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On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions. II: The zero-viscosity and dispersion limits. (English) Zbl 0834.35085
In these two papers the authors study in great detail the nonlinear hyperbolic equation $(u_t+ uu_x )_x= u^2_x/2$; it arises as the canonical asymptotic equation for weakly nonlinear solutions of some hyperbolic equations derived from variational principles (e.g., in the dynamics of liquid crystals). In the first paper two kinds of weak solutions are defined: they are called dissipative and conservative solutions, according to energy decays rapidly or is conserved; they agree for smooth solutions. In both classes it is proved the global existence of Hölder continuous solutions (in spite of the fact that derivatives may blow up) provided that the derivative of the initial data has bounded variation and compact support. Proofs are obtained by means of a careful analysis of some approximate explicit solutions. It is also shown that conservative solutions can be obtained as limits of solutions constructed with the so-called regularized method of characteristics; moreover dissipative solutions are asymptotic, for large time, to a special piecewise linear solution. The paper also includes detailed comparaisons with Burgers’ equation. The second paper deals with the convergence of solutions obtained by viscous or dispersive regularizations. The authors consider as initial data a simple step function giving blow up of the derivative of the solution and show that solutions obtained by viscous regularization tend to a unique global weak solution of the equation without viscosity. At last, some numerical computations are given which suggest that solutions obtained by dispersive regularization converge, for vanishing dispersion, to a conservative solution which is different from the preceding one.

35L70Nonlinear second-order hyperbolic equations
35Q53KdV-like (Korteweg-de Vries) equations
35B40Asymptotic behavior of solutions of PDE
35D05Existence of generalized solutions of PDE (MSC2000)
Full Text: DOI
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