Volume 35, Issue 3
On the Well-Posedness of Weak Solutions

Taiping Liu

Ann. Appl. Math., 35 (2019), pp. 309-316.

Published online: 2020-08

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  • Abstract

This is to comment on the well-posedness of weak solutions for the initial value problem for partial differential equations. In recent decades, and particularly in recent years, there have been substantial progresses on construction by convex integration for the study of non-uniqueness of solutions for incompressible Euler equations, and even for compressible Euler equations. This prompts the question of whether it is possible to give a sense of well-posedness, which is narrower than the canonical Hadamard sense, so that the evolutionary equations are well-posed. We give a brief and partial review of the related results and offer some thoughts on this fundamental topic.

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@Article{AAM-35-309, author = {Liu , Taiping}, title = {On the Well-Posedness of Weak Solutions}, journal = {Annals of Applied Mathematics}, year = {2020}, volume = {35}, number = {3}, pages = {309--316}, abstract = {

This is to comment on the well-posedness of weak solutions for the initial value problem for partial differential equations. In recent decades, and particularly in recent years, there have been substantial progresses on construction by convex integration for the study of non-uniqueness of solutions for incompressible Euler equations, and even for compressible Euler equations. This prompts the question of whether it is possible to give a sense of well-posedness, which is narrower than the canonical Hadamard sense, so that the evolutionary equations are well-posed. We give a brief and partial review of the related results and offer some thoughts on this fundamental topic.

}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aam/18084.html} }
TY - JOUR T1 - On the Well-Posedness of Weak Solutions AU - Liu , Taiping JO - Annals of Applied Mathematics VL - 3 SP - 309 EP - 316 PY - 2020 DA - 2020/08 SN - 35 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/aam/18084.html KW - nonlinear partial differential equations, well-posedness theory, weak solutions. AB -

This is to comment on the well-posedness of weak solutions for the initial value problem for partial differential equations. In recent decades, and particularly in recent years, there have been substantial progresses on construction by convex integration for the study of non-uniqueness of solutions for incompressible Euler equations, and even for compressible Euler equations. This prompts the question of whether it is possible to give a sense of well-posedness, which is narrower than the canonical Hadamard sense, so that the evolutionary equations are well-posed. We give a brief and partial review of the related results and offer some thoughts on this fundamental topic.

Liu , Taiping. (2020). On the Well-Posedness of Weak Solutions. Annals of Applied Mathematics. 35 (3). 309-316. doi:
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