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In this article, we study the break-down of smooth and continuous solution toisentropic Euler system in multi dimension. Sideris [Comm. Math. Phys. 1985]proved the blow up of smooth solutions when initial data satisfies an `integralcondition'. We show that a $C^1$ solution of isentropic Euler equation breaksdown if (i) gradient of initial velocity has an eigenvalue with negativereal-part at some point $x_0\in\mathbb{R}^d$ and (ii) Hessian of initialdensity satisfies a smallness condition in Sobolev space. Our proof also worksfor the data which fails to satisfy the above-mentioned `integral condition'.Further, we prove the global existence of smooth solution when (i) gradient ofinitial velocity has an eigenvalue with non-negative real-part in whole$\mathbb{R}^d$ and (ii) initial density satisfies a smallness condition. Thisextends the global existence result of [Grassin, Indiana Univ. Math. J. 1998].Another goal of this article is to study the breakdown of continuous weaksolutions of isentropic Euler equations. We are able to show that the `integralcondition' of Sideris can cause the breakdown of continuous solutions in finitetime. This improves the blow up result of Sideris from $C^1$ to continuousspace.

On blow up of $C^1$ solutions of isentropic Euler system