By Jian-Guo Liu, and Athanasios Tzavaras Eitan Tadmor, Eitan Tadmor, Jian-guo Liu, Athanasios E. Tzavaras (ed.)

The foreign convention on Hyperbolic difficulties: concept, Numerics and functions, ``HYP2008'', was once held on the college of Maryland from June 9-13, 2008. This was once the 12th assembly within the bi-annual foreign sequence of HYP meetings which originated in 1986 at Saint-Etienne, France, and during the last 20 years has develop into one of many best quality and such a lot winning convention sequence in utilized arithmetic. This e-book, the second one in a two-part quantity, includes greater than sixty articles in response to contributed talks given on the convention. The articles are written by way of top researchers in addition to promising younger scientists and canopy a various variety of multi-disciplinary issues addressing theoretical, modeling and computational matters coming up below the umbrella of ``hyperbolic PDEs''. This quantity will convey readers to the vanguard of study during this such a lot lively and significant region in utilized arithmetic

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**Additional info for Hyperbolic Problems: Theory, Numerics and Applications, Part 2: Contributed Talks**

**Example text**

1) + ∇P (ρ) + ρ∇V = 2 ρ∇ √ρ ∂t J + div J⊗J ρ ⎪ ⎩ −ΔV = ρ, (p+1)/2 , 1 ≤ p < 5. 1. 1) are originated by a term of the form Re ∇ψ ⊗ ∇ψ since formally 2 Re ∇ψ ⊗ ∇ψ = 2 J ⊗J √ √ . 1). 1) is done with Cauchy data of the form (ρ0 , J0 ) = (|ψ0 |2 , Im(ψ0 ∇ψ0 )), for some ψ0 ∈ H 1 (R3 ; C). 1. 1), (ρ0 , J0 ) := (|ψ0 |2 , Im(ψ0 ∇ψ0 )). 1) in the space-time slab [0, T ) × R3 . 6) is conserved for all times t ∈ [0, T ). The idea behind the proof of this Proposition is the following. 2) with ψ(0) = ψ0 ; then it is globally well-posed for initial data in H 1 (R3 ) (see [3]), and the solution satisﬁes ψ ∈ C 0 (R; H 1 (R3 )).

15] G. Crippa & C. De Lellis: Estimates and regularity results for the DiPerna–Lions ﬂow. J. Reine Angew. Math. 616 (2008), 15–46. [16] C. Dafermos: Hyperbolic conservation laws in continuum physics. Second edition. Grundlehren der Mathematischen Wissenschaften 325, Springer-Verlag, Berlin, 2005. [17] C. De Lellis: Notes on hyperbolic systems of conservation laws and transport equations. Handbook of Diﬀerential Equations: Evolutionary Equations, vol. III. Edited by C. M. Dafermos and E. Feireisl.

For the proof we refer to [2]. See the end of the ﬁrst section for the notion of rectiﬁable set. 1. Let H ∈ Lipc (R2 ). e. h ∈ R the following statements hold. e. x ∈ Eh ; the function 1/|∇H| belongs to L1 (Eh , H1 ). (ii) The family Ch is countable and H1 (Eh \ ∪C∈Ch C) = 0. (iii) Every C ∈ Ch is a closed simple curve. e. s ∈ [α, β]∗ , where we denote by [α, β]∗ the quotient space consisting of the interval [α, β] with identiﬁed endpoints, endowed with the distance dist [α,β]∗ (x, y) = min |x − y|, (β − α) − |x − y| .