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12 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES The corresponding Cauchy problem admits the trivial solution (2.13) again, as well as the following one-parameter family of solutions: ⎧ 1, x < −λt, ⎪⎨ −v, −λt
3. SINGULAR LIMITS AND THE ENTROPY INEQUALITY 13 3. Singular limits and the entropy inequality Examples 2.4 and 2.5 show that weak solutions of the Cauchy problem (1.1) and (1.2) are generally non-unique. To single out the solution of interest, we restrict attention to weak solutions realizable as limits (ε → 0) of smooth solutions of an augmented system ∂ t u ε + ∂ x f(u ε )=R ε , u ε = u ε (x, t), (3.1) where ε>0 represents a small-scale parameter corresponding, in the applications, to the viscosity, capillarity, etc. of the physical medium under consideration. The right-hand side of (3.1) may contain a singular regularization R ε depending upon u ε ,εu ε x,ε 2 u ε xx,... and (in a sense clarified by Definition 3.1 below) vanishing when ε → 0. We always assume that the system of conservation laws (1.1) is endowed with a strictly convex entropy pair (U, F ) and that the singular limit u = lim u ε (3.2) ε→0 exists in a sufficiently strong sense. Precisely, there exists a constant C>0 independent of ε such that (U contains the closed ball with center 0 and radius C and) ‖u ε ‖ L∞ ( I R× I R +) ≤ C (3.3) and the convergence (3.2) holds almost everywhere in (x, t). To arrive to a well-posed Cauchy problem for the hyperbolic system (1.1) we attempt to derive some conditions satisfied by the limit u, which are expected to characterize it among all of the weak solutions of (1.1). In the present section, under some natural conditions on the smoothing term R ε , we derive the so-called entropy inequality associated with the entropy pair (U, F ) (that is, (3.8) below). The entropy inequality plays a fundamental role in the mathematical theory for (1.1). As will be further discussed in Section 5, it does not always completely characterize the limit of (3.1), however. First ( of all, we wish ) that the limit u be a weak solution of (1.1). For each function θ ∈ Cc ∞ I R × (0, +∞) , relying on (3.1)–(3.3) we find ∫∫ ( u∂t θ + f(u) ∂ x θ ) ( dxdt = lim u I R× I R ε→0 + ∫∫R ε ∂ t θ + f(u ε ) ∂ x θ ) dxdt I × I R + = lim R ε→0 ∫∫R ε θdxdt. I × I R + Therefore, we arrive at the following condition on R ε which is necessary (and sufficient) for the limit u to be weak solution of (1.1). Definition 3.1. (Conservative regularization.) The right-hand side R ε of (3.1) is said to be conservative (in the limit ε → 0) if lim ε→0 ∫∫ R ε θdxdt=0, I R× I R + θ ∈ C ∞ c ( R I × (0, +∞) ) . (3.4) □ Next, we take advantage of the existence of an entropy pair (U, F ). Multiplying (3.1) by ∇U(u ε ) we observe that, according to Definition 1.7, the left-hand side admits a conservative form, namely ∂ t U(u ε )+∂ x F (u ε )=∇U(u ε ) · R ε . (3.5)
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CHAPTER IV EXISTENCE THEORY FOR THE
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1. CLASSICAL ENTROPY SOLUTIONS FOR
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4. REFINED ESTIMATES 113 In turn, w
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1. A CLASS OF LINEAR HYPERBOLIC EQU
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PART 2 SYSTEMS OF CONSERVATION LAWS
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140 CHAPTER VI. THE RIEMANN PROBLEM
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168 CHAPTER VII. CLASSICAL ENTROPY
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260 APPENDIX if its total variation
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262 APPENDIX vector-valued Radon me
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BIBLIOGRAPHICAL NOTES Chapter I. Sm
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BIBLIOGRAPHICAL NOTES 267 with the
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BIBLIOGRAPHICAL NOTES 269 The exist
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BIBLIOGRAPHY Abeyaratne R. and Know
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BIBLIOGRAPHY 273 Baiti P., LeFloch
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BIBLIOGRAPHY 275 Chen G.-Q. and Kan
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BIBLIOGRAPHY 277 in “Nonlinear An
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BIBLIOGRAPHY 279 Fonseca I. (1989b)
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BIBLIOGRAPHY 281 Hattori H. (1986a)
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BIBLIOGRAPHY 283 Jenssen H.K. (2000
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BIBLIOGRAPHY 285 LeFloch P.G. (1988
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BIBLIOGRAPHY 287 Liu T.-P. (1974),
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BIBLIOGRAPHY 289 Oleinik O. (1957),
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BIBLIOGRAPHY 291 Serrin J. (1979),
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BIBLIOGRAPHY 293 Temple B. (1982),
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