Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to N-waves

Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions

“…On the contrary, numerous works are devoted to the study of the large-time behaviour of the entropy solution of a scalar conservation law posed on the whole space (Cauchy problem). See in particular the results of decay of the entropy solution to N -waves, as studied in [Lax57,IO60,DiP75,LP84,Daf85,Kim03]. Consider also the studies on the stability of profiles for scalar conservation laws on the line, in particular in [KM85,MN94,LN97,FS01,Ser04].…”

An Oleinik-type estimate for a convection–diffusion equation and convergence to N-waves

“…Since max x juðx; tÞj ¼ Oð1= ffiffi t g p Þ (see [15]), these two estimates, (6) and (8), are almost equivalent for a large time tb1: On the other hand, since ð ffiffi t g p Þ 2Àg -0 as t-0; we may say (8) is weaker than (6) for small t40: Note that, for the Burgers equation ðg ¼ 2Þ we show that these two are identical with C ¼ 1: Using this Oleinik-type estimate, we show our main result: This papers is organized as follows. In Section 2, we introduce a technique to obtain the uniform estimates of similarity solutions to the inviscid problem which is based on the Oleinik estimate.…”

“…The steady state of the inviscid problem (14), (15) is an N-wave which is a member of the two-parameter family of single variable functions…”

“…Since g 0 ð0Þ ¼ 0 for 1ogo2; the estimate (18) implies that w x ðx 0 ; sÞp0: & Remark 3. Let wðx; sÞ be the solution of (14) that satisfies the entropy condition (15). Then, since the solution w satisfies the (transformed) Oleinik estimate (17) and that, if 1ogo2 and uðx 0 ; tÞ ¼ 0; @ x uðx 0 ; tÞp0:…”

“…Since this method is not applicable under the presence of diffusion, we do not pursue the convergence in this paper. We refer readers to [12,15,18] for the asymptotic convergence of inviscid cases.…”