever collide. More precisely, for each time interval [t p ....

D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math., 11 (1973), pp. 368-386.

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D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math., 11 (1973), pp. 368--386.

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D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math., 11 (1973), pp. 368--386.

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Schaeffer D., A regularity theorem for conservation laws, Adv. in Math. 11 (1973), 368--386.

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