AN LJ24 FLC ATTEMPTING TO CLB FROM FL410 TO FL430 FOR WX AVOIDANCE NEAR SAT EXPERIENCED A DUAL ENG FLAMEOUT.

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WE WERE CRUISING AT FL410 AND SAW SOME TSTMS BUILDING UP IN THE DISTANCE. WE DECIDED TO CLB UP TO FL430. WE RECEIVED CLRNC TO CLB TO FL430. THE FO INCREASED PWR TO CLB AND INITIATED THE CLB. WE BARELY LEFT FL410 WHEN THE L ENG FLAMED OUT; FOLLOWED +/-5 SECONDS LATER FOLLOWED BY THE R ENG. BOTH PLTS THEN DONNED OXYGEN MASKS. I ASKED FOR AND RECEIVED A CLRNC TO DSND. THE INITIAL CLRNC WAS FOR FL330; THIS BECAUSE OF TFC. BEING UNABLE TO MAINTAIN ALT; WE DECLARED AN EMER AND RECEIVED FURTHER CLRNC TO FL240 AND A TURN FOR THE DSCNT TO 090 DEGS. THIS WAS FOR TFC AVOIDANCE. MEANWHILE THE FO REMAINED IN CTL OF THE AIRPLANE WHILE I RAN THE CHKLIST. DSNDING OUT OF FL250; USING THE AIRSTART CHKLIST; I MANAGED TO RESTART THE R ENG; FOLLOWED WITHIN 2 MINS BY THE L ENG. BOTH ENGS RAN NORMAL AFTER THE AIRSTART. WE NEVER DID DSND BELOW OUR CLRNC LIMIT OF FL240. AFTER ABOUT 5 MINS AT FL240 WE DETERMINED ALL ENG PARAMETERS WERE WITHIN NORMAL OPERATING RANGE. WE CANCELED THE EMER AND ASKED FOR AND RECEIVED A CLRNC TO FL330. THE ENGS CONTINUED TO OPERATE NORMALLY AND WE CLBED TO FL370 AND GOT CLRED DIRECT TO OUR DEST; FOLLOWED BY A NORMAL DSCNT AND LNDG. A VISUAL CHK OF THE ENGS REVEALED NOTHING. THE CAUSE OF THE FLAMEOUT REMAINS UNKNOWN. IT MIGHT HAVE BEEN WINDSHEAR/TURB ASSOCIATED WITH THE PRESENCE OF CUMULUS CLOUDS. THE ENG PWR SETTINGS WERE WITHIN NORMAL PARAMETERS. I BELIEVE THERE WAS NO WAY TO AVOID THIS OCCURRENCE. THE OCCURRENCE WAS SOLVED BY USING CREW COORD AND THE LEARJET CHKLIST. THERE WAS NO DAMAGE VISIBLE TO THE ENGS AND NO FLUID LEAKS WERE PRESENT.