Why are lower cost retailers going bankrupt but premium chains are not? In recent months, there has been a lot of talk of a "retail apocalypse". The increasing purchases on the internet have (supposedly) put pressure on storefront retailers. Payless is declaring bankruptcy and Sears has made public statements that suggest they may be facing bankruptcy shortly.

Nevertheless, luxury retailers like Nordstrom and the Hudson Bay Company, exactly the kind of companies I would expect to be most affected by internet sales cutting into their business, seem to be doing well.

Why would this be the case?  

Note that my impression that higher-income consumers are more likely to use is the internet is born out by statistics (Pew Research report on internet use by income level.)
I think your anecdotal evidence is not enough to support your generalisation. In many markets there are just a few big firms (well-known to many), and many small ones (lesser-known). You could imagine a distribution of firm size being left-skewed (like a Chi-Squared or Beta distribution). Actually, this is a table showing the distribution of size by employment for 2014 for the US:

 

(Source here)

Thus, even if some big companies are closing down, what about the so many other smaller ones? Maybe some of the are also closing down, but this is less salient because they are less known (and less big thus less "important").

Now, if your claim were to be true, several factor could contribute to it. Most of shoppers in mass-market stores want quality, but also cost-efficiency. Conversely, luxury chains rely particularly on the value of brands. Their shoppers want quality, but also status. Consumers of the former are mainly worried about cost, and therefore loyalty to brand/stores is more or less inexistent. If they can buy cheaper online many would do so. Thus, internet has certainly increased competition for these stores. Conversely, consumers of exclusive shops care about store experience, relations, visibility, things which Amazon cannot give. There might even be an issue of power. You would not spend \$5,000 on a dress buying it online. You would go to the store, where others can see you, and buy it, so other can see your purchasing power.

Greater inequality might be a factor too. Inequality in the US went up since the 1980's. This has been "driven" particularly by a stagnation of income for most of households and a fast increase in top 10 and top 1% income shares. You can imagine that this means great pressure on prices for mass retailers, without an equivalent effect for exclusive retailers. 

Finally, big (ergo popular) stores and small (ergo exclusive) stores are not direct competitors. Since brands are precisely a mean of differentiation, many luxury goods face lower competition than mass-market oriented goods (I am sure you can name more brand of cars directly competing with Ford than with Aston Martin). This, together with the status element mentioned earlier, mean that luxury producers (and retailers) are likely to face a higher price-elasticity of demand. Under this setting, they can change higher markups than standard goods producers. This increases the profitability of their business, giving their brand/company higher life expectancy. That is why they are less likely to go bankrupt.