Article Contents
Article Contents

# Local well-posedness for the Zakharov system in dimension d ≤ 3

• The Zakharov system in dimension $d\leqslant 3$ is shown to be locally well-posed in Sobolev spaces $H^s \times H^l$, extending the previously known result. We construct new solution spaces by modifying the $X^{s,b}$ spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.

Mathematics Subject Classification: Primary: 35Q55.

 Citation:

• Figure 1.  Region of well-posedness for $d = 3$ in [2]

Figure 2.  New region of well-posedness for $d\leq 3$

Table 1.  $H \times L\rightarrow L$

 $|\tau_1|$ $|\tau_2|$ $|\tau_0|$ Conclusion $\lesssim \lambda_1$ $\lesssim \lambda_1$ $\lesssim \lambda_1$ $\lambda_0^2\lesssim \lambda_1$ $\lesssim \lambda_1$ $\lambda_1 \ll \cdot \ll \lambda_1^2$ $\lambda_1 \ll \cdot \ll \lambda_1^2$ $|{\tau _W}|\sim|{\tau _0}|,\;\;{\lambda _1} \ll \lambda _0^2$ $\lambda_1\ll\cdot \ll \lambda_1^2$ $\lesssim \lambda_1$ $\lambda_1 \ll \cdot \ll \lambda_1^2$ $|\tau_S|\sim |\tau_0|, \lambda_1 \ll \lambda_0^2$ $\lambda_1 \ll\cdot \ll \lambda_1^2$ $\lambda_1 \ll \cdot \ll \lambda_1^2$ $\lambda_1 \ll \cdot \ll \lambda_1^2$ $|\tau_S|,|\tau_W|\sim \lambda_0^2, \lambda_1 \ll \lambda_0^2$ $\lambda_1 \ll\cdot \ll \lambda_1^2$ $\lambda_1 \ll \cdot \ll \lambda_1^2$ $\ll \lambda_1$ $\lambda_0^2 \ll \lambda_1$
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