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Representation of dissipative solutions to a nonlinear variational wave equation
 Comm. Math. Sci
"... The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuou ..."
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The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov’s compactness theorem. Reverting to the original variables, one recovers a solution to the nonlinear wave equation where the total energy is a monotone decreasing function of time. 1
On the lifespan of and the blowup mechanism for smooth solutions to a class of 2D nonlinear wave equations with small initial data
, 2012
"... ar ..."
Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation
, 2015
"... The paper is concerned with conservative solutions to the nonlinear wave equation utt−c(u) c(u)ux x = 0. For an open dense set of C3 initial data, we prove that the solution is piecewise smooth in the tx plane, while the gradient ux can blow up along finitely many characteristic curves. The analysi ..."
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The paper is concerned with conservative solutions to the nonlinear wave equation utt−c(u) c(u)ux x = 0. For an open dense set of C3 initial data, we prove that the solution is piecewise smooth in the tx plane, while the gradient ux can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom’s transversality theorem. 1
CONTINUOUS SOLUTIONS OF A BALANCE EQUATION
"... Abstract. In this paper we provide a characterization of intrinsic Lipschitz graphs in the subRiemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φy + [φ 2 /2]t = w, where w is ..."
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Abstract. In this paper we provide a characterization of intrinsic Lipschitz graphs in the subRiemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φy + [φ 2 /2]t = w, where w is a bounded function.
5. Equivalence among Distributional PDE and Intrinsic Lipschitz Condition. 14
"... Abstract. In this paper we provide a characterization of intrinsic Lipschitz graphs in the subRiemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φy + [φ2/2]t = w, where w is a ..."
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Abstract. In this paper we provide a characterization of intrinsic Lipschitz graphs in the subRiemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φy + [φ2/2]t = w, where w is a bounded function depending on φ.