论文标题
$γ= 3 $的等质Euler方程的无条件规律性和跟踪结果
Unconditional regularity and trace results for the isentropic Euler equations with $γ= 3$
论文作者
论文摘要
在本文中，我们研究了具有$γ= 3 $的等质欧拉方程的有界熵溶液的规律性特性。首先，我们使用爆破技术来获得所有此类解决方案的新痕量定理。其次，我们在动力学配方上使用修改后的De Giorgi型迭代，以显示Riemann不变性的新局部规律性结果。我们能够得出结论，实际上，对于任何有界的熵解决方案$ u $，密度$ρ$几乎到处都是上半连续的，远离真空。据我们所知，这是非线性双曲系统的第一个例子，该系统未能是寺庙类别，但具有通用$ l^\ infty $初始数据的属性产生了有界熵解决方案，并具有近乎经典的规律性。这提供了一个例子，即$ 2 \ times 2 $双曲系统可以具有一些更引人注目的正规化效果，该效果已知，这些效果通常在真正的非线性，多维标量设置中。虽然我们无法使用规律性结果显示无条件的唯一性，但结果大大降低了当前凸集成方法可以在这种情况下使用的可能性。
In this paper, we study the regularity properties of bounded entropy solutions to the isentropic Euler equations with $γ= 3$. First, we use a blow-up technique to obtain a new trace theorem for all such solutions. Second, we use a modified De Giorgi type iteration on the kinetic formulation to show a new partial regularity result on the Riemann invariants. We are able to conclude that in fact for any bounded entropy solution $u$, the density $ρ$ is almost everywhere upper semicontinuous away from vacuum. To our knowledge, this is the first example of a nonlinear hyperbolic system, which fails to be Temple class, but has the property that generic $L^\infty$ initial data give rise to bounded entropy solutions with a form of near classical regularity. This provides one example that $2\times 2$ hyperbolic systems can possess some of the more striking regularizing effects known to hold generically in the genuinely nonlinear, multidimensional scalar setting. While we are not able to use our regularity results to show unconditional uniqueness, the results substantially lower the likelihood that current methods of convex integration can be used in this setting.
