# Improvement on the rough solutions of 3D compressible Euler equations

#### Autoren: Huali Zhang, Lars Andersson (2022)

Wang proved the local existence and uniqueness of solutions for 3D isentropic compressible Euler equations by vector-field approach if the initial velocity $$v_0$$, logarithmic density $$\rho_0$$ and specific vorticity $$w_0$$ satisfy $$(v_0, \rho_0, w_0) \in H^s\times H^s \times H^{s_0} (2<s_0<s)$$, and the author proposed a conjecture that the local solutions of 3D isentropic, compressible Euler equations is well posed if $$(v_0, \rho_0,w_0) \in H^{2+}\times H^{2+}\times H^{\frac{3}{2}+}$$. Inspired by the pioneer work, we study the local existence, uniqueness, and continuous dependence of rough solutions of 3D compressible Euler equations with dynamic vorticity and entropy.
We firstly prove the local well-posedness of solutions via Smith-Tataru's approach when the initial velocity $$v_0$$, logarithmic density $$\rho_0$$, entropy $$h_0$$, and specific vorticity $$w_0$$ satisfy $$(v_0, \rho_0,h_0,w_0) \in H^s \times H^s \times H^{s_0+1}\times H^{s_0}(2<s_0<s)$$. So this recovers Wang's result. This result has three distinct features: (i) a Strichartz estimate of linear wave equations endowed with a acoustic metric, (ii) a continuous dependence on initial data with low regularity, (iii) a type of Strichartz estimates of solutions with a regularity of the velocity 2+, density 2+, entropy $$\frak52+$$ and specific vorticity $$\frak32+$$. Secondly, based on these key Strichartz estimates, we obtain the existence and uniqueness of solutions of 3D compressible Euler equations if the initial data satisfies $$(v_0, \rho_0, h_0, w_0) \in H^{\frac52} \times H^{\frac52} \times H^{\frac52+}\times H^{\frac32+}$$. Furthermore, we prove the existence and uniqueness of solutions of 3D isentropic compressible Euler equations if $$(v_0, \rho_0, w_0) \in H^{2+} \times H^{2+} \times H^{2}$$.

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