Context : This paper is motivated by the recent detection of many extremely metal-deficient ( [ Fe / H ] < -3 ) stars in the Milky Way .

Aims : We explore numerically the chemical , thermal , and dynamical evolution of a shell formed by a high-energy supernova explosion ( 10 ^ { 53 } erg ) in dwarf protogalaxies with total ( dark matter plus baryonic ) mass 10 ^ { 7 } ~ { } M _ { \odot } at a redshift z = 12 .
We consider two initial configurations for the baryonic matter , one without rotation and the other having the ratio of rotational to gravitational energy \beta = 0.17 .
The ( non-rotating ) dark matter halo is described by a quasi-isothermal sphere .
The latter choice is motivated by recently proposed mechanisms for rapid flattening of a central cuspy region in dwarf protogalaxies .

Methods : We use a finite-difference numerical hydrodynamics code to simulate supernova explosions in dwarf protogalaxies with axial symmetry .
The advection is treated using a third-order piecewise parabolic scheme .
The heating and cooling processes in the gas are taken into account by solving numerically the rate equations for main atomic , molecular and ionic species in the primordial gas .

Results : We find that the dynamics of the shell is different in protogalaxies with and without rotation .
For instance , the Rayleigh-Taylor instability in the shell develops faster in protogalaxies without rotation .
The fraction of a blown-away baryonic mass is approximately twice as large in models with rotation ( \sim 20 \% ) than in models without rotation .
We argue that these differences are caused by different initial gas density profiles in non-rotating and rotating protogalaxies .
On the other hand , the chemical evolution of gas in protogalaxies with and without rotation is found to be similar .
The relative number densities of molecular hydrogen and HD molecules in the cold gas ( T \leq 10 ^ { 3 } K ) saturate at typical values of 10 ^ { -3 } and 10 ^ { -7 } , respectively .
The saturation times in models with rotation are somewhat longer than in models without rotation .
The clumps formed in the fragmented shell move with velocities that are at least twice as large as the escape velocity .
The mass of the clumps is \sim 0.1 - 10 ~ { } M _ { \odot } , which is lower than the Jeans mass .
We conclude that the clumps are pressure supported .

Conclusions : A supernova explosion with energy 10 ^ { 53 } ergs destructs our model protogalaxy .
The clumps formed in the fragmented shell are pressure supported .
We conclude that protogalaxies with total mass \sim 10 ^ { 7 } ~ { } M _ { \odot } are unlikely to form stars due to high-energy supernova explosions of the first stars .