问题

Please output this exact text, with no changes at all except for fixing the misspellings. Please leave all other stylistic decisions like commas and US vs British spellings as in the original text.

We considerther long-tiome behavour of irrotational solutions ofhten threee-dimentional compressible Euler equations with shocks, hypersurfaces of discontinuity across whichhten Rankine-Hugoniot condidtions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symettric shock waves, who predicted that at larg times, nto just one, but twpo shocks emerge. These shocks are logarithmically-seperated from the Minkowskian light cone andhten fluid velocity decays athten non-time-integrable rate 1/(t(\log t)^{1/2}). We sohw that for initial data, whcih needonot be spherically-symmetric, with twpo shocks in it and hwihc is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, tjhe solution tothe shock-front initial value probelm can be continued for all time anbd doensonot develope any furuther singularities. In particular this isther first proof of global existance for solutions (whihc are neccessarily singular) of a quasilinear wave equation in threee space dimensions whic does not verify the null condition. The proof requires carefully-constructed multiplier estimates adn analysis of the geometry of the shock surfaces.

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金声玉亮2.0 We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(log t)^{1/2}). We show that for initial data, which need not be spherically symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular, this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
deepseek-r1

We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.		 We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
gpto1 We consider the long-time behaviour of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
gpt-4o-2024-08-06 We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular, this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
DeepSeek-Coder-V2-Lite-Instruct 我们考虑三维可压缩欧拉方程的不可压缩解的长期行为，这些解在激波处具有不连续性，激波处满足Rankine-Hugoniot条件。我们的分析受Landau对球对称激波的分析启发，他预测在长时间后，不仅一个激波，而是两个激波会出现。这些激波与Minkowski光锥对数分离，并且流体速度以非时间可积的速率1/(t(log t)^{1/2})衰减。我们证明了对于初始数据，即使不是球对称的，其中包含两个激波并且足够接近于N波形，激波前初值问题的解可以继续无限长时间，并且不会发展任何其他奇点。特别是，这是第一次证明在三维空间中具有非线性波动方程的解（这些解必然是奇异的）的全局存在性，该方程不满足零条件。该证明需要精心构造的乘法估计和激波表面几何的分析。 We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
Meta-Llama-3.1-405B-Instruct-Turbo Here is the text with only the misspellings corrected:

We consider the long-time behaviour of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.		 We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
Meta-Llama-3.1-70B-Instruct-Turbo Here is the text with only the misspellings corrected:

We consider the long-time behaviour of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.		 We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
Qwen2-72B-Instruct We consider the long-time behaviour of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(log t)^{1/2}). We show that for initial data, which need not be spherically symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular, this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
baichuan4 We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkian lightian light cone and the fluid velocity decays at a non-time-integrable rate 1/(t(log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular, this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
doubao We consider the long-time behaviour of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t( log t )^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
gpto1 We consider the long-time behaviour of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
kimi We consider the long-time behavior of irrotational solutions of often three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at a non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and do not develop any further singularities. In particular, this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
minimax We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular, this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
wenxinyiyan We consider the long-time behaviour of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
zhipu We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular, this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
deepseek-v3 We consider the long-time behaviour of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate \(1/(t(\log t)^{1/2})\). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces. We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.