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Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated using the Navier–Stokes equations (NSEs) with time-fractional derivatives of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. In <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="bold">H</mi><mrow><mi>γ</mi><mo>,</mo><mo>℘</mo></mrow></msup></semantics></math></inline-formula>, we prove the existence and uniqueness of local and global mild solutions by using fuzzy techniques. Meanwhile, we provide a local moderate solution in Banach space. We further show that classical solutions to such equations exist and are regular in Banach space.