The Experts below are selected from a list of 4260 Experts worldwide ranked by ideXlab platform
Jared Speck - One of the best experts on this subject based on the ideXlab platform.
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Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity
Inventiones mathematicae, 2018Co-Authors: Jonathan Luk, Jared SpeckAbstract:We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple plane wave solutions, starting from smooth initial data, is stable under perturbations of the initial data that break the plane symmetry. Moreover, we provide a sharp asymptotic description of the singularity formation. The new feature of our work is that the perturbed solutions are allowed to have small but non-zero vorticity, even at the location of the shock. Thus, our results provide the first constructive description of the vorticity near a singularity formed from compression. Specifically, the vorticity remains Uniformly bounded, while the vorticity divided by the density exhibits even more regular behavior: the ratio remains Uniformly Lipschitz relative to the standard Cartesian coordinates. To control the vorticity, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou in his groundbreaking, sharp proof of shock formation in vorticity-free regions. In particular, we rely on a new formulation of the compressible Euler equations (derived in a companion article) exhibiting remarkable structures. To derive estimates, we construct an eikonal function adapted to the acoustic characteristics (which correspond to sound wave propagation) and a related set of geometric coordinates and differential operators. Thanks to the remarkable structure of the equations, the same set of coordinates and differential operators can be used to analyze the vorticity, whose characteristics are transversal to the acoustic characteristics. In particular, our work provides the first constructive description of shock formation without symmetry assumptions in a system with multiple speeds.
Vladimir Mazya - One of the best experts on this subject based on the ideXlab platform.
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on the boundedness of first derivatives for solutions to the neumann laplace problem in a convex domain
Journal of Mathematical Sciences, 2009Co-Authors: Vladimir MazyaAbstract:It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are Uniformly Lipschitz. Applications of this result to some aspects of the regularity of solutions to the Neumann problem on convex polyhedra are given. Bibliography: 27 titles.
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boundedness of the gradient of a solution to the neumann laplace problem in a convex domain
Comptes Rendus Mathematique, 2009Co-Authors: Vladimir MazyaAbstract:Abstract It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n -dimensional domain are Uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).
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boundedness of the gradient of a solution to the neumann laplace problem in a convex domain
arXiv: Analysis of PDEs, 2008Co-Authors: Vladimir MazyaAbstract:It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are Uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given.
Jonathan Luk - One of the best experts on this subject based on the ideXlab platform.
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Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity
Inventiones mathematicae, 2018Co-Authors: Jonathan Luk, Jared SpeckAbstract:We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple plane wave solutions, starting from smooth initial data, is stable under perturbations of the initial data that break the plane symmetry. Moreover, we provide a sharp asymptotic description of the singularity formation. The new feature of our work is that the perturbed solutions are allowed to have small but non-zero vorticity, even at the location of the shock. Thus, our results provide the first constructive description of the vorticity near a singularity formed from compression. Specifically, the vorticity remains Uniformly bounded, while the vorticity divided by the density exhibits even more regular behavior: the ratio remains Uniformly Lipschitz relative to the standard Cartesian coordinates. To control the vorticity, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou in his groundbreaking, sharp proof of shock formation in vorticity-free regions. In particular, we rely on a new formulation of the compressible Euler equations (derived in a companion article) exhibiting remarkable structures. To derive estimates, we construct an eikonal function adapted to the acoustic characteristics (which correspond to sound wave propagation) and a related set of geometric coordinates and differential operators. Thanks to the remarkable structure of the equations, the same set of coordinates and differential operators can be used to analyze the vorticity, whose characteristics are transversal to the acoustic characteristics. In particular, our work provides the first constructive description of shock formation without symmetry assumptions in a system with multiple speeds.
Simone M Bruschi - One of the best experts on this subject based on the ideXlab platform.
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concentrated terms and varying domains in elliptic equations Lipschitz case
Mathematical Methods in The Applied Sciences, 2016Co-Authors: Gleiciane S Aragao, Simone M BruschiAbstract:In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is Uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in H1an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.
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concentrated terms and varying domains in elliptic equations Lipschitz case
arXiv: Analysis of PDEs, 2014Co-Authors: Gleiciane S Aragao, Simone M BruschiAbstract:In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is Uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain. We prove that this family of solutions converges to the solutions of a limit problem in H^1 , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.
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very rapidly varying boundaries in equations with nonlinear boundary conditions the case of a non Uniformly Lipschitz deformation
Discrete and Continuous Dynamical Systems-series B, 2010Co-Authors: Jose M Arrieta, Simone M BruschiAbstract:We continue the analysis started in [3] and announced in [2], studying the behavior of solutions of nonlinear elliptic equations in e with nonlinear boundary conditions of type , when the boundary of the domain varies very rapidly. We show that if the oscillations are very rapid, in the sense that, roughly speaking, its period is much smaller than its amplitude and the function is of a dissipative type, that is, it satisfies , then the boundary condition in the limit problem is , that is, we obtain a homogeneus Dirichlet boundary condition. We show the convergence of solutions in and norms and the convergence of the eigenvalues and eigenfunctions of the linearizations around the solutions. Moreover, if a solution of the limit problem is hyperbolic (non degenerate) and some extra conditions in are satisfied, then we show that there exists one and only one solution of the perturbed problem nearby.
Haihao Lu - One of the best experts on this subject based on the ideXlab platform.
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relative continuity for non Lipschitz nonsmooth convex optimization using stochastic or deterministic mirror descent
Informs Journal on Optimization, 2019Co-Authors: Haihao LuAbstract:The usual approach to developing and analyzing first-order methods for nonsmooth (stochastic or deterministic) convex optimization assumes that the objective function is Uniformly Lipschitz continu...
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relative continuity for non Lipschitz non smooth convex optimization using stochastic or deterministic mirror descent
arXiv: Optimization and Control, 2017Co-Authors: Haihao LuAbstract:The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is Uniformly Lipschitz continuous with parameter $M_f$. However, in many settings the non-differentiable convex function $f(\cdot)$ is not Uniformly Lipschitz continuous -- for example (i) the classical support vector machine (SVM) problem, (ii) the problem of minimizing the maximum of convex quadratic functions, and even (iii) the univariate setting with $f(x) := \max\{0, x\} + x^2$. Herein we develop a notion of "relative continuity" that is determined relative to a user-specified "reference function" $h(\cdot)$ (that should be computationally tractable for algorithms), and we show that many non-differentiable convex functions are relatively continuous with respect to a correspondingly fairly-simple reference function $h(\cdot)$. We also similarly develop a notion of "relative stochastic continuity" for the stochastic setting. We analysis two standard algorithms -- the (deterministic) mirror descent algorithm and the stochastic mirror descent algorithm -- for solving optimization problems in these two new settings, and we develop for the first time computational guarantees for instances where the objective function is not Uniformly Lipschitz continuous. This paper is a companion paper for non-differentiable convex optimization to the recent paper by Lu, Freund, and Nesterov, which developed similar sorts of results for differentiable convex optimization.
